The On-Line Encyclopedia of Integer Sequences, Recent Additions

This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

It shows the most recently added sequences in reverse chronological order.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    demo1.html:         Demos 
    Sindx.html:         Index 
    WebCam.html:        WebCam
    Submit.html:        Contribute new sequence or comment 
    eishelp1.html:      Internal format 
    eishelp2.html:      Beautified format
    transforms.html:    Transforms 
    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/
    

The WebCam at www.research.att.com/~njas/sequences/WebCam.html
is another way to browse the recent additions.

[If the database has just been resorted into lexicographic order,
the present file will be empty, but the WebCam will still work.]

(start)

%I A082745
%S A082745 2,5,10,14,20,28,33,37,43,57,61,67,74,81,89,100,107,115,128,134,138,151
%N A082745 Duplicate of A064955.
%K A082745 dead
%O A082745 1,1

%I A110375
%S A110375 11269,11566,12376,12430,12700,12754,15013,17589,17797,18181,18421,
%T A110375 18453,18549,18597,18885,18949,18997
%N A110375 Numbers n such that Maple 9.5, Maple 10, Maple 11 and Maple 12 give the wrong answers for the number of partitions of n.
%C A110375 Based on various postings on the Web, sent to njas by R. J. Mathar.  Thanks to several correspondents who sent information about other versions of Maple.
%C A110375 Mathematica 6.0, DrScheme and pari-2.3.3 all give the correct answers.
%C A110375 Comment from Robert Gerbicz, May 13 2008: Ramanujan's congurence says that numbpart(5*k+4)==0 mod 5, so numbpart(11269)=...851==1 mod 5 can't be correct.
%e A110375 From PARI, the correct answer:
%e A110375 numbpart(11269)
%e A110375 2311391772313039755144117876494556289590601993601099725578515191051551761\
%e A110375 80318215891795874905318274163248033071850
%e A110375 From Maple 11, incorrect:
%e A110375 combinat[numbpart](11269);
%e A110375 2311391772313039755144117876494556289590601993601099725578515191051551761\
%e A110375 80318215891795874905318274163248033071851
%e A110375 On the other hand, the old Maple 6 gives the correct answer.
%Y A110375 Cf. A000041.
%K A110375 nonn,new
%O A110375 1,1
%A A110375 njas, May 13 2008
%E A110375 The list of terms shown here is believed to be complete as far as it goes, but there are also many larger terms in the sequence.

%I A038048
%S A038048 1,3,8,42,144,1440,5760,75600,524160,6531840,43545600,1117670400,
%T A038048 6706022400,149448499200,2092278988800,40537905408000,376610217984000,
%U A038048 13871809695744000,128047474114560000,5109094217170944000
%N A038048 a(n) = (n-1)! * sum {d|n} d.
%C A038048 Or, a(n) = Sum_{ d divides n } n!/d. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 24 2005
%C A038048 Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).
%D A038048 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
%D A038048 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).
%H A038048 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b038048.txt">Table of n, a(n) for n=1..100</a>
%H A038048 H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions for branched covers of elliptic curves and quasi-modular forms</a>
%F A038048 a(p) = (p+1)*(p-1)! if p is a prime. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 24 2005
%F A038048 E.g.f.: log(f(x)), where f(x) = o.g.f. for partitions (A000041), Product_{k=1..inf} 1/(1-x^k) - njas.
%F A038048 E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 27 2005
%e A038048 a(6) = 6!{1/1 +1/2 +1/3 + 1/6}=1440.
%p A038048 with(numtheory): a:=proc(n) local div: div:=divisors(n): n!*sum(1/div[j],j=1..tau(n)) end: seq(a(n),n=1..23); (Deutsch)
%Y A038048 Left edge of triangle in A008298. Cf. A058892.
%Y A038048 Cf. A057625.
%Y A038048 Cf. A110373, A110374.
%K A038048 easy,nonn,nice,new
%O A038048 1,2
%A A038048 Christian G. Bower (bowerc(AT)usa.net)
%E A038048 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 24 2005
%E A038048 Edited by njas, May 12 2008 at the suggestion of Joerg Arndt.

%I A138562
%S A138562 1,3,38,588
%N A138562 Number of "squashed-tree" graphs with n central nodes, the labeled case, allowing the direct link between L and R.
%C A138562 These are simple connected graphs with n+2 nodes labeled L, R, 1, 2, ..., n. The subgraph on nodes 1..n is a forest (no loops). Nodes L and R are both connected to some subset of 1..n and perhaps to each other.
%C A138562 These are the graphs that can arise when one starts with a tree with m >= n+2 labeled nodes, some of which are colored blue, some are colored red and the remaining n nodes are uncolored. Then all the blue nodes are coalesced into a single node L and all the red nodes into a single node R.
%F A138562 Although we have not written out all the details of the proof, it appears that a(n) ~ 2^n*n^(n-2).
%e A138562 a(0) = 1: L--R.
%e A138562 a(1) = 3: L--1--R, 1--L--R, L--R--1 and the 3-cycle L--1--R--L.
%e A138562 a(2) = 38: the 14 examples shown in A138460 plus the same set with an edge joining L and R: 28 in all, plus the following 10 graphs, for a total of 38.
%e A138562 =====
%e A138562 . 1
%e A138562 ./..
%e A138562 L---R (number = 2)
%e A138562 .\..
%e A138562 . 2
%e A138562 =====
%e A138562 . 1
%e A138562 ./..
%e A138562 L---R (number = 2)
%e A138562 .../
%e A138562 . 2
%e A138562 =====
%e A138562 . 1
%e A138562 ./|.
%e A138562 L-|-R (number = 2)
%e A138562 .\|.
%e A138562 . 2
%e A138562 =====
%e A138562 . 1
%e A138562 ./|.
%e A138562 L-|-R (number = 4)
%e A138562 ..|.
%e A138562 . 2
%e A138562 =====
%Y A138562 Cf. A138560. A001187(n+2) is an upper bound.
%K A138562 nonn,more,new
%O A138562 1,2
%A A138562 Nadia Heninger (nadiah(AT)cs.princeton.edu) and njas, May 10 2008
%E A138562 a(3) should be rechecked - computed by hand.

%I A138560
%S A138560 0,1,14,231
%N A138560 Number of "squashed-tree" graphs with n central nodes, the labeled case, not allowing the direct link between L and R.
%C A138560 These are simple connected graphs with n+2 nodes labeled L, R, 1, 2, ..., n. The subgraph on nodes 1..n is a forest (no loops). Nodes L and R are both connected to some subset of 1..n but not to each other.
%F A138560 Although we have not written out all the details of the proof, it appears that a(n) ~ 2^n*n^(n-2).
%e A138560 a(1) = 1: L--1--R.
%e A138560 a(2) = 14:
%e A138560 =====
%e A138560 . 1
%e A138560 ./.\
%e A138560 L . R (number = 1)
%e A138560 .\./
%e A138560 . 2
%e A138560 =====
%e A138560 . 1
%e A138560 ./.\
%e A138560 L . R (number = 4)
%e A138560 .\..
%e A138560 . 2
%e A138560 =====
%e A138560 . 1
%e A138560 ./|\
%e A138560 L | R (number = 1)
%e A138560 .\|/
%e A138560 . 2
%e A138560 =====
%e A138560 . 1
%e A138560 ./|\
%e A138560 L | R (number = 4)
%e A138560 .\|.
%e A138560 . 2
%e A138560 =====
%e A138560 . 1
%e A138560 ./|\
%e A138560 L | R (number = 2)
%e A138560 . |.
%e A138560 . 2
%e A138560 =====
%e A138560 . 1
%e A138560 . |\
%e A138560 L | R (number = 2)
%e A138560 .\|.
%e A138560 . 2
%e A138560 =====
%e A138560 Total = 14
%Y A138560 Cf. A138562.
%K A138560 nonn,more,new
%O A138560 1,2
%A A138560 Nadia Heninger (nadiah(AT)cs.princeton.edu) and njas, May 10 2008
%E A138560 a(3) should be rechecked - computed by hand.

%I A138780
%S A138780 1,6,21,56,126,252,2,462,18,792,90,1287,330,2002,990,3003,2574,3,4368,
%T A138780 6006,36,6188,12870,234,8568,25740,1092,11628,48620,4095,15504,87516,
%U A138780 13104,4,20349,151164,37128,60
%N A138780 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k+2) (n>=7, 1<=k<=(n-2)/5).
%C A138780 Row n contains floor((n-2)/5) terms.
%C A138780 Row sums yield A137361.
%p A138780 T:=proc(n,k) options operator, arrow: k*binomial(n-2*k, 3*k+2) end proc: for n from 7 to 23 do seq(T(n,k),k=1..(n-2)*1/5) end do; # yields sequence in triangular form
%Y A138780 Cf. A137361.
%K A138780 nonn,tabf,new
%O A138780 7,2
%A A138780 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A138779
%S A138779 1,5,15,35,70,126,2,210,16,330,72,495,240,715,660,1001,1584,3,1365,3432,
%T A138779 33,1820,6864,198,2380,12870,858,3060,22880,3003,3876,38896,9009,4,4845,
%U A138779 63648,24024,56,5985,100776,58344,420
%N A138779 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k+1) (n>=6, 0<=k<=(n-1)/5).
%C A138779 Row n contains floor((n-1)/5) terms.
%C A138779 Row sums yield A137360.
%p A138779 T:=proc(n,k) options operator, arrow: k*binomial(n-2*k,3*k+1) end proc: for n from 6 to 23 do seq(T(n,k),k=1..(n-1)*1/5) end do; # yields sequence in triangular form
%Y A138779 Cf. A137360.
%K A138779 nonn,tabf,new
%O A138779 6,2
%A A138779 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A138778
%S A138778 1,4,10,20,35,56,2,84,14,120,56,165,168,220,420,286,924,3,364,1848,30,
%T A138778 455,3432,165,560,6006,660,680,10010,2145,816,16016,6006,4,969,24752,
%U A138778 15015,52,1140,37128,34320,364
%N A138778 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k) (n>=5, 1<=k<=n/5).
%C A138778 Row n contains floor(n/5) terms.
%C A138778 Row sums yield A137359.
%p A138778 T:=proc(n,k) options operator, arrow: k*binomial(n-2*k,3*k) end proc: for n from 5 to 22 do seq(T(n,k),k=1..(1/5)*n) end do; # yields sequence in triangular form
%Y A138778 Cf. A137359.
%K A138778 nonn,tabf,new
%O A138778 5,2
%A A138778 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A138556
%S A138556 1,2,3,4,5,6,7,8,9,10,11,12,13,14,17,18,19,21,22,23,26,27,28,29,31,32,
%T A138556 34,36,37,38,40,41,42,43,46,47,48,49,53,54,55,57,58,59,60,61,62,63,67,
%U A138556 71,73,74,75
%N A138556 Positive integers n for which values of A039649(n) are primes.
%C A138556 For every prime p,the numbers p and 2p are terms of this sequence.
%Y A138556 Cf. A039649, A000010 A000040.
%K A138556 nonn,new
%O A138556 1,2
%A A138556 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138539
%S A138539 2,3,5,7,13,19,37,41,61,73,97
%N A138539 Primes p_n for which A140141(n)<2p_n.
%Y A138539 Cf. A140141, A039649, A000010, A000040.
%K A138539 nonn,new
%O A138539 1,1
%A A138539 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138537
%S A138537 11,17,23,29,31,43,47,53,59,67,71,79,83,89
%N A138537 Primes p_n for which 140141(n)=2p_n.
%Y A138537 Cf. A140141, A039649, A000010, A000040.
%K A138537 nonn,new
%O A138537 1,1
%A A138537 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138536
%S A138536 3,5,7,11,19,23,29,37,43,47,53,59,67,71,79,83,97,101,103,107,109,127,
%T A138536 137,139,149,157,163,167,173,179,181,191,197,223,227,239,251,257,263,
%U A138536 269,271,277,283,293,311,317,331,347,349
%N A138536 Odd primes p_n for which A140140(p_n)=(p_n-1)/2,where p_n=n-th odd prime.
%Y A138536 Cf. A140140, A137576, A000040.
%K A138536 nonn,new
%O A138536 1,1
%A A138536 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138535
%S A138535 13,17,31,41,61,73,89,113,131,151,193,199,211,229,233,241,281,307,313,
%T A138535 337,379,397
%N A138535 Odd primes p_n for which A140140(p_n)<(p_n-1)/2.
%Y A138535 Cf. A987654, A987655.
%K A138535 nonn,new
%O A138535 1,1
%A A138535 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138509
%S A138509 2,1,4,3,12,11,10,9,8,7,6,5,20,19,18,17,16,15,14,13,38,37,36,35,34,33,
%T A138509 32,31,30,29,28,27,26,25,24,23,22,21,56,55,54,53,52,51,50,49,48,47,46,
%U A138509 45,44,43,42,41,40,39,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89
%N A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.
%F A138509 There are respectively A137583 = 2, 2, 8, 8, 18, 18, 32, 32 terms from right to left.
%K A138509 nonn,fini,full,new
%O A138509 1,1
%A A138509 Paul Curtz (bpcrtz(AT)free.fr), May 10 2008

%I A138508
%S A138508 4,6,3,1,5,1,2,3,1,1,4,2,3,4,1,2,2,5,2,5,1,1,3,2,1,1,1,2,3,3,5,2,1,1,1,
%T A138508 4,4,1,1,4,4,2,2,8,2,1,2,4,1,3,1,11,3,3,1,7,1,1,2,2,2,1,4,2,1,2,4,4,1,2,
%U A138508 2,2,3,1,3,6,1,5,2,2
%N A138508 Semiprime analogue of Riesel problem: start with n; repeatedly double and add 1 until reach a semiprime. Sequence gives number of steps to reach a semiprime or 0 if no prime is ever reached.
%C A138508 This to A050412 as semiprimes A001358 are to primes A000040.
%F A138508 a(n) = smallest m >= 1 such that n*2^m - 1 is semiprime (or 0 if no such semiprime exists).
%e A138508 a(52) = 11 because there are 11 steps in the iteration until a semiprime, namely 52; 2*423+1 = 847 = 7 * 11^2; 2*847+1 = 1695 = 3 * 5 * 113; 2*1695+1 = 3391 (prime); 2*3391+1 = 6783 = 3 * 7 * 17 * 19; 2*6783+1 = 13567 (prime); 2*13567+1 = 27135 = 3^4 * 5 * 67; 2*27135+1 = 54271 = 7 * 7753, which is semiprime.
%Y A138508 Cf. A001358, A050412.
%K A138508 easy,more,nonn,new
%O A138508 1,1
%A A138508 Jonathan Vos Post (jvospost3(AT)gmail.com), May 10 2008

%I A138494
%S A138494 1,3,4,5,7,8,11,13,13,14,15,19,20,21,21,23,26,29,29,28,35,33,34,37,37,
%T A138494 41,40,41,45,44,51,49,51,54,49,57,54,63,59,56,65,65,71,68,65,73,72,77,
%U A138494 75,79,78,75,83,80,91,85,89,88,91,95,94,97,99,96,99,99,105,110,103,109
%N A138494 a and b are integers > 0, aa + bb = cc. The number of integer solutions for a,b between successive c integers. Integer solutions for c (Pythagorean triples) are not included.
%o A138494 (Qbasic) OPEN "PYTH.TXT" FOR OUTPUT AS #1
%o A138494 FOR C = 1 TO 100
%o A138494 N = 0
%o A138494 FOR A = 1 TO C
%o A138494 FOR B = 1 TO C
%o A138494 D = SQR(A * A + B * B)
%o A138494 IF D &gt; C AND D &lt; C + 1 THEN N = N + 1
%o A138494 NEXT B
%o A138494 NEXT A
%o A138494 PRINT #1, N;
%o A138494 NEXT C
%o A138494 CLOSE
%K A138494 easy,nonn,new
%O A138494 1,2
%A A138494 Rick Walcott (rick(AT)campbellsci.com), May 09 2008

%I A138473
%S A138473 0,21,987,46368,2178309,102334155,4807526976,225851433717,
%T A138473 10610209857723,498454011879264,23416728348467685,1100087778366101931,
%U A138473 51680708854858323072,2427893228399975082453,114059301025943970552219
%N A138473 Fibonacci(8n).
%C A138473 21*A049668 a(n)
%o A138473 Mupad: numlib::fibonacci(8*n) $ n = 0..25;
%Y A138473 Cf. A049668, A134498.
%K A138473 nonn,new
%O A138473 0,2
%A A138473 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008

%I A138469
%S A138469 5,6,7,8,9,10,13,14,15,16,17,18,31,32,33,34,35,36,49,50,51,52,53,54,81,
%T A138469 82,83,84,85,86,113,114,115,116,117,118
%N A138469 6 X 6 square of orbital p (mixed partially pure metals, nonmetals and noble gases) in Mendeleyev-Seaborg or Janet-Tarantola periodic table.
%Y A138469 Cf. A130517, A131603.
%K A138469 nonn,tabf,fini,full,new
%O A138469 1,1
%A A138469 Paul Curtz (bpcrtz(AT)free.fr), May 09 2008

%I A138467
%S A138467 1,2,3,3,4,5,6,7,7,8,9,10,11,11,12,13,14,15,15,16,17,18,18,19,20,21,22,
%T A138467 22,23,24,25,26,26,27,28,29,30,30,31,32,33,34,34,35,36,37,37,38,39,40,
%U A138467 41,41,42,43,44,45,45,46,47,48,49,49,50,51,52,53,53,54,55,56,56,57,58
%N A138467 a(1)=1, then for n>=2 a(n)=n-floor((1/3)*a(a(n-1))).
%D A138467 B. Cloitre, On some generalisations of A005206, in preparation 2008
%F A138467 For n>=1, a(n)=floor(r*(n+1)) where r=(3/2)*(sqrt(7/3)-1)
%o A138467 (PARI) a(n)=floor((3/2)*(sqrt(7/3)-1)*(n+1))
%Y A138467 Cf. A005206, A135414.
%K A138467 nonn,new
%O A138467 1,2
%A A138467 Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2008

%I A138466
%S A138466 1,2,2,3,4,5,5,6,7,8,8,9,10,10,11,12,13,13,14,15,16,16,17,18,19,19,20,
%T A138466 21,21,22,23,24,24,25,26,27,27,28,29,30,30,31,32,32,33,34,35,35,36,37,
%U A138466 38,38,39,40,40,41,42,43,43,44,45,46,46,47,48,49,49,50,51,51,52,53,54
%N A138466 a(1)=1, then for n>=2 a(n)=n-floor((1/2)*a(a(n-1))).
%F A138466 For n>=1, a(n)=floor((sqrt(3)-1)*(n+1))
%o A138466 (PARI) a(n)=floor((sqrt(3)-1)*(n+1))
%K A138466 nonn,new
%O A138466 1,2
%A A138466 Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2008

%I A138465
%S A138465 1,2,7,45,650,24520,2625117,836488618,818230288201,2513135860300849,
%T A138465 24686082394548211147,787959836124458000837941,
%U A138465 82905574521614049485027140026
%N A138465 Number of n X n binary matrices with both rows and columns, considered as binary numbers, in nondecreasing order.
%C A138465 Ordering only rows gives A060690
%K A138465 nonn,new
%O A138465 0,2
%A A138465 Ron Hardin (rhh(AT)cadence.com), May 08 2008

%I A140119
%S A140119 2,4,8,8,22,6,72,92,266,426,838,1172,1432,398,3614,15140,41274,95126,
%T A140119 195698,370876,652384,1063442,1570116,1961852,1560168,1401888,11023226,
%U A140119 36000318,93408538,214275608,450374202,879254356,1599245876,2695464868
%V A140119 2,4,8,8,22,-6,72,-92,266,-426,838,-1172,1432,-398,-3614,15140,-41274,95126,-195698,
%W A140119 370876,-652384,1063442,-1570116,1961852,-1560168,-1401888,11023226,-36000318,93408538,
%X A140119 -214275608,450374202,-879254356,1599245876,-2695464868,4138070460,-5539280974
%N A140119 Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.
%C A140119 Construct the least-degree polynomial p(x) which fits the first n primes (p has degree n - 1 or less). Then predict the next prime by evaluating p(n + 1).
%C A140119 a(n) = sum_1_n p_i (-1)^(n - i) binomial(n, i - 1) where p_i are the primes.
%C A140119 Can anything be said about the pattern of positive and negative values?
%H A140119 Jonathan Wellons, <a href="http://www.research.att.com/~njas/sequences/a140119.txt">Table of n, a(n) for n = 1..1500</a>
%e A140119 The lowest-order polynomial having points (1,2), (2,3), (3,5) and (4,7) is f(x) = 1/6 (-x^3 + 9x^2 - 14x + 18). When evaluated at x = 5, f(5) = 8.
%Y A140119 Cf. A140118.
%K A140119 sign,new
%O A140119 1,1
%A A140119 Jonathan Wellons (wellons(AT)gmail.com), May 08, 2008

%I A140118
%S A140118 3,7,9,19,3,49,39,151,189,381,371,219,991,4059,11473,26193,53791,100639,
%T A140118 175107,281581,410979,506757,391647,401587,2962157,9621235,24977199,
%U A140118 57408111,120867183,236098467,428880285,719991383,1096219131,1442605443
%V A140118 3,7,9,19,3,49,-39,151,-189,381,-371,219,991,-4059,11473,-26193,53791,-100639,175107,
%W A140118 -281581,410979,-506757,391647,401587,-2962157,9621235,-24977199,57408111,-120867183,
%X A140118 236098467,-428880285,719991383,-1096219131,1442605443,-1401210665,99178397,4340546667
%N A140118 Extrapolation for (n + 1)-st odd prime made by fitting least-degree polynomial to first n odd primes.
%C A140118 Construct the least-degree polynomial p(x) which fits the first n odd primes (p has degree n - 1 or less). Then predict the next prime by evaluating p(n + 1).
%C A140118 a(n) = sum_1_n p_i (-1)^(n - i) binomial(n, i - 1) where p_i are the primes.
%C A140118 Can anything be said about the pattern of positive and negative values?
%C A140118 How many of these terms are the correct (n + 1)th prime?
%C A140118 How many terms are prime?
%H A140118 Jonathan Wellons, <a href="http://www.research.att.com/~njas/sequences/a140118.txt">Table of n, a(n) for n = 1..1500</a>
%e A140118 The lowest-order polynomial having points (1,3), (2,5), (3,7) and (4,11) is f(x) = 1/3 (x^3 - 6x^2 + 17x - 3). When evaluated at x = 5, f(5) = 19.
%Y A140118 Cf. A140119.
%K A140118 sign,new
%O A140118 1,1
%A A140118 Jonathan Wellons (wellons(AT)gmail.com), May 08, 2008

%I A140141
%S A140141 2,4,8,9,22,21,34,27,46,58,62,57,55,86,94,106,118,77,134,142,91,158,166,
%T A140141 178,119
%N A140141 Positions of second appearances of primes in A039649.
%C A140141 The first occurrence of a prime p in A039649 is not interesting because for an odd prime p it is evidently p. Since phi(2p)= p-1 then for the n-th prime p_n we have p_n<=a(n)<=2p_n.
%Y A140141 Cf. A039649, A000010, A000040.
%K A140141 nonn,new
%O A140141 1,1
%A A140141 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A138777
%S A138777 1,3,6,10,15,21,1,28,6,36,21,45,56,55,126,66,252,1,78,462,9,91,792,45,
%T A138777 105,1287,165,120,2002,495,136,3003,1287,1,153,4368,3003,12,171,6188,
%U A138777 6435,78,190,8568,12870,364
%N A138777 Triangle read by rows: T(n,k)=binomial(n-2k,3k+2) (n>=2, 0<=k<=(n-2)/5).
%C A138777 Row n contains floor((n+3)/5) terms.
%C A138777 Row sums yield A137358.
%p A138777 T:=proc(n,k) options operator, arrow: binomial(n-2*k,3*k+2) end proc: for n from 2 to 20 do seq(T(n,k),k=0..(n-2)*1/5) end do; # yields sequence in triangular form
%Y A138777 Cf. A137358.
%K A138777 nonn,tabf,new
%O A138777 2,2
%A A138777 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A138776
%S A138776 1,2,3,4,5,6,1,7,5,8,15,9,35,10,70,11,126,1,12,210,8,13,330,36,14,495,
%T A138776 120,15,715,330,16,1001,792,1,17,1365,1716,11,18,1820,3432,66,19,2380,
%U A138776 6435,286,20,3060,11440,1001
%N A138776 Triangle read by rows: T(n,k)=binomial(n-2k,3k+1) (n>=1, 0<=k<=(n-1)/5).
%C A138776 Row n contains floor((n+4)/5) terms.
%C A138776 Row sums yield A137357.
%p A138776 T:=proc(n,k) options operator, arrow: binomial(n-2*k, 3*k+1) end proc: for n to 20 do seq(T(n,k),k=0..(n-1)*1/5) end do; # yields sequence in triangular form
%Y A138776 Cf. A137357.
%K A138776 nonn,tabf,new
%O A138776 1,2
%A A138776 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A138775
%S A138775 1,1,1,1,1,1,1,1,4,1,10,1,20,1,35,1,56,1,1,84,7,1,120,28,1,165,84,1,220,
%T A138775 210,1,286,462,1,1,364,924,10,1,455,1716,55,1,560,3003,220,1,680,5005,
%U A138775 715,1,816,8008,2002,1
%N A138775 Triangle read by rows: T(n,k)=binomial(n-2k,3k) (n>=0, 0<=k<=n/5).
%C A138775 Row n contains 1+floor(n/5) terms.
%C A138775 Row sums yield A137356.
%p A138775 T:=proc(n,k) options operator, arrow: binomial(n-2*k, 3*k) end proc: for n from 0 to 20 do seq(T(n,k),k=0..(1/5)*n) end do; # yields sequence in triangular form
%Y A138775 Cf. A137356.
%K A138775 nonn,tabf,new
%O A138775 0,9
%A A138775 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008

%I A139225
%S A139225 2,14,310,5334,22361430,5726491990,91625444694
%N A139225 M(M-1)/3, where M is Mersenne prime A000668(n).
%F A139225 a(n)=A000668(n)*(A000668(n)-1)/3.
%Y A139225 Cf.A000668, A036689, A139115, A139116, A139223, A139224, A139226.
%K A139225 more,nonn,new
%O A139225 1,1
%A A139225 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A139226
%S A139226 1,7,155,2667,11180715,2863245995,45812722347
%N A139226 M(M-1)/6, where M is Mersenne prime A000668(n).
%C A139226 Perfect number A000396(n) minus Mersenne prime A000668(n), divided by 3.
%H A139226 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos"</a>.
%F A139226 a(n)=A000668(n)*(A000668(n)-1)/6 = A139223(n)/6 = A139224(n)/3. Also a(n)=(A000396(n)-A000668(n))/3.
%Y A139226 Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139224, A139225, A139256, A139306, .
%K A139226 more,nonn,new
%O A139226 1,1
%A A139226 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A139224
%S A139224 3,21,465,8001,33542145,8589737985,137438167041
%N A139224 M(M-1)/2, where M is Mersenne prime A000668(n).
%C A139224 Perfect number A000396(n) minus Mersenne prime A000668(n).
%H A139224 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos"</a>.
%F A139224 a(n)=A000668(n)*(A000668(n)-1)/2. Also a(n)=A000396(n)-A000668(n).
%Y A139224 Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139225, A139226, A139256, A139306, .
%K A139224 more,nonn,new
%O A139224 1,1
%A A139224 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A139223
%S A139223 6,42,930,16002,67084290,17179475970,274876334082
%N A139223 M(M-1), where M is Mersenne prime A000668(n).
%H A139223 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos"</a>.
%F A139223 a(n)=A000668(n)*(A000668(n)-1).
%Y A139223 Cf. A000668, A036689, A139115, A139116, A139224, A139225, A139226, A139256.
%K A139223 more,nonn,new
%O A139223 1,1
%A A139223 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A140140
%S A140140 1,2,3,5,4,7,9,11,14,10,18,12,21,23,26,29,17,35,28,39,41,42,48,50,51,53,
%T A140140 54,43,63,46,68,69,74,38,78,81,83,86,89,90,95,59,98,85
%N A140140 Positions of first appearances of odd primes in A137576.
%C A140140 a(n)<=(p_n-1)/2 , where p_n is the n-th odd prime.
%Y A140140 Cf. A137576, A000040.
%K A140140 nonn,new
%O A140140 1,2
%A A140140 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 10 2008

%I A139116
%S A139116 1,3,10,21,78,136,171,465,1830,3916,5671,8001,135460,183921,817281,
%T A139116 2425503,2600340,5172936,9041878,9779253,46933516,49406770,62860078
%N A139116 p(p-1)/2, where p is A000043(n).
%F A139116 a(n)=A000043(n)*(A000043(n)-1)/2.
%Y A139116 Cf. A000043, A000217, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.
%K A139116 nonn,new
%O A139116 1,1
%A A139116 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A139115
%S A139115 2,6,20,42,156,272,342,930,3660,7832,11342,16002,270920,367842,1634562,
%T A139115 4851006,5200680,10345872,18083756,19558506,93867032,98813540,125720156
%N A139115 p(p-1), where p is A000043(n).
%F A139115 a(n)=A000043(n)*(A000043(n)-1).
%Y A139115 Cf. A000043, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.
%K A139115 nonn,new
%O A139115 1,1
%A A139115 Omar E. Pol (info(AT)polprimos.com), May 10 2008

%I A140139
%S A140139 1,2,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,
%T A140139 51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81
%N A140139 Binomial transform of [1, 1, 2, -3, 4, -5, 6, -7,...].
%F A140139 A007318 * [1, 1, 2, -3, 4, -5, 6, -7,...]. Sums of antidiagonal terms of the following array: 1, 1, 1, 1, 1,... 1, 3, 5, 7, 9,... 1, 1, 1, 1, 1,...
%e A140139 a(4) = 7 = (1, 3, 3, 1) dot (1, 1, 2, -3) = (1 + 3 + 6 - 3).
%K A140139 nonn,new
%O A140139 1,2
%A A140139 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 09 2008

%I A140046
%S A140046 1,1,3,10,41,186,922,4911,27830,166656,1049410,6922476,47698148,
%T A140046 342483885,2557538781,19829608532,159393394129,1326509171669,
%U A140046 11415703608635,101473987987073,930688926616454,8798656042121634
%N A140046 G.f. satisfies: A(x) = x/(1 - A(x+x^2)).
%e A140046 G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 41*x^5 + 186*x^6 + 922*x^7 +...
%e A140046 A(x+x^2) = x + 2*x^2 + 5*x^3 + 20*x^4 + 90*x^5 + 454*x^6 + 2488*x^7 +...
%e A140046 Let B(x) = x + x^2; define B_{n+1}(x) = B( B_{n}(x) ) with B_0(x)=x;
%e A140046 then g.f. A(x) equals the continued fraction:
%e A140046 A(x) = x/(1 - B(x)/(1 - B_2(x)/(1 - B_3(x)/(1 - B_4(x)/(1 - ...)))))
%e A140046 where B_{n}(x) begin:
%e A140046 B_2(x) = x + 2*x^2 + 2*x^3 + x^4 ;
%e A140046 B_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8 ;
%e A140046 B_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 + 188*x^7 +...;
%e A140046 B_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 + 1656*x^7 +...
%o A140046 (PARI) {a(n)=local(A=x);if(n==0,A=x,for(i=1,n,A=x/(1-subst(A,x,x+x^2 +x*O(x^n))))); polcoeff(A,n)}
%Y A140046 Cf. A127782.
%K A140046 nonn,new
%O A140046 1,3
%A A140046 Paul D. Hanna (pauldhanna(AT)juno.com), May 09 2008

%I A139575
%S A139575 70368744177664,8862938119652501095929
%N A139575 Numbers with 47 divisors.
%C A139575 46th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
%H A139575 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139575 a(n)=A000040(n)^(47-1)=A000040(n)^46.
%Y A139575 Cf. A000005, A000040, A001248, A030514, A030516, A030529, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574.
%K A139575 bref,more,nonn,new
%O A139575 1,1
%A A139575 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A139574
%S A139574 4398046511104,109418989131512359209,227373675443232059478759765625
%N A139574 Numbers with 43 divisors.
%C A139574 42nd powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
%H A139574 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139574 a(n)=A000040(n)^(43-1)=A000040(n)^42.
%Y A139574 Cf. A000005, A000040, A001248, A030514, A030516, A030529, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139575.
%K A139574 bref,more,nonn,new
%O A139574 1,1
%A A139574 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A140135
%S A140135 16,24,54,54,54,90,140,140,140,140,210,315,315,315,315,315,315,462,550,
%T A140135 550,550,650,858,858,858,858,858,858,858,1155,1190,1330,1330,1330,1482,
%U A140135 1794,1794
%N A140135 Product of largest semiprime <= n and smallest semiprime >= n.
%C A140135 This is to A030664 as semiprimes A001358 are to primes A000040. Subset of A014613.
%F A140135 a(n) = MAX{j in A001358 and j <= n} * MIN{j in A001358 and j >= n}
%e A140135 a(10) = 140 because the largest semiprime <= 10 is 10, the smallest semiprime >= 10 is 140, and 10*14 = 140.
%Y A140135 Cf. A001358, A014613, A030664.
%K A140135 easy,more,nonn,new
%O A140135 4,1
%A A140135 Jonathan Vos Post (jvospost3(AT)gmail.com), May 09 2008

%I A140136
%S A140136 1,1,1,1,7,7,1,1,20,75,75,20,1,1,42,364,1001,1001,364,42,1,1,75,1212,
%T A140136 6720,15288,15288,6720,1212,75,1,1,121,3223,30723,127908,255816,255816,
%U A140136 127908,30723,3223,121,1,1,182,7371,109538,737737,2510508
%N A140136 Numerator coefficients for generators of lattice path enumeration square array A111910.
%C A140136 sum{k=0..n, T(n,k)x^k}/(1-x)^(3n+1) generates row n of A111910.
%D A140136 G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
%F A140136 Triangle T(q,n) where T(n,q)=sum{j=0..n, (-1)^j*C(3q+1,j)*K(n-j,q)} with K(p,q)=A111910(p,q).
%e A140136 Triangle begins
%e A140136 1,
%e A140136 1,1,
%e A140136 1,7,7,1,
%e A140136 1,20,75,75,20,1,
%e A140136 1,42,364,1001,1001,364,42,1,
%e A140136 1,75,1212,6720,15288,15288,6720,1212,75,1
%K A140136 easy,nonn,new
%O A140136 0,5
%A A140136 Paul Barry (pbarry(AT)wit.ie), May 09 2008

%I A139588
%S A139588 1,4,9,16,24,25,30,40,42,49,54,56,66,70,78,81,88,102,104,105,110,114,
%T A139588 121,128,130,135,136,138
%N A139588 Nonprime numbers with Fibonacci number of divisors.
%C A139588 A000005(a(n)) is a Fibonacci number.
%Y A139588 Cf. A000005, A000045, A018252, A123193, A134805, A139589, A139590.
%K A139588 easy,nonn,new
%O A139588 1,2
%A A139588 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A139590
%S A139590 8,21,34,55,144,377,2584,4181,6765
%N A139590 Fibonacci numbers with non-Fibonacci number of divisors.
%C A139590 A000005(a(n)) is a non-Fibonacci number A001690.
%Y A139590 Cf. A000005, A000045, A001690, A063375, A133021, A134805.
%K A139590 more,nonn,new
%O A139590 1,1
%A A139590 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A139589
%S A139589 1,1,2,3,5,13,89,233,610,987,1597
%N A139589 Fibonacci numbers with Fibonacci number of divisors.
%C A139589 A000005(a(n)) is a Fibonacci number.
%Y A139589 Cf. A000005, A000045, A063375, A133021, A134805.
%K A139589 more,nonn,new
%O A139589 0,3
%A A139589 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A139118
%S A139118 1,6,8,10,12,14,15,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,40,
%T A139118 42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,65,66,68,69,70,72,74,
%U A139118 75,76,77,78,80,82,84,85,86,87,88,90,91
%N A139118 Numbers with nonprime number of divisors.
%C A139118 A000005(a(n)) is nonprime. Complement of A009087. Also, nonprime numbers with nonprime number of divisors.
%Y A139118 Cf. A000005, A009087, A018252.
%K A139118 easy,nonn,new
%O A139118 1,2
%A A139118 Omar E. Pol (info(AT)polprimos.com), May 09 2008

%I A140132
%S A140132 0,1,2,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7,
%T A140132 8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,
%U A140132 10,11,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7,8,3,4,5,9,10,11,6,7
%N A140132 a(n)=Sum_digits{a(n-1)+a(n-2)+Sum_digits[a(n-1)]+Sum_digits[a(n-2)]}, with a(0)=0 and a(1)=1.
%C A140132 After the first three terms the sequence is periodic: 6,7,8,3,4,5,9,10,11.
%F A140132 a(n)=(1/12)*{-3*(n mod 9)+[(n+1) mod 9]+[(n+2) mod 9]+9*[(n+3) mod 9]+[(n+4) mod 9]+[(n+5) mod 9]+9*[(n+6) mod 9]+[(n+7) mod 9]+[(n+8) mod 9]}-9*{[C(2*n,n) mod 2]+[C((n+1)^2,n+3) mod 2]+[C((n+12)^4,n+14) mod 2]}, with n>=0
%p A140132 P:=proc(n) local a,b,i,k,w,x,y; a:=0; b:=1; print(a); print(b); for i from 1 by 1 to n do w:=0; k:=a; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=0; k:=b; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; c:=b; y:=0; k:=a+b+w+x; while k>0 do y:=y+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=b; b:=y; print(y); od; end: P(100);
%Y A140132 Cf. A016052, A047892, A047897-A047900, A047902, A047903, A055263, A134268, A135210, A140131.
%K A140132 easy,nonn,new
%O A140132 0,3
%A A140132 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), May 09 2008

%I A140131
%S A140131 0,1,2,6,16,35,66,121,203,333,550,902,1473,2401,3896,6330,10264,16619,
%T A140131 26919,43588,70562,114198,184804,299051,483906,783013,1266971,2050038,
%U A140131 3317059,5367143,8684259,14051473,22735799,36787341,59523223,96310634
%N A140131 a(n)=a(n-1)+a(n-2)+Sum_digits[a(n-1)]+Sum_digits[a(n-2)], with a(0)=0 and a(1)=1.
%p A140131 P:=proc(n) local a,b,c,i,k,w,x; a:=0; b:=1; print(a); print(b); for i from 1 by 1 to n do w:=0; k:=a; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=0; k:=b; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; c:=b; b:=a+b+w+x; a:=c; print(b); od; end: P(100);
%Y A140131 Cf. A016052, A047892, A047897-A047900, A047902, A047903, A055263, A134268, A135210, A140132.
%K A140131 easy,nonn,new
%O A140131 0,3
%A A140131 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), May 09 2008

%I A140123
%S A140123 4,12,36,180,1260,252,252,2772,69300,900900,900900,15315300,15315300,
%T A140123 290990700,290990700,6692786100,46849502700,46849502700,46849502700,
%U A140123 46849502700
%N A140123 Denominator of Sum_{k=1..n} (-1)^k / semiprime(k).
%e A140123 The first 10 values of A140122(n)/a(n) = -1/4, -1/12, -7/36, -17/180, -209/1260, -25/252, -37/252, -281/2772, -9797/69300, -92711/900900. The 10th term of the sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26) = -92711/900900 hence a(10) = 900900. The 20th term of the alternating sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26)-(1/33)+(1/34)-(1/35)+(1/38)-(1/39)+(1/46)-(1/49)+(1/51)-(1/55)+(1/57) = -5218865543/46849502700, hence a(20) = 46849502700.
%Y A140123 Cf. A001358, A002110, A024530, A140122.
%K A140123 easy,frac,more,nonn,new
%O A140123 1,1
%A A140123 Jonathan Vos Post (jvospost3(AT)gmail.com), May 09 2008

%I A140122
%S A140122 1,1,7,17,209,25,37,281,9797,92711,120011,2027317,30861373,38322673,
%T A140122 735926129,6107595203,5188977503,6040786643,5218865543
%N A140122 Negative of numerator of Sum_{k=1..n} (-1)^k / semiprime(k).
%e A140122 The first 10 values of a(n)/A140123(n) = -1/4, -1/12, -7/36, -17/180, -209/1260, -25/252, -37/252, -281/2772, -9797/69300, -92711/900900. The 10th term of the sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26) = -92711/900900 hence a(10) = -(-92711) = 92711. The 20th term of the alternating sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26)-(1/33)+(1/34)-(1/35)+(1/38)-(1/39)+(1/46)-(1/49)+(1/51)-(1/55)+(1/57) = -5218865543/46849502700, hence a(20) = 5218865543.
%Y A140122 Cf. A001358, A002110, A024530, A140123.
%K A140122 easy,frac,more,nonn,new
%O A140122 1,3
%A A140122 Jonathan Vos Post (jvospost3(AT)gmail.com), May 09 2008

%I A139568
%S A139568 32,8192,8589934592,2305843009213693952,
%T A139568 191561942608236107294793378393788647952342390272950272
%N A139568 Bisection of ultraperfect numbers A139306.
%Y A139568 Cf. A099057, A099058, A139306, A139563, A139564, A139567.
%K A139568 more,nonn,new
%O A139568 1,1
%A A139568 Omar E. Pol (info(AT)polprimos.com), May 08 2008

%I A139567
%S A139567 8,512,33554432,137438953472,2658455991569831745807614120560689152
%N A139567 Bisection of ultraperfect numbers A139306.
%Y A139567 Cf. A099057, A099058, A139306, A139563, A139564, A139568.
%K A139567 more,nonn,new
%O A139567 1,1
%A A139567 Omar E. Pol (info(AT)polprimos.com), May 08 2008

%I A139117
%S A139117 3,6,15,28,91,153,190,496,1891,4005,5778,8128,135981,184528,818560,
%T A139117 2427706,2602621,5176153,9046131,9783636,46943205,49416711,62871291,
%U A139117 198751953,235477551,269340445,990013753,3718970646,6105511756
%N A139117 Triangular numbers (A000217) with indices A000043.
%F A139117 a(n)=A000217(A000043(n)).
%e A139117 a(4)=28 because A000043(4)=7 and the 7th triangular number A000217(7) is 28.
%Y A139117 Cf. A000217, A000396, A034953.
%K A139117 nonn,new
%O A139117 1,1
%A A139117 Omar E. Pol (info(AT)polprimos.com), May 08 2008

%I A140114
%S A140114 0,0,1,3,2,4,3,5,4,8,5,8,6,13,7,10,13,10,12,9,14
%N A140114 a(n)=number of pseudoprimes >(n-1)^2 and <n^2.
%C A140114 A first submission seems to have not been received.
%e A140114 The first pseudoprimes are 6,10,14,15,21,22,26
%e A140114 None is <4, hence a(1)=a(2)=0
%e A140114 One only is <9, hence a(3) = 1
%e A140114 Three more, 10,14,and 15 are <16, hence a(4)=3
%K A140114 easy,nonn,new
%O A140114 1,4
%A A140114 Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008

%I A140117
%S A140117 2,4,8,15,16,31,32,33,34,63,64,65,66,68,72,125,126,127,128,129,130,131,
%T A140117 132,136,144,160,192,247,251,253,254,255,256,257,258,259,260,264,272,
%U A140117 288,320,384,503,507,509,510,511,512,513,514,515,516,517,518,520,521
%N A140117 Numbers n for which A140116(n) uses fewer symbols than A007088(n).
%C A140117 The representation of n given by A140116(n) is more compact than the standard binary form of n for these numbers consisting of a sufficiently large proportion of 0s or of 1s among their bits.
%e A140117 See also examples for A140116. 2180 is a term here as 2180 = 100010000100 (base 2), which uses 12 symbols, but A140116(2180) = 10112111210, which uses only 11 symbols including the two delimiter symbols (for which 2 is used).
%Y A140117 Cf. A140116, A007088.
%K A140117 nonn,new
%O A140117 1,1
%A A140117 Rick L. Shepherd (rshepherd2(AT)hotmail.com), May 08 2008

%I A138193
%S A138193 9,15,25,27,33,39,49,55,57,63,81,87,95,111,119,121,125,135,143,153,159,
%T A138193 161,169,175,177,183,201,207,209,225,243,249,287,289,295,297,303,319,
%U A138193 321,329,335,343,351,361,369,375,391,393,407,415,417,423,447,489,497
%N A138193 Odd composite numbers n for which A137576((n-1)/2)-1 is divisible by phi(n).
%C A138193 If p is an odd prime then A137576((p-1)/2)=p. Therefore the composite numbers n may be considered as quasiprimes. In particular, if (m,n)=1 we have a natural generalization of the little Fermat theorem: m^(A137576((n-1)/ 2)-1)=1 mod n.
%H A138193 Ray Chandler, <a href="http://www.research.att.com/~njas/sequences/b138193.txt"> Table of n, a(n) for n=1..1239</a>
%e A138193 a(1)=9: A137576(4)=13 and 13-1 is divisible by phi(9)=6.
%Y A138193 Cf. A137576, A002326, A006694.
%K A138193 nonn,new
%O A138193 1,1
%A A138193 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 04 2008
%E A138193 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 08 2008

%I A138227
%S A138227 21,35,45,51,65,69,75,77,85,91,93,99,105,115,117,123,129,133,141,145,
%T A138227 147,155,165,171,185,187,189,195,203,205,213,215,217,219,221,231,235,
%U A138227 237,245,247,253,255,259,261,265,267,273,275,279,285,291,299,301,305
%N A138227 Odd positive integers n for which A137576((n-1)/2)-1 is not a multiple of A000010(n).
%C A138227 All terms are composite numbers since if p is an odd prime then A137576((p-1)/2)-1=p-1=A000010(p).
%C A138227 Conjecture. This sequence is infinite.
%H A138227 Ray Chandler, <a href="http://www.research.att.com/~njas/sequences/b138227.txt"> Table of n, a(n) for n=1..6500</a>
%Y A138227 Cf. A137576. A000010, A002326, A006694.
%K A138227 nonn,new
%O A138227 1,1
%A A138227 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 05 2008
%E A138227 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 08 2008

%I A138217
%S A138217 1,3,5,7,9,11,13,15,17,19,23,25,27,29,31,33,37,39,41,43,47,49,53,55,57,
%T A138217 59,61,63,67,71,73,79,81,83,87,89,95,97,101,103,107,109,111,113,119,121,
%U A138217 125,127,131,135,137,139,143,149,151,153,157,159,161,163,167,169,173
%N A138217 Odd numbers n for which A137576((n-1)/2)-1 is a multiple of A000010(n).
%C A138217 The sequence contains all odd primes. Indeed, if p is a prime then A137576((p-1)/2)-1=p-1=A000010(p).
%C A138217 Conjecture: the sequence contains infinitely many composite numbers.
%H A138217 Ray Chandler, <a href="http://www.research.att.com/~njas/sequences/b138217.txt"> Table of n, a(n) for n=1..3501</a>
%Y A138217 Cf. A137576, A000010, A002326, A006694.
%K A138217 nonn,new
%O A138217 1,2
%A A138217 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 05 2008
%E A138217 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 08 2008

%I A137576
%S A137576 1,3,5,7,13,11,13,17,17,19,31,23,41,55,29,31,41,61,37,49,41,43,85,47,85,
%T A137576 57,53,81,73,59,61,73,73,67,111,71,73,141,151,79,217,83,89,113,89,109,
%U A137576 131,145,97,211,101,103,169,107,109,145,113,221,133,193,221,141,301,127
%N A137576 a(n)=A002326(n)*A006694(n)+1.
%C A137576 Conjecture. a(n)=2n+1 if and only if 2n+1 is 1 or a prime, otherwise a(n)>2n+1. If the conjecture is true then the sequence contains all odd primes. Moreover each odd prime p first appears for n<=(p-1)/2.
%C A137576 Conjecture as stated is not true - a(n)=2n+1 for composites 2047, 3277, 4033, and 8321. But if 2n+1 is prime, then a(n)=2n+1 appears to be sufficient for other statements following conjecture to be true. - Chandler
%H A137576 Ray Chandler, <a href="http://www.research.att.com/~njas/sequences/b137576.txt"> Table of n, a(n) for n=0..10000</a>
%H A137576 Vladimir Shevelev, <a href="http://arxiv.org/abs/0804.3682">Exact exponent of remainder term of Gelfond's digit theorem in binary case</a>
%Y A137576 Cf. A002326, A006694.
%Y A137576 Adjacent sequences: A137573 A137574 A137575 this_sequence A137577 A137578 A137579.
%Y A137576 Sequence in context: A070334 A137700 A071810 this_sequence A111745 A098957 A018205.
%K A137576 nonn,new
%O A137576 0,2
%A A137576 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 26 2008, Apr 28 2008
%E A137576 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 08 2008

%I A138464
%S A138464 1,1,1,1,3,3,1,6,15,16,1,10,45,110,125
%N A138464 Triangle read by rows: T(n,k) = number of forests on n labeled nodes with k edges (n>=1, 0<=k<=n-1).
%e A138464 Triangle begins:
%e A138464 1
%e A138464 1 1
%e A138464 1 3 3
%e A138464 1 6 15 16
%e A138464 1 10 45 110 125
%Y A138464 Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.
%K A138464 nonn,tabl,more,new
%O A138464 1,5
%A A138464 njas, May 09 2008

%I A136605
%S A136605 1,1,1,1,1,1,1,1,2,2,1,1,2,3,3,1,1,2,4,6,6,1,1,2,4,7,11,11,1,1,2,4,
%T A136605 8,14,23,23,1,1,2,4,8,15,29,46,47,1,1,2,4,8,16,32,60,99,106,1,1,2,4,
%U A136605 8,16,33,66,128,216,235,1,1,2,4,8,16,34,69,143,284,488,551,1,1,2,4
%N A136605 Triangle read by rows: T(n,k) = number of forests on n unlabeled nodes with k edges (n>=1, 0<=k<=n-1).
%D A136605 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 58-59.
%e A136605 Triangle begins:
%e A136605 1
%e A136605 1,1
%e A136605 1,1,1
%e A136605 1,1,2,2
%e A136605 1,1,2,3,3
%e A136605 1,1,2,4,6,6 <- T(6,3) = 4 forests on 6 nodes with 3 edges.
%e A136605 1,1,2,4,7,11,11
%e A136605 1,1,2,4,8,14,23,23
%e A136605 1,1,2,4,8,15,29,46,47
%e A136605 1,1,2,4,8,16,32,60,99,106
%e A136605 1,1,2,4,8,16,33,66,128,216,235
%e A136605 1,1,2,4,8,16,34,69,143,284,488,551
%e A136605 1,1,2,4,8,16,34,70,149,315,636,1121,1301
%e A136605 1,1,2,4,8,16,34,71,152,330,710,1467,2644,3159
%Y A136605 Row sums give A005195. Rightmost diagonal gives A000055. Cf. A001858, A138464.
%K A136605 nonn,tabl,new
%O A136605 1,9
%A A136605 njas, May 09 2008

%I A138463
%S A138463 2,22,394,8558,206098,5293446,142078746,3937603038,111818026018,3236724317174,
%T A138463 95149655201962
%N A138463 A bisection of A006318.
%K A138463 nonn,new
%O A138463 0,1
%A A138463 njas, May 08 2008

%I A138462
%S A138462 1,6,90,1806,41586,1037718,27297738,745387038,20927156706,600318853926,
%T A138462 17518619320890,518431875418926
%N A138462 A bisection of A006318.
%K A138462 nonn,new
%O A138462 0,2
%A A138462 njas, May 08 2008

%I A138461
%S A138461 0,1,2,4,6,11,14,29,26,85,12,320,312,1639,3190,10484,25822,75005,200488,
%T A138461 564662,1555804,4363139,12184456,34267931,96435100,272390561,770734846,
%U A138461 2186278294,6213111234
%V A138461 0,1,-2,4,-6,11,-14,29,-26,85,-12,320,312,1639,3190,10484,25822,75005,200488,
%W A138461 564662,1555804,4363139,12184456,34267931,96435100,272390561,770734846,
%X A138461 2186278294,6213111234
%N A138461 Inverse binomial transform of A000957.
%H A138461 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%Y A138461 Cf. A000957, A138415.
%K A138461 sign,new
%O A138461 0,3
%A A138461 njas, May 08 2008

%I A138415
%S A138415 0,1,2,4,10,31,110,421,1686,6961,29392,126292,550360,2426503,10803802,48507844,
%T A138415 219377950,998436793,4569488372,21016589074,97090411020,450314942683,2096122733212,
%U A138415 9788916220519,45850711498860,215348942668681,1013979873542690,4785437476592806
%N A138415 Binomial transform of A000957.
%H A138415 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%Y A138415 Cf. A000957, A138461.
%K A138415 nonn,new
%O A138415 0,3
%A A138415 njas, May 08 2008

%I A138414
%S A138414 1,1,6,57,622,7338,91144,1174281,15548694,210295326,2892818244,40347919626,
%T A138414 569274150156,8110508473252
%N A138414 A bisection of A000957.
%Y A138414 Cf. A000957, A138413.
%K A138414 nonn,easy,more,new
%O A138414 0,3
%A A138414 njas, May 08 2008

%I A138413
%S A138413 0,0,2,18,186,2120,25724,325878,4260282,57048048,778483932,10786724388,
%T A138413 151355847012,2146336125648,30711521221376
%N A138413 A bisection of A000957.
%Y A138413 Cf. A000957, A138414.
%K A138413 nonn,easy,more,new
%O A138413 0,3
%A A138413 njas, May 08 2008

%I A138400
%S A138400 38261,5639533,16575101,19521091,22364431,82787233,86405131
%N A138400 Ennea-Primes. Prime Numbers n as a Sum of 9 prime numbers (8 twin primes and single prime number in - between) are primes.
%C A138400 {5 penta-} {9 ennea-} Table of non-technical numeric prefixes -- http://en.wikipedia.org/wiki/Numerical_prefix
%e A138400 {4229,4231} {4241,4243} {4253} {4259,4261} {4271,4273} Sum = 38261.
%t A138400 a={};Do[p0=Prime[n];a1=Prime[n-4];a2=Prime[n-3];b1=Prime[n-2];b2=Prime[n-1];c1=Prime[n+1];c2=Prime[n+2];d1=Prime[n+3];d2=Prime[n+4];sp=a1+a2+b1+b2+p0+c1+c2+d1+d2;If[PrimeQ[sp]&&a2-a1==2&&b2-b1==2&&c2-c1==2&&d2-d1==2,AppendTo[a,sp]],{n,5,10^6}];a
%K A138400 nonn,uned,new
%O A138400 1,1
%A A138400 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 08 2008

%I A138399
%S A138399 4253,626617,1841681,2169007,2484931,9198577,9600571
%N A138399 Ennea-Primes. Prime Numbers n (single prime number in-between 8 twin primes) such that Sum of 9 prime numbers (8 twin primes and single prime number in-between) are primes.
%C A138399 {5 penta-} {9 ennea-} Table of non-technical numeric prefixes -- http://en.wikipedia.org/wiki/Numerical_prefix
%e A138399 {4229,4231} {4241,4243} {4253} {4259,4261} {4271,4273} Sum = 38261.
%t A138399 a={};Do[p0=Prime[n];a1=Prime[n-4];a2=Prime[n-3];b1=Prime[n-2];b2=Prime[n-1];c1=Prime[n+1];c2=Prime[n+2];d1=Prime[n+3];d2=Prime[n+4];sp=a1+a2+b1+b2+p0+c1+c2+d1+d2;If[PrimeQ[sp]&&a2-a1==2&&b2-b1==2&&c2-c1==2&&d2-d1==2,AppendTo[a,p0]],{n,5,10^6}];a
%K A138399 nonn,uned,new
%O A138399 1,1
%A A138399 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 08 2008

%I A138398
%S A138398 1,2,4,6,7,12,14
%N A138398 Values of records in A140067.
%e A138398 The value of 7 in this sequence means that A137849(8) = 7 with 8 being the corresponding value in A140067
%Y A138398 Cf. A140067, A137849.
%K A138398 more,nonn,new
%O A138398 1,2
%A A138398 J. Lowell, (jhbubby(AT)mindspring.com), May 08 2008

%I A138397
%S A138397 181,331,1381,6481,6551,8069,10499,11633,17669,17851,27551
%N A138397 Penta-Primes. Prime Numbers n as a Sum of 5 prime numbers (four twin primes and single prime number in - between) are primes.
%e A138397 {29, 31} {37} {41, 43} Sum = 181
%t A138397 a={};Do[p0=Prime[n];a1=Prime[n-2];a2=Prime[n-1];b1=Prime[n+1];b2=Prime[n+2];sp=a1+a2+p0+b1+b2;If[PrimeQ[sp]&&a2-a1==2&&b2-b1==2,AppendTo[a,sp]],{n,3,10^3}];a
%K A138397 nonn,uned,new
%O A138397 1,1
%A A138397 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 08 2008

%I A138396
%S A138396 37,67,277,1297,1307,1613,2099,2333,3533,3571,5507
%N A138396 Penta-Primes. Prime numbers n (single prime number in between four twin primes) such that Sum of 5 prime numbers (four twin primes and single prime number in - between) are primes.
%e A138396 {29, 31} {37} {41, 43} Sum = 181.
%t A138396 a={};Do[p0=Prime[n];a1=Prime[n-2];a2=Prime[n-1];b1=Prime[n+1];b2=Prime[n+2];sp=a1+a2+p0+b1+b2;If[PrimeQ[sp]&&a2-a1==2&&b2-b1==2,AppendTo[a,p0]],{n,3,10^3}];a
%K A138396 nonn,uned,new
%O A138396 1,1
%A A138396 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 08 2008

%I A138394
%S A138394 1,2,3,4,6,8,9,10,16,20,24,30,48,60,64,72,80,84,100,108,126
%N A138394 Smallest number with a(n) divisors is in A134865.
%e A138394 8 is in this sequence because the smallest number with 8 divisors (24) is a member of A134865.
%Y A138394 Cf. A134865, A005179.
%K A138394 more,nonn,new
%O A138394 1,2
%A A138394 J. Lowell, (jhbubby(AT)mindspring.com), May 08 2008

%I A138389
%S A138389 1,2,3,4,5,6,7,8,9,10,11,12,13,15,17,19,20,21,23,24,25,29,31,33,35,37,
%T A138389 41,43,47,49,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,
%U A138389 131,137,139,143,149,151,157,163,167,169,173
%N A138389 Binomial primes: positive integers n such that every i not exceeding n/2 for which (n,i)>1 does not divide binomial(n-i-1,i-1).
%C A138389 Every i not exceeding n/2 for which (n,i)=1 divides binomial(n-i-1,i-1). For n>24,a(n) is either prime or square of a prime or a product of twin primes.
%D A138389 V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, International J. of Number Theory, 3,no.1(2007),119-139.
%Y A138389 Cf. A000040, A001248, A077800, A037074.
%K A138389 nonn,new
%O A138389 1,2
%A A138389 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 08 2008

%I A138384
%S A138384 3,2,1,1,0,34,21,13,8,5,377,233,144,89,55,4181,2584,1597,987,610,46368,
%T A138384 28657,17711,10946,6765
%N A138384 Reverse groups of five Fibonascci numbers.
%F A138384 a(n)=11a(n-5)+a(n-10).
%Y A138384 Cf. A102312 (and A049666) , A099100, A134889, A134890, A134891.A000045.
%K A138384 nonn,new
%O A138384 0,1
%A A138384 Paul Curtz (bpcrtz(AT)free.fr), May 08 2008

%I A138383
%S A138383 3,3,9,13,38,25,62,37,86,159,61,207,158,85,182,303,339,121,387,278,145,459,
%T A138383 326,519,748,398,205,422,217,446,1687,518,807,277,1445,301,927,963,662,
%U A138383 1023,1059,361,1865,385,782,397,2466,2610,902,457,926,1419,481,2465,1527
%N A138383 If p(i) = i-th prime, a(n) = p(n)+1 + p(n)+2 + ... + p(n+1). a(0) = 3 by convention.
%F A138383 a(n) = (p(n+1)-p(n))*(p(n+1)+p(n)+1)/2 for n >= 1. - njas, May 08 2008
%e A138383 3 = 1 + 2;
%e A138383 3 = 3;
%e A138383 9 = 4 + 5;
%e A138383 13 = 6 + 7;
%e A138383 38 = 8 + 9 + 10 + 11;
%e A138383 ...
%Y A138383 Cf. A000040.
%K A138383 easy,nonn,new
%O A138383 0,1
%A A138383 Odimar Fabeny (aifab(AT)yahoo.com.br), May 08 2008

%I A138382
%S A138382 1,3,1,2,4,12,4,8,16,48,16,32,64,192,64,128,256,768,256,512,1024
%V A138382 1,-3,-1,-2,-4,12,4,8,16,-48,-16,-32,-64,192,64,128,256,-768,-256,-512,-1024
%N A138382 Second differences of A138382.
%F A138382 a(n)=-4a(n-4).
%K A138382 sign,new
%O A138382 0,6
%A A138382 Paul Curtz (bpcrtz(AT)free.fr), May 08 2008

%I A138381
%S A138381 50,85,125,130,170,185,205,221,250,265,290,305,338,365,370,377,410,425,
%T A138381 445,493,53,0,533,545,578,610,629,650,685,725,754,845,850,890,905,962,
%U A138381 965,970,986,1010,1037,1073,1105,1130,1145,1165,1205,1250,1258,1313
%N A138381 Numbers which are the sum of a strictly positive square and a square of a prime in two or more ways.
%H A138381 Stephan Daniel, <a href="http://journals.cambridge.org/action/displayAbstract?aid=88659">On the sum of a square and a square of a prime</a>, Math. Proc. Cambr. Phil. Soc. vol. 131. no. 1 (2001) 1-22.
%H A138381 <a href="http://research.att.com/~njas/sequences/Sindx_Su.html#ssq">Index entries for sequences related to sums of squares</a>
%e A138381 50 = 1^2+7^2 = 5^2+5^2. 85 = 6^2+7^2 = 9^2+2^2. 125 = 2^2+11^2 = 10^2+5^2.
%e A138381 425 = 8^2+19^2 = 16^2+13^2 = 20^2+5^2 .
%K A138381 easy,nonn,new
%O A138381 1,1
%A A138381 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 08 2008

%I A138380
%S A138380 1,2,1,2,4,8,4,8,16,32,16,32,64,128,64,128,256,512,256,512,1024,2048,
%T A138380 1024,2048,4096,8192,4096,8192,16384,32768
%V A138380 1,2,-1,-2,-4,-8,4,8,16,32,-16,-32,-64,-128,64,128,256,512,-256,-512,-1024,-2048,1024,
%W A138380 2048,4096,8192,-4096,-8192,-16384,-32768
%N A138380 First differences of A138377.
%K A138380 sign,new
%O A138380 0,2
%A A138380 Paul Curtz (bpcrtz(AT)free.fr), May 08 2008

%I A138379
%S A138379 1,6,60,888,18120,485280,16445520,685722240,34411184640,2041544736000,
%T A138379 141106965753600,11223409849344000,1016591564596608000,
%U A138379 103921686070339737600,11896153817325313536000
%N A138379 Number of embedded coalitions in an n-person game where the position of the individual player is important.
%C A138379 See A138378 for references and links.
%F A138379 a(1) = 1 * SUM(i=0 to 0)combination(1,i) = 1, a(2) = 2 * SUM(i=0 to 1)combination(2,i) = 6, a(n) = n![SUM(i=2 to n-1) combination(n,i) * {SUM(j=1 to i-1) * a(j)/j! } + SUM(i=0 to n-1) combination(n,i)], for n > 2.
%e A138379 a(1) = 1 * SUM(i=0 to 0)combination(1,i) = 1,
%e A138379 a(2) = 2 * SUM(i=0 to 1)combination(2,i) = 6,
%e A138379 a(3) = 3![combination(3,2) * a(1)/1! + combination(3,2) + combination(3,1) + combination(3,0)] = 60,
%e A138379 a(4) = 4![combination(4,3) * {a(2)/2! + a(1)/1!} + combination(4,2) * a(1)/1! + combination(4,3) + combination(4,2) + combination(4,1) + combination(4,0)] = 888,
%e A138379 a(5) = 5![combination(5,4) * {a(3)/3! + a(2)/2! + a(1)/1!} + combination(5,3) * {a(2)/2! + a(1)/1!} + combination(5,2) * a(1)/1! + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)] = 18120.
%Y A138379 Cf. A138378.
%K A138379 nonn,new
%O A138379 1,2
%A A138379 David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008

%I A138378
%S A138378 1,3,10,37,151,674,3263,17007,94828,562595,3535027,23430840,163254885,
%T A138378 1192059223,9097183602,72384727657,599211936355,5150665398898,
%U A138378 45891416030315,423145657921379
%N A138378 Number of embedded coalitions in an n-person game.
%C A138378 The strategic behavior of players depends crucially on the coalition structures of a game.
%D A138378 DAVID W. K. YEUNG, 2008, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10(1), 129-136.
%D A138378 E. T. Bell, 1934, Exponential numbers, American Mathematical Monthly 41,411-419.
%D A138378 Conway, J. H. and Guy, R. K. [1995] The Book of Numbers (Springer-Verlag, NewYork).
%H A138378 DAVID W. K. YEUNG, <a href="http://www.worldscinet.com/cgi-bin/details.cgi?id=pii:S0219198908001819&type=html">Perfect Numbers</a>, International Game Theory Review 10(2008), 129-136.
%F A138378 a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.
%e A138378 a(1) = combination(1,0) = 1,
%e A138378 a(2) = combination(2,1) + combination(2,0)= 3,
%e A138378 a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,
%e A138378 a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,
%e A138378 a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.
%Y A138378 Cf. A138379.
%K A138378 easy,nonn,new
%O A138378 1,2
%A A138378 David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008

%I A138377
%S A138377 0,1,3,2,0,4,12,8,0,16,48,32,0,64,192,128,0,256,768,512,0
%V A138377 0,1,3,2,0,-4,-12,-8,0,16,48,32,0,-64,-192,-128,0,256,768,512,0
%N A138377 a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2; thereafter a(n)=-4a(n-4).
%C A138377 First and third differences have only 2^n's.
%Y A138377 Cf. A138380, A138382.
%K A138377 sign,easy,more,new
%O A138377 0,3
%A A138377 Paul Curtz (bpcrtz(AT)free.fr), May 08 2008

%I A138376
%S A138376 0,1,3,0,4,1,7,0,8,1,2,0,3,1,6,0,7,1,10,0,2,1,5,0,6,1,9,0,10,1,4,0,5,1,
%T A138376 8,0,9,1,12,0,4,1,7,0,8,1,11,0,12,1,6,0,7,1,10,0,11,1,14,0,6,1,9,0,10,1,
%U A138376 13,0,14,1,8,0,9,1,12,0,13,1,16,0,8,1,11,0,12,1,15,0,16,1,10,0,11,1,14
%N A138376 a(n+1) = abs[ a(n) + Sum_of_digits_of(n+1)], with a(0)=0.
%C A138376 a(4*k)=0 , with k>=1
%C A138376 a(4*k-2)=1, with k>=1
%F A138376 a(n+1) = abs[ a(n) + Sum_of_digits_of(n+1)], with a(0)=0.
%p A138376 P:=proc(n) local a,i,k,w; a:=0; print(a); for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=abs(a+(-1)^i*w); print(a); od; end: P(100);
%Y A138376 Cf. A037123.
%K A138376 easy,nonn,new
%O A138376 0,3
%A A138376 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), May 08 2008

%I A138375
%S A138375 13,17,29,113,157,269,337,797,1613,1777,1949,2129,3613,4637,5197,6737,
%T A138375 7069,7757,8849,11677,12113,13009,13469,14897,15889,20177,20749,23117,
%U A138375 24977
%N A138375 Primes of the form n^2+13.
%t A138375 Intersection[Table[n^2+13,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=13,i<=13,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138375 nonn,new
%O A138375 1,1
%A A138375 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138368
%S A138368 13,37,61,181,373,541,853,1237,1381,1693,1861,2221,3037,3733,7237,7933,
%T A138368 8293,9421,12781,14173,14653,16141,19333,25933,27901,28573
%N A138368 Primes of the form n^2+12.
%t A138368 Intersection[Table[n^2+12,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=12,i<=12,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138368 nonn,new
%O A138368 1,1
%A A138368 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138362
%S A138362 11,47,587,911,1307,2927,8111,9227,13007,14411,15887,20747,22511,32411,
%T A138362 34607,41627,44111,54767,60527,72911,90011
%N A138362 Primes of the form n^2+11.
%t A138362 Intersection[Table[n^2+11,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=11,i<=11,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138362 nonn,new
%O A138362 1,1
%A A138362 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138355
%S A138355 11,19,59,131,179,739,971,1531,2411,2819,3259,3491,5051,5939,6571,6899,
%T A138355 8291,9419,9811
%N A138355 Primes of the form n^2+10.
%t A138355 Intersection[Table[n^2+10,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=10,i<=10,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138355 nonn,new
%O A138355 1,1
%A A138355 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138353
%S A138353 13,73,109,409,1033,1453,1609,2713,3373,3853,4909,6733,7753,9613,10009,
%T A138353 12109,12553,13933,19609,20173,25609
%N A138353 Primes of the form n^2+9.
%t A138353 Intersection[Table[n^2+9,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=9,i<=9,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138353 nonn,new
%O A138353 1,1
%A A138353 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138352
%S A138352 1,1,0,1,1,0,1,2,1,0,1,2,5,1,0,1,2,8,8,1,0,1,2,9,20,14,1,0,1,2,9,29
%N A138352 Table read by antidiagonals: T(n,k) is the number of transitive directed multigraphs with loops with n arcs and k vertices.
%C A138352 Partial sums of the rows of A139623, i.e., T(n,k) = sum(T139623(n,p),p=0..k).
%C A138352 If k>=2n, T(n,k) = A139628(n).
%Y A138352 Cf. A138107.
%K A138352 nonn,tabl,new
%O A138352 0,8
%A A138352 Benoit Jubin (benoit_jubin(AT)yahoo.fr), May 07 2008

%I A138338
%S A138338 17,89,233,449,1097,2609,3257,6569,7577,12329,13697,15137,16649,18233,
%T A138338 19889,21617
%N A138338 Primes of the form n^2+8.
%t A138338 Intersection[Table[n^2+8,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=8,i<=8,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
%K A138338 nonn,new
%O A138338 1,1
%A A138338 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138323
%S A138323 8,251,78376,1977405119,34524689549050,8650450444070886983,
%T A138323 239081086135595395734136,257829867026393862843621801395
%N A138323 Sum_{k = 1..n} p(k)^p(k + 1), where p(k) = k - th prime.
%e A138323 2^3=8
%e A138323 2^3+3^5=8+243=251
%e A138323 2^3+3^5+5^7=8+243+78125=78376
%t A138323 P3[n_] := Sum[Prime[i]^Prime[i + 1], {i, 1, n}]; Table[P3[n], {n, 1, 8}]
%K A138323 nonn,new
%O A138323 1,1
%A A138323 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 07 2008

%I A138302
%S A138302 1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,28,29,
%T A138302 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,55,
%U A138302 57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81
%N A138302 Products of distinct relatively prime terms of A084400.
%C A138302 These numbers are called "compact integers". They consist of 1 and the positive integers for which all exponents of primes in its prime power factorization are nonnegative powers of 2.
%C A138302 The density of this sequence exists and equals 0.872497...
%C A138302 There exist only seven compact factorials A000142(n) for n=1,2,3,6,7,10 and 11.
%D A138302 V. Shevelev, Compact integers and factorials. Acta Arithmetica,126,no.3(2007), 195-236.
%H A138302 S. Litsyn and V. Shevelev, <a href="http://www.integers-ejcnt.org/">On Factorization of Integers with Restrictions on the Exponents</a>, INTEGERS: The Electronic Journal ofCombinatorial Number Theory 7(2007),#A33.
%Y A138302 Cf. A084400, A050376, A005117.
%K A138302 nonn,new
%O A138302 1,2
%A A138302 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 07 2008

%I A138300
%S A138300 2,8,8,8,8,8,8,8,8,8,8,8,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,
%T A138300 18,18,18,18,18,18,18,18,18,18,18,18,32,32,32,32,32,32,32,32,32,32,32,
%U A138300 32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32
%N A138300 Differences of each column for atomic numbers of Mendeleyev-Seaborg 7*32 elements periodic table,first extension,A138096 table.86 terms.Horizontal lecture.
%e A138300 First column differences: 2, 8, 8, 18, 18, 32; second: 8, 8, 18, 18, 32.
%e A138300 Table: 1 2 , 11 8's , 27 18's , 47 32's
%e A138300 2.............................................................................................8
%e A138300 8..8...........................................................................8..8..8..8..8..8
%e A138300 8..8..........................................................................18.18.18.18.18.18
%e A138300 18.18.18...........................................18.18.18.18.18.18.18.18.18.18.18.18.18.18.18
%e A138300 18.18.18...........................................32.32.32.32.32.32.32.32.32.32.32.32.32.32.32
%e A138300 32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32.32
%Y A138300 Cf. A093907, A137508, A137575, A137583, A138102.
%K A138300 nonn,uned,new
%O A138300 0,1
%A A138300 Paul Curtz (bpcrtz(AT)free.fr), May 07 2008

%I A138298
%S A138298 0,2,10,21,110,233,1220,2584,13530,28657,150050,317811,1664080,3524578
%N A138298 Among Fibonacci numbers.We complete A137976 (3, 13, 34, 144) with two leading 1's.Definition must be changed:a(n) is differences.
%C A138298 For A134490:b(n)=11b(n-1)+b(n-2).Odd 1, 1, A137976 last digit = 1, 3, 4, 7, 1, 8, 9, 7, must be period 12: repeat 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2 = A130893.
%F A138298 Mix 2*A102312 or 10*A049666, A134490.
%K A138298 nonn,uned,new
%O A138298 0,2
%A A138298 Paul Curtz (bpcrtz(AT)free.fr), May 07 2008

%I A138296
%S A138296 5,13,7,35,29,5,97,133,13,9,275,641,35,53,8,793,3157,97,351,34,5,2315,
%T A138296 15689,275,2417,152,13,7,6817,78253,793,16839,706,35,29,10,20195,390881,
%U A138296 2315,117713,3368,97,133,58,13,60073,1953637,6817,823671,16354,275,641
%N A138296 Table T(k,n) read along antidiagonals: sum of the kth powers of the distinct prime factors of A024619(n).
%C A138296 Row k=1 is A109353. Rows k=2,3 and 4 are subsequences of A005063-A005065.
%H A138296 J.-M de Koninck, F. Luca, <a href="http://dx.doi.org/10.1016/j.jnt.2007.01.010">Integers divisible by sums of powers of their prime factors</a>, J. Num. Theory vol 128 (2008) 557-563.
%F A138296 T(k,n) = sum_{d in A000040, d| A024619(n)} d^k.
%e A138296 Upper left corner of the table starting at row k=1, column n=1:
%e A138296 1|......5.......7.......5.......9.......8.......5.......7.
%e A138296 2|.....13......29......13......53......34......13......29.
%e A138296 3|.....35.....133......35.....351.....152......35.....133.
%e A138296 4|.....97.....641......97....2417.....706......97.....641.
%e A138296 5|....275....3157.....275...16839....3368.....275....3157.
%e A138296 6|....793...15689.....793..117713...16354.....793...15689.
%e A138296 7|...2315...78253....2315..823671...80312....2315...78253.
%e A138296 8|...6817..390881....6817.5765057..397186....6817..390881.
%p A138296 A001221 := proc(n) nops(numtheory[factorset](n)) ; end: A024619 := proc(n) local a; if n = 1 then RETURN(6); else for a from A024619(n-1)+1 do if A001221(a) > 1 then RETURN(a) ; fi ; od: fi ; end: T := proc(n,j) local f,beta ; beta := 0 ; for f in ifactors( A024619(n) )[2] do beta := beta+op(1,f)^j ; od: RETURN(beta) ; end: for d from 1 to 10 do for n from 1 to d do printf("%d,",T(n,d-n+1)) ; od: od: # R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 07 2008
%K A138296 easy,nonn,tabl,new
%O A138296 1,1
%A A138296 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 07 2008

%I A138280
%S A138280 0,110,2212110,332322132212110,4434332433232214332322132212110,
%T A138280 554544354434332544343324332322154434332433232214332322132212110
%N A138280 Recursive sequence of integer lists involved with the game Tag Deal.
%D A138280 Barry Cipra's game of Tag Deal
%F A138280 a(n) = 2*a(n-1) + n + a(n-1). Here addition means concatnation of numbers into a list; i.e. 2 + 3 = 23
%Y A138280 Cf. A987654, A987655.
%K A138280 nonn,base,new
%O A138280 0,2
%A A138280 Trevor G. Hyde (trivial34(AT)hotmail.com), May 06 2008

%I A138170
%S A138170 2,3,5,23,31,61,83,89,149,179,239,251,263,269,353,367,419,433,449,503,
%T A138170 557,569,571,587,653,701,733,761,839,941,983,991,1109,1123,1187,1193
%N A138170 Prime numbers p1 such that p1*p2+mod[p2,p1] is a prime, where p2 is the next prime after p1.
%e A138170 2 prime, 3 next-prime, 2*3+Mod[3,2]=2*3+1=7
%e A138170 3 prime, 5 next-prime, 3*5+Mod[5,3]=3*5+2=17
%e A138170 5 prime, 7 next-prime, 5*7+Mod[7,5]=5*7+2=37
%t A138170 a={};Do[p1=Prime[n];p2=Prime[n+1];e=p1*p2+Mod[p2,p1];If[PrimeQ[e],AppendTo[a,p1]],{n,10^2*2}];a
%K A138170 nonn,new
%O A138170 1,1
%A A138170 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138111
%S A138111 2,3,7,19,23,43,53,79,127,211,229,233,337,397,443,463,467,499,503,601,
%T A138111 631,661,967,991,1009,1129,1153,1213
%N A138111 Prime numbers p1 such that p1*p2-mod[p2,p1] is a prime, where p2 is the next prime after p1.
%e A138111 2 prime, 3 next-prime, 2*3-Mod[3,2]=2*3-1=5 prime.
%e A138111 3 prime, 5 next-prime, 3*5-Mod[5,3]=3*5-2=13 prime.
%e A138111 7 prime, 11 next-prime, 7*11-Mod[11,7]=7*11-4=73 prime.
%t A138111 a={};Do[p1=Prime[n];p2=Prime[n+1];e=p1*p2-Mod[p2,p1];If[PrimeQ[e],AppendTo[a,p1]],{n,10^2*2}];a
%K A138111 nonn,new
%O A138111 1,1
%A A138111 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138091
%S A138091 25,43,55,109,131,379,389,587,617,649,683,697,719,971,1013,1279,1291,
%T A138091 1727,1735,1823,1853,2005,2059,2087,2167,2395,2399,2561,2647,2783,2957,
%U A138091 2983
%N A138091 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11+(n+12)^12+(n+13)^13 is a prime.
%t A138091 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11+(n+12)^12+(n+13)^13],AppendTo[a,n]],{n,10^3*3}];a
%K A138091 nonn,new
%O A138091 1,1
%A A138091 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138089
%S A138089 2,13,42,46,57,102,104,111,167,171,182,189,191,207,244,268,298,312,322,
%T A138089 349,366,368,371,442,466,477,508,517,553,661,674,686,728,737,748,776
%N A138089 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11+(n+12)^12 is a prime.
%t A138089 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11+(n+12)^12],AppendTo[a,n]],{n,10^2*8}];a
%K A138089 nonn,new
%O A138089 1,1
%A A138089 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138065
%S A138065 26,36,40,96,152,178,246,262,276,310,360,496,568,572,586,646,654,694,
%T A138065 706,738,808,822,828,852,898,976,988
%N A138065 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11 is a prime.
%t A138065 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9+(n+10)^10+(n+11)^11],AppendTo[a,n]],{n,10^3}];a
%K A138065 nonn,new
%O A138065 1,1
%A A138065 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138059
%S A138059 123,135,201,363,987,1485,1545,1593,1713,1947,2211,2391,2571,2577,2751,
%T A138059 3093,3273,3375,3381,3693,3801,4155,4407,4521,4587
%N A138059 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9 is a prime.
%t A138059 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8+(n+9)^9],AppendTo[a,n]],{n,10^3*5}];a
%K A138059 nonn,new
%O A138059 1,1
%A A138059 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A138023
%S A138023 1,5,8,22,24,34,48,52,58,59,69,73,92,110,134,148,157,167,181,188,226,
%T A138023 233,268,303,307,321,332,337,349,376,381,415,500,503,549,558,590
%N A138023 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8 is a prime.
%t A138023 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7+(n+8)^8],AppendTo[a,n]],{n,10^2*6}];a
%K A138023 nonn,new
%O A138023 1,2
%A A138023 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A137980
%S A137980 4,24,28,90,112,232,310,346,480,492,522,564,592,648,666,690,694,766,802,
%T A137980 856,868,900,930,960,990
%N A137980 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7 is a prime.
%t A137980 a={};Do[If[PrimeQ[n^70+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4+(n+5)^5+(n+6)^6+(n+7)^7],AppendTo[a,n]],{n,10^3}];a
%K A137980 nonn,new
%O A137980 1,1
%A A137980 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A137950
%S A137950 1,3,4,5,7,11,14,21,22,23,28,31,33,47,50,53,56,59,70,72,82,88,92,99,106,
%T A137950 120,122,124,135,140,149,157,159,162,166,169,172,179,182
%N A137950 Numbers n such that n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4 is a prime.
%t A137950 a={};Do[If[PrimeQ[n^0+(n+1)^1+(n+2)^2+(n+3)^3+(n+4)^4],AppendTo[a,n]],{n,10^2*2}];a
%K A137950 nonn,new
%O A137950 1,2
%A A137950 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A137850
%S A137850 2,3,5,17,191,257,1009,4561,4591,21601,57601,54121,86677,176401,
%T A137850 415801,291721
%N A137850 Smallest prime which is a palindrome in n bases (or numbering systems, including unary).
%C A137850 a(n) = A000040(A137779^(-1)(n)).
%C A137850 The sequence is not monotonic: a(10) > a(11) = 54121.
%C A137850 a(17) > 675221.
%e A137850 a(12) = 54121 because 54121 is a palindrome in 12 different bases, including base 1 and base 54120.
%Y A137850 Cf. A137779.
%K A137850 hard,more,nonn,new
%O A137850 1,1
%A A137850 Attila Olah (jolafix(AT)gmail.com), May 06 2008, corrected May 08 2008

%I A137848
%S A137848 2,6,8,17,22,39,89,143,167,312,334,357,414
%N A137848 Numbers n such that 10^n + 81 is prime.
%e A137848 10^2+81=100+81=181 is prime
%t A137848 q=81;s="";For[a=q,a<=q,s="10^n+"<>ToString[a]<>":";n=0;For[i=1,i< 10^3,If[PrimeQ[10^i+a],n=1;s=s<>ToString[i]<>","];i++ ];If[n>0,Print[s]];a++ ]
%K A137848 nonn,new
%O A137848 1,1
%A A137848 Vladimir Orlovsky (4vladimir(AT)gmail.com), May 06 2008

%I A137779
%S A137779 1,2,3,3,2,3,4,2,3,3,4,3,3,3,2,2,3,3,4,3,4,2,3,3,3,3,2,4,4,3,4,3,2,
%T A137779 2,2,4,4,2,2,4,2,4,5,3,4,3,4,2,4,3,3,3,4,3,6,2,2,4,4,3,2,2,4,2,5,2,
%U A137779 3,5,2,3,5,2,2,6,5,3,2,3,4,4,4,5,3,4,2,5,3,4,4,4,3,3,4,2,3,3,3,4,4
%N A137779 Number of bases (numbering systems, including unary) in which the nth prime is a palindrome having at least two digits.
%C A137779 Each prime p > 2 is palindrome in at least base 1 and base p-1, since p = 1*(p-1)^1 + 1*(p-1)^0 and p = 1*1^(p-1) + 1*1(p-2) + ... + 1*1^1 + 1*1^0.
%H A137779 Attila Olah, <a href="http://www.research.att.com/~njas/sequences/b137779.txt">Table of n, a(n) for n = 1..10000</a>
%e A137779 a(621) = 9 because the 621th prime (4591) is a palindrome in 9 bases: base 1, 19, 20, 24, 33, 37, 51, 54 and 4590 (4591 = 1*4590^1 + 1*4590^0).
%Y A137779 Cf. A137850.
%K A137779 easy,base,nonn,new
%O A137779 1,3
%A A137779 Attila Olah (jolafix(AT)gmail.com), May 06 2008, corrected May 08 2008

%I A137775
%S A137775 0,3,6,45,252,1935,16146,153657,1616760,18699579,235498590,3207570597,
%T A137775 46968796404,735689606535,12272343940458,217191191400945,
%U A137775 4064131571557104,80166987477918963,1662468879466624950
%N A137775 Number of triples of permutations on n letters such that for each j, exactly one of the permutations fixes j and the other two have the same image on j.
%C A137775 This sequence arises in a calculation of the fourth moments of the volumes of random polytopes in certain very symmetric convex bodies.
%D A137775 M. Meckes, Volumens of symmetric random polytopes, Arch. Math. 82 (2004) 85--96.
%F A137775 a(n) = n(a(n-1)+3*a(n-2)) with a(0)=1; exponential generating function = exp(-3x)/(1-x)^3.
%e A137775 a(2)=3 because one of the permutations must be the identity and the other two are the transposition (1 2); there are three ways to pick which is the identity.
%Y A137775 Cf. A000166.
%K A137775 nonn,new
%O A137775 1,2
%A A137775 Mark W. Meckes (mark.meckes(AT)case.edu), May 06 2008

%I A129620
%S A129620 1,1,0,1,1,0,1,2,1,0,1,2,5,1,0,1,2,8,9,1,0,1,2,8,24,17,1,0,1,2,8,32
%N A129620 Table read by antidiagonals: T(n,k) is the number of connected directed multigraphs with loops with n arcs and k vertices.
%C A129620 Partial sums of the rows of A139621, i.e., T(n,k) = sum(T139621(n,p),p=0..k).
%C A129620 T(n,2) = T138107(n,2) - floor(n/2).
%C A129620 If k >= n+1, T(n,k) = A137975(n).
%Y A129620 Cf. A138107.
%K A129620 nonn,tabl,new
%O A129620 0,8
%A A129620 Benoit Jubin (benoit_jubin(AT)yahoo.fr), May 06 2008

%I A140116
%S A140116 0,20,1,21,10,21021,21020,210,11,1120,1121,211210,11210,21121,21120,211,
%T A140116 100,10020,10021,2100211210,100210,210021121,210021120,2100211,100211,
%U A140116 210021021,210021020,2100210,21002120,210021,210020,2100,101,10120
%N A140116 Numbers encoded in an alternate, sometimes more compact, binary system with a third, dual-purpose, prefix/delimiter symbol not always required.
%C A140116 In general, this representation is (sometimes much) more compact for numbers having almost all 0s or 1s in their binary representations.
%C A140116 Definition: i) 0 is encoded as 0. ii) For n>0, first convert the number to its standard binary representation n_2 (A007088(n)).
%C A140116 If, and only if, n_2 has more 1s than 0s (i.e., n is a term of A072600), a(n) begins with the prefix symbol 2.
%C A140116 For determining bit positions below, the least significant bit position is counted as position 0. In all cases, a(n) next includes the bit position of the leading 1 in n_2.
%C A140116 For the last step, the target bit type depends upon whether the prefix symbol was used: if so, target is 0 else 1. Finally, scan n_2 left-to-right for all bits of the target type.
%C A140116 For each such bit found, append the delimiter symbol 2 followed by that bit's position p in standard binary; i.e., append 2, then A007088(p).
%C A140116 PARI program to do encoding is available; may be posted later.
%C A140116 Some ideas for improvement (and other sequences):
%C A140116 1) Introduce a fourth symbol, 3, to serve as a second delimiter (usually meaning "switch to 0-positions now") and save a symbol position by avoiding the need for a prefix -- except that in the cases where the numbers are of the form 2^k-1 (all 1s, which would still need to be distinguished from a 1 followed by all 0s) the new "delimiter" would have to be used as the final symbol instead.
%C A140116 2) Use a type of run length encoding on the bit positions to make representations more compact also when there are blocks of 0s or 1s.
%C A140116 3) Use the fact that the present method may encode the 1s complement of a particular number more compactly.
%e A140116 a(63) = 2101 as 63 = 111111 (base 2) has more 1s than 0s; the leading 1 is in position 5 = 101 (base 2). a(64) = 110 as 64 = 1000000 (base 2) does not have more 1s than 0s; the leading 1 is in position 6 = 110 (base 2). a(65) = 11020 as 65 = 1000001 (base 2) does not have more 1s than 0s; 1s are in positions 6 = 110 (base 2) and 0 = 0 (base 2).
%Y A140116 Cf. A140117, A007088, A072600.
%K A140116 base,nonn,new
%O A140116 0,2
%A A140116 Rick L. Shepherd (rshepherd2(AT)hotmail.com), May 08 2008

%I A139409
%S A139409 1,11037,158712,904117,906373
%N A139409 Numbers n such that n=d_1!!^2+d_2!!^2+...+d_k!!^2 where d_1d_2...d_k is the decimal expansion of n.
%e A139409 906373=9!!^2+0!!^2+6!!^2+3!!^2+7!!^2+3!!^2.
%K A139409 base,fini,full,nonn,new
%O A139409 1,2
%A A139409 Farideh Firoozbakht (mymontain(AT)yahoo.com), May 08 2008

%I A140115
%S A140115 0,0,0,2,6,8,12,18,20,27,30,43,44,60,58,76,75,88,112,106,124,160
%N A140115 a(n)=number of "pseutriprimes (product of 3 distinct primes) >(n-1)^3 and < n^3.
%e A140115 The first pseutriprimes are 30,42,66,70,78,102,110,114,130
%e A140115 None is <27, hence a(1)=a(2)=a(3)=0
%e A140115 30 and 42 are < 64, hence a(4)=2
%e A140115 6 more are <125, hence a(5)=6
%Y A140115 Cf. A140114.
%K A140115 easy,nonn,new
%O A140115 0,4
%A A140115 Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008

%I A140113
%S A140113 1,5,8,24,29,65,72,136,145,245,256,400,413,609,624,880,897,1221,1240,
%T A140113 1640,1661,2145,2168,2744,2769,3445,3472,4256,4285,5185,5216,6240,6273,
%U A140113 7429,7464,8760,8797,10241,10280,11880,11921,13685,13728,15664,15709
%N A140113 a(1)=1, a(n)=a(n-1)+n if n odd, a(n)=a(n-1)+ n^2 if n is even.
%t A140113 a = {}; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m + (Cos[Pi m/2]^2) m^2, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
%Y A140113 Cf. A136047.
%K A140113 nonn,new
%O A140113 1,2
%A A140113 Artur Jasinski (grafix(AT)csl.pl), May 08 2008

%I A140095
%S A140095 1,1,5,41,437,5513,78477,1225865,20644021,370334137,7017055933,
%T A140095 139562915193,2899946191077,62722686552841,1408033260333581,
%U A140095 32729098457253417,786224322656857941,19486950945070339801
%N A140095 G.f. satisfes: A(x) = x/[1 - A(A(A(A(x))))] = Series_Reversion[x - x*A(A(A(x)))].
%F A140095 Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
%F A140095 then A_{n}(x) = A_{n-1}/[1 - A_{n+3}(x)] ;
%F A140095 thus A_{n}(x) = 1 - A_{n-4}(x) / A_{n-3}(x).
%e A140095 G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 437*x^5 + 5513*x^6 + 78477*x^7 +...
%e A140095 Self-compositions A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
%e A140095 A_2(x) = 1 - Series_Reversion[A_2(x)]/Series_Reversion[A(x)] where
%e A140095 A_2(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1220*x^5 + 16028*x^6 +...
%e A140095 A_3(x) = 1 - Series_Reversion[A(x)]/x where
%e A140095 A_3(x) = x + 3*x^2 + 21*x^3 + 207*x^4 + 2489*x^5 + 34259*x^6 +...
%e A140095 A_4(x) = 1 - x/A(x) where
%e A140095 A_4(x) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4408*x^5 + 63776*x^6 +...
%e A140095 A_5(x) = 1 - A(x)/A_2(x) where
%e A140095 A_5(x) = x + 5*x^2 + 45*x^3 + 525*x^4 + 7165*x^5 + 109125*x^6 +...
%e A140095 A_6(x) = 1 - A_2(x)/A_3(x) where
%e A140095 A_6(x) = x + 6*x^2 + 60*x^3 + 756*x^4 + 10972*x^5 + 175948*x^6 +...
%e A140095 A_7(x) = 1 - A_3(x)/A_4(x) where
%e A140095 A_7(x) = x + 7*x^2 + 77*x^3 + 1043*x^4 + 16065*x^5 + 271103*x^6 +...
%e A140095 A_8(x) = 1 - A_4(x)/A_5(x) where
%e A140095 A_8(x) = x + 8*x^2 + 96*x^3 + 1392*x^4 + 22704*x^5 + 402784*x^6 +...
%e A140095 Self-compositions are also related by continued fractions:
%e A140095 A(x) = x/(1 - A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 -...)))) ;
%e A140095 A_2(x) = A(x)/(1 - A_4(x)/(1 - A_7(x)/(1 - A_10(x)/(1 -...)))) ;
%e A140095 A_3(x) = A_2(x)/(1 - A_5(x)/(1 - A_8(x)/(1 - A_11(x)/(1 -...)))).
%o A140095 (PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n,B=subst(A, x, A); A=x/(1-subst(B, x, B))); polcoeff(A, n))}
%Y A140095 Cf. A140094, A088714.
%K A140095 nonn,new
%O A140095 1,3
%A A140095 Paul D. Hanna (pauldhanna(AT)juno.com), May 08 2008

%I A140094
%S A140094 1,1,4,25,199,1855,19387,221407,2717782,35455981,487672243,7029980797,
%T A140094 105732907498,1653377947393,26805765569863,449568735630517,
%U A140094 7785116448484318,138980739891821269,2554369130466577138
%N A140094 G.f. satisfes: A(x) = x/[1 - A(A(A(x)))] = Series_Reversion[x - x*A(A(x))].
%F A140094 Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
%F A140094 then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
%F A140094 thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
%e A140094 G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
%e A140094 Self-compositions A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
%e A140094 A_2(x) = 1 - Series_Reversion[A(x)]/x where
%e A140094 A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...
%e A140094 A_3(x) = 1 - x/A(x) where
%e A140094 A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
%e A140094 A_4(x) = 1 - A(x)/A_2(x) where
%e A140094 A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...
%e A140094 A_5(x) = 1 - A_2(x)/A_3(x) where
%e A140094 A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...
%e A140094 A_6(x) = 1 - A_3(x)/A_4(x) where
%e A140094 A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...
%e A140094 Self-compositions are also related by continued fractions:
%e A140094 A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
%e A140094 A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).
%o A140094 (PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, subst(A, x, A)))); polcoeff(A, n))}
%Y A140094 Cf. A140095, A088714.
%K A140094 nonn,new
%O A140094 1,3
%A A140094 Paul D. Hanna (pauldhanna(AT)juno.com), May 08 2008

%I A139573
%S A139573 1099511627776,12157665459056928801,9094947017729282379150390625
%N A139573 Numbers with 41 divisors.
%C A139573 40th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
%H A139573 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139573 a(n)=A000040(n)^(41-1)=A000040(n)^40.
%Y A139573 Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572.
%K A139573 bref,more,nonn,new
%O A139573 1,1
%A A139573 Omar E. Pol (info(AT)polprimos.com), May 07 2008

%I A139572
%S A139572 68719476736,150094635296999121,14551915228366851806640625,
%T A139572 2651730845859653471779023381601
%N A139572 Numbers with 37 divisors.
%C A139572 36th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
%H A139572 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139572 a(n)=A000040(n)^(37-1)=A000040(n)^36.
%Y A139572 Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139573.
%K A139572 more,nonn,new
%O A139572 1,1
%A A139572 Omar E. Pol (info(AT)polprimos.com), May 07 2008

%I A139571
%S A139571 1073741824,205891132094649,931322574615478515625,
%T A139571 22539340290692258087863249,17449402268886407318558803753801
%N A139571 Numbers with 31 divisors.
%C A139571 30th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
%H A139571 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F A139571 a(n)=A000040(n)^(31-1)=A000040(n)^30.
%Y A139571 Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139572, A139573.
%K A139571 more,nonn,new
%O A139571 1,1
%A A139571 Omar E. Pol (info(AT)polprimos.com), May 07 2008

%I A140079
%S A140079 254540,310155,378014,421134,432795,483405,486590,486794,488565,489345,
%T A140079 507129,522444,545258,549185,558789,558830,567644,577940,584154,591260,
%U A140079 598689,627095,634809,637329,663585,666995,667029,678755,687939,690234
%N A140079 Numbers n such that n and n+1 have 5 distinct prime factors.
%C A140079 Smallest number r such that r and r+1 have n distinct prime factors see A093548
%H A140079 D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim., <a href="http://front.math.ucdavis.edu/0803.2636">Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers</a>.
%t A140079 a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
%Y A140079 Cf. A074851, A140077, A140078.
%K A140079 nonn,new
%O A140079 1,1
%A A140079 Artur Jasinski (grafix(AT)csl.pl), May 07 2008

%I A140077
%S A140077 230,285,429,434,455,494,560,594,609,615,644,645,650,665,740,741,759,
%T A140077 804,805,819,825,854,860,884,902,935,945,969,986,987,1001,1014,1022,
%U A140077 1034,1035,1044,1064,1065,1070,1085,1104,1105,1130,1196,1209,1220,1221
%N A140077 Numbers n such that n and n+1 have 3 distinct prime factors.
%H A140077 D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim., <a href="http://front.math.ucdavis.edu/0803.2636">Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers</a>.
%t A140077 a = {}; Do[If[Length[FactorInteger[n]] == 3 && Length[FactorInteger[n + 1]] == 3, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
%Y A140077 Cf. A074851, A140078, A140079 .
%K A140077 nonn,new
%O A140077 1,1
%A A140077 Artur Jasinski (grafix(AT)csl.pl), May 07 2008

%I A140078
%S A140078 7314,8294,8645,9009,10659,11570,11780,11934,13299,13629,13845,14420,
%T A140078 15105,15554,16554,16835,17204,17390,17654,17765,18095,18290,18444,
%U A140078 18920,19005,19019,19095,19227,20349,20405,20769,21164,21489,21735
%N A140078 Numbers n such that n and n+1 have 4 distinct prime factors.
%C A140078 For numbers n such that n and n+1 have k distinct prime factors see:
%C A140078 k=2 A074851
%C A140078 k=3 A140077
%C A140078 k=4 A140078
%C A140078 k=5 A140079
%H A140078 D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim., <a href="http://front.math.ucdavis.edu/0803.2636">Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers</a>.
%t A140078 a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
%Y A140078 Cf. A074851, A140077, A140079 .
%K A140078 nonn,new
%O A140078 1,1
%A A140078 Artur Jasinski (grafix(AT)csl.pl), May 07 2008

%I A140049
%S A140049 1,1,5,55,1005,26601,941863,42372177,2336926665,153927536545,
%T A140049 11869936146891,1055015092106889,106731589524249517,
%U A140049 12163935655214359329,1548324822731892094191,218516875165035215308801
%N A140049 E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
%F A140049 a(n) = A140054(n+1)/(n+1).
%F A140049 E.g.f.: A(x) = exp(G(x)) where G(x) = e.g.f. of A140055.
%F A140049 E.g.f. satisfies: A(x) = exp( x*A(x) * A(x*A(x)) ).
%e A140049 A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
%e A140049 Log(A(x)) = G(x) = e.g.f. of A140055:
%e A140049 Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
%o A140049 (PARI) {a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n+1)}
%o A140049 (PARI) {a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(A,n+1)}
%Y A140049 Cf. A140054, A140055.
%K A140049 nonn,new
%O A140049 0,3
%A A140049 Paul D. Hanna (pauldhanna(AT)juno.com), May 06 2008

%I A139826
%S A139826 1,2,3,5,6,7,10,13,15,21,22,30,33,37,42,57,58,70,78,85,93,102,105,130,
%T A139826 133,165,177,190,210,253,273,330,345,357,385,462,1365
%N A139826 Squarefree idoneal numbers (A000926).
%K A139826 fini,full,nonn,new
%O A139826 1,2
%A A139826 T. D. Noe (noe(AT)sspectra.com), May 06 2008

%I A140074
%S A140074 1,1,1,0,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,1,0,1,1,
%T A140074 0,1,0,0,0,1,1,0,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0,
%U A140074 1,0,1,1,1,1,0,0,1,0,0,1,1,0,1,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,1
%N A140074 Excess over the asymptote of the number of perfect squares between cubes.
%C A140074 There are always at least two squares between positive consecutive cubes, starting with the perfect squares 1 and 4 between the perfect cubes 1 (included) and 8 (excluded).
%C A140074 The number of squares between the cube of n (included) and the cube of n+1 (excluded) is always one of the two integers bracketing 3*sqrt(n)/2.
%C A140074 The number a(n) in the sequence is 0 if the correct count is the lower number or 1 if the actual count is the higher number.
%F A140074 a(n) = floor(sqrt((n+1)^3-1)) - ceiling(sqrt(n^3)) + 1 - floor(1.5 sqrt(n))
%e A140074 The sequence starts with a(0)=1 for n=0 because there is just one perfect square (0) between the cube of 0 (included) and the cube of 1 (excluded).
%e A140074 This exceeds by a(0)=1 the asymptotic expression floor(1.5*sqrt(n)) for the value n=0.
%K A140074 easy,nonn,new
%O A140074 0,1
%A A140074 Gerard P. Michon (g.michon(AT)att.net), May 06 2008

%I A092947
%S A092947 3,5,181,2521,552553,39070081,4180176001,1483048828801,672375473078401,
%T A092947 106890247271808001,40812700642879334401,37716399337002946560001,
%U A092947 18266163370859189769984001,9553876078552184850831360001
%N A092947 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the primes arising as the product of the terms of groups +1.
%C A092947 See A092944 for additional clarification of definition.
%Y A092947 Cf. A092944, A092945, A092946.
%K A092947 nonn,new
%O A092947 1,1
%A A092947 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 23 2004
%E A092947 Edited, corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 07 2008

%I A092946
%S A092946 2,5,19,29,73,113,167,269,431,509,673,977,1193,1423,1861,1993,2467,3041,
%T A092946 3391,4003,4523,5309,6011,6833,7993,9239,10093,10909,12157,13417,15199,
%U A092946 16651,17971,19477,21517,23197,25121,27799,29537,31891,34583,37189
%N A092946 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the primes arising as the sum of the terms of groups.
%C A092946 See A092944 for additional clarification of definition.
%Y A092946 Cf. A092944, A092945, A092947.
%K A092946 nonn,new
%O A092946 1,1
%A A092946 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 23 2004
%E A092946 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 07 2008

%I A092945
%S A092945 2,4,10,9,23,28,29,47,115,71,88,214,215,188,341,133,220,372,250,321,227,
%T A092945 311,281,310,592,857,691,406,470,483,904,903,707,601,876,727,726,1299,
%U A092945 828,925,1217,1254,963,1426,1213,1394,2108,1356,1448,1286,1564,1455
%N A092945 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product + 1 is prime. Sequence contains the last term of each group.
%C A092945 Sequence is underdefined. The original author may have intended to say that the terms must be distinct and minimal; perhaps he should clarify. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 22 2006
%C A092945 See A092944 for additional clarification of definition.
%e A092945 a(6) = 28 because 15+16+17+18+19+28 = 113 is prime and 15*16*17*18*19*28 + 1 = 39070081 is prime.
%Y A092945 Cf. A092944, A092946, A092947.
%K A092945 nonn,new
%O A092945 1,1
%A A092945 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 23 2004
%E A092945 a(6) from Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 22 2006
%E A092945 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 07 2008

%I A092944
%S A092944 2,1,3,5,11,15,20,27,36,44,54,64,76,89,102,117,132,149,166,184,204,228,
%T A092944 249,272,296,323,349,376,403,432,461,493,524,556,589,625,660,697,737,
%U A092944 775,814,855,898,943,987,1031,1076,1122,1169,1219,1269,1321,1373,1427
%N A092944 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the first member of the groups.
%C A092944 The n-th group is selected such that it is lexicographically minimal.
%C A092944 First n-2 numbers are the least previously unused numbers.
%C A092944 The (n-1)st number is chosen with the additional condition that if the product is odd, the sum is also odd (to avoid an impossible situation in picking the n-th number).
%C A092944 The n-th number is chosen as the least unused number that meets the two prime conditions.
%C A092944 In the 3rd group, 6 is selectd as the 2nd number rather than 5, else no 3rd number could be found to meet the prime conditions.
%e A092944 Table begins:
%e A092944 2
%e A092944 1,4
%e A092944 3,6,10
%e A092944 5,7,8,9
%e A092944 11,12,13,14,23
%e A092944 15,16,17,18,19,28
%Y A092944 Cf. A092945, A092946, A092947.
%K A092944 nonn,new
%O A092944 1,1
%A A092944 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 23 2004
%E A092944 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 07 2008

%I A136349
%S A136349 4,6,30,2310,30030,6,30,2310,30030,304250263527210,
%T A136349 23768741896345550770650537601358310,
%U A136349 19361386640700823163471425054312320082662897612571563761906962414215012369856637179096947335243680669607531475629148240284399976570
%N A136349 Numbers n of the form product( prime(i), i=1..k) for some k such that n-1 is prime.
%C A136349 Next term is too large to be included: see A006794. - M. F. Hasler, May 02 2008
%C A136349 This sequence is different from A121069 and A002110.
%F A136349 a(n) = A057705(n) + 1 = A034386( A006794(n) ). - M. F. Hasler, May 02 2008
%e A136349 30 is a member (corresponding to k=3) since2*3*5 - 1 = 30 - 1 = 29 is prime.
%o A136349 (PARI) c=0;t=1;vector(7,n,until( ispseudoprime( -1+t*=prime(c++)),);t)
%Y A136349 Cf. A136350 A136351 A136352 A002110 A121069.
%K A136349 nonn,new
%O A136349 1,1
%A A136349 Enoch Haga (Enokh(AT)comcast.net), Dec 25 2007
%E A136349 Edited by M. F. Hasler (www.univ-ag.fr/~mhasler), May 02 2008

%I A137507
%S A137507 1,3,5,7,9,11,13,15,17,19,300,320,340,360,380,400,420,440,460,480,14000,
%T A137507 14200,14400,14600,14800,15000,15200,15400,15600,15800,1060000,1062000,
%U A137507 1064000,1066000,1068000,1070000
%N A137507 100^[n/10] + 2 n 10^[n/10]: inspired by Engel expansion of Pi.
%o A137507 (PARI) a(n)= (n+5 + 10^(floor((n+5)/10)))^2 - (n+5)^2 for (n=1, 30, print1(" "a(n)" ")) \\ changed n+5 to n. - M.F.Hasler, May 02 2008
%o A137507 (PARI) vector(30,n, n--; 2*n*10^(n\10) + 100^(n\10)) \\ - M.F.Hasler, May 02 2008
%Y A137507 Cf. A006784.
%K A137507 nonn,new
%O A137507 0,1
%A A137507 Alexander R. Povolotsky (pevnev(AT)juno.com), Apr 23 2008
%E A137507 Edited by M. F. Hasler (www.univ-ag.fr/~mhasler) and njas, May 02 2008

%I A136258
%S A136258 1,5,8,6,4,20,32,24,16,80,128,96,64,320,512,384,256,1280,2048,1536,1024,
%T A136258 5120,8192,6144,4096,20480,32768,24576,16384,81920,131072,98304,65536,
%U A136258 327680,524288,393216,262144,1310720,2097152,1572864,1048576
%V A136258 1,5,8,6,-4,-20,-32,-24,16,80,128,96,-64,-320,-512,-384,256,1280,2048,1536,-1024,-5120,
%W A136258 -8192,-6144,4096,20480,32768,24576,-16384,-81920,-131072,-98304,65536,327680,524288,
%X A136258 393216,-262144,-1310720,-2097152,-1572864,1048576
%N A136258 a(n)=2a(n-1)-2a(n-2), with a(0)=1, a(1)=5.
%C A136258 Sequence opposite in sign to its second differences.
%C A136258 Binomial transform of 1, 4, -1, -4.
%C A136258 A bisection gives A135520.
%F A136258 a(4n+1)=5*(-4)^n, a(4n+3)=6*(-4)^n. - M. F. Hasler, May 01 2008
%o A136258 (PARI) vector(100,n,t=if(n<3,[t1=1,5][n],-2*t1+2*t1=t)) \\ - M. F. Hasler, May 01 2008
%K A136258 sign,easy,new
%O A136258 1,2
%A A136258 Paul Curtz (bpcrtz(AT)free.fr), Mar 18 2008
%E A136258 Edited and extended by M. F. Hasler (www.univ-ag.fr/~mhasler), May 01 2008

%I A136410
%S A136410 9,15,16,21,25,27,28,33,36,39,40,45,49,51,52,57,63,64,65,66,69,75,76,81,
%T A136410 85,87,88,91,93,96,99,100,105,111,112,117,120,121,123,124,125,126,129,
%U A136410 133,135,136,141,144,145,147,148
%N A136410 Numbers n having a proper divisor d > 2 such that d-1 divides n-1.
%e A136410 E.g. consider n=91: we can take d=7, 7 divides 91 and 6 divides 90, so 91 is in the sequence.
%K A136410 nonn,new
%O A136410 1,1
%A A136410 J. Perry (johnandruth(AT)jrperry.orangehome.co.uk), Apr 13 2008
%E A136410 Definition, terms and offset corrected by M. F. Hasler (www.univ-ag.fr/~mhasler), May 01 2008
%E A136410 Edited by njas, May 10 2008

%I A129340
%S A129340 1,2,3,6,8,11,22,28,36,47,94,116,144,180,227,454,548,664,808,988,1215,
%T A129340 2430,2884,3432,4096,4904,5892,7107,14214,16644,19528,22960,27056,31960,
%U A129340 37852,44959,89918,104132,120776,140304,163264,190320,222280
%N A129340 Triangular array read by rows: for n, k >= 1, a(n+1, 1) = 2*a(n, n); a(n+1, k+1) = a(n, k)+a(n+1, k).
%C A129340 Main diagonal is A035009. First column is A001861.
%F A129340 a(n, n) = A035009(n). For k < n, a(n, k) = 2*sum_{i = 1..k} binomial(k-1, i-1)*A035009(n-i).
%Y A129340 Cf. A001861, A035009.
%K A129340 nonn,tabl,easy,new
%O A129340 1,2
%A A129340 Paul Curtz (bpcrtz(AT)free.fr), May 28 2007
%E A129340 Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), May 02 2008

%I A137849
%S A137849 1,2,2,4,2,6,2,7,5,7,2,12,2,9,8,14,2,17,2,17,10,10,2,24,9,12,12,23,2,28,
%T A137849 2,25,13,14,12,35,2,15,14,34,2,36,2,30,23,17,2,48,14,33,18,34,2,46,18,
%U A137849 45,19,19,2,60,2,21,30,49,20,54,2,41,22,47,2,71,2,24,36,45,21,63,2,67
%N A137849 Number of integers m from 1 through n inclusive such that d_i(n)<=d_i(m) for 1<=i<=Min(d(n),d(m)) where d_i(n) denotes the ith smallest divisor of n and d(n) denotes the number of divisors of n (A000005).
%C A137849 In other words, number of integers m in {1,...,n} such that the i-th divisor of m is >= the i-th divisor of n, for i=1,...,min(A000005(m),A000005(n)).
%F A137849 a(n) = 2 iff n is prime.
%e A137849 a(10) = 7 because there are 7 integers, 1, 2, 3, 5, 7, 9, and 10, whose divisors meet the criterion for n = 10 (4 does not meet this criterion in that 4's 3rd smallest divisor is 4 and 10's third smallest divisor is 5; similarly 6 and 8 do not meet the criterion).
%o A137849 (PARI) A137849(n)={ local( d=divisors(n), d2 ); sum( m=1,n, d2=divisors(m); vecmin( vector(min(#d,#d2),i,d2[i]-d[i]))>=0 )} \\ - M. F. Hasler, May 01 2008
%Y A137849 Cf. A094783 (numbers where a(n) = n).
%K A137849 nonn,new
%O A137849 1,2
%A A137849 J. Lowell (jhbubby(AT)mindspring.com), Apr 29 2008
%E A137849 Edited and extended by M. F. Hasler (www.univ-ag.fr/~mhasler) and Ray Chandler (rayjchandler(AT)sbcglobal.net), May 01 2008

%I A137506
%S A137506 141,59,265,35,389,11,513,13,637,11,761,13,885,11,1009,13,1133,11,1257,
%T A137506 13,1381,11,1505,13,1629,11,1753,13,1877,11,2001,13,2125,11,2249,13,
%U A137506 2373,11,2497,13,2621,11,2745,13,2869,11,2993,13,3117,11,3241,13,3365
%N A137506 a(2n+1)=141+124n, a(2n+2)=|a(2n)-24| with a(2)=59, thus a(4,6,8,...)=35,11,13,11,13...
%C A137506 The definition of this sequence is inspired by the first 3+2+3+2 decimals of Pi, which have the property that 141+59=200, 265+35=300. The following 3+2 digits don't share such a property, but are followed by digits 238,462 with sum 700... - M. F. Hasler, May 01 2008
%C A137506 Using the given formula for a(n) we could construct d(n)= sum(k=1,n,(81 - (37*(-1)^k)*k + 2*(-1)^k + 25*k)/10^(1/4-1/4*(-1)^k+5/2*k)).
%C A137506 E.g. sum(k=1,6,(81 + (37*(-1)^(k+1) + 25)*k + 2*(-1)^k)/10^(1/4*(1 + (-1)^(k+1) + 10*k))) = 0.141592653538911 and sum(k=1..oo,(81 - (37*(-1)^k)*k + 2*(-1)^k + 25*k)/10^(1/4-1/4*(-1)^k+5/2*k)) = 0.1415926535389115128763663759...
%F A137506 G.f.: (22*x^9 + 26*x^7 - 83*x^5 - 17*x^4 - 24*x^3 + 124*x^2 + 59*x +141) /(x^6 -x^4 - x^2 + 1)
%F A137506 a(n) = a(n-2) + a(n-4) - a(n-6) a(n) = 81 - 37*(-1)^n*n + 2*(-1)^n + 25*n
%o A137506 (PARI) a(n) = {local(a = vector(n)); a[1]=141; a[2]=59; for(m=3,n, if((Mod(m,2))==0, a[m]=abs(a[m-3]+a[m-2]+100-a[m-1])); if((Mod(m,2))!=0, a[m]=a[m-2]+124; ); ); a; }
%o A137506 (PARI) A137506(n)=if( n%2, 141+n\2*124, if( n<6, [59,35][n\2], [11,13][1+!(n%4)])) \\ - M. F. Hasler, May 01 2008
%K A137506 nonn,base,new
%O A137506 1,1
%A A137506 Alexander R. Povolotsky (pevnev(AT)juno.com), Apr 23 2008
%E A137506 Edited & extended by M. F. Hasler (www.univ-ag.fr/~mhasler), May 01 2008

%I A137510
%S A137510 0,0,0,2,0,2,3,0,2,4,3,2,5,0,2,3,4,6,0,2,7,3,5,2,4,8,0,2,3,6,9
%N A137510 Triangle read by rows in which row n lists the divisors of n in the range 1 < d < n; or 0 if there are no such divisors.
%C A137510 a(n) = 0 if n is 1 or a prime.
%Y A137510 Cf. A070824, A027750, A027751.
%K A137510 nonn,tabf,easy,more,new
%O A137510 1,4
%A A137510 njas, May 08 2008

%I A137438
%S A137438 1,0,3,3,3,7,9,12,14,22,30,39,41,57,86,87,121,179,164,225,300,362,433,
%T A137438 571,624,846,968,1134,1391,1902,1992,2407,3043,3688,4321
%N A137438 Number of conjugate-congruent partitions of n.
%C A137438 See reference for precise definition.
%D A137438 A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
%K A137438 nonn,more,new
%O A137438 1,3
%A A137438 njas (njas(AT)research.att.com), May 07 2008

%I A132181
%S A132181 1,2,6,1,28,1,1,58,1,708,1,1,2,1,2836,1,1,22696,1,1,1,590122,1,12,1,1,2,
%T A132181 1,1180246,1,9441976,1,1,1,169955586,1,2,1,2,1,2719289392,1,1,1,1,
%U A132181 5438578786,1,32631472722,1,2,1,391577672676,1,1,2,1,1566310690708,1,1
%N A132181 a(n)=smallest positive integer such that product{k=1 to n}(1+1/a(k)) has a prime numerator.
%H A132181 Owen Whitby, <a href="http://www.research.att.com/~njas/sequences/b132181.txt">Table of n, a(n) for n = 1..200</a>
%F A132181 Comments from Owen Whitby (whitbyo(AT)acm.org), May 07 2008 (Start): Successive terms a(.) can be calculated using the following recurrences for the numerator n(.) and denominator d(.) of the product.
%F A132181 a(1)=1; n(1)=1, d(1)=1 ==> a(2)=1, n(2)=2, d(2)=1 ( to start things off );
%F A132181 n(i)=2, d(i)= odd ==> a(i+1)=q-1, n(i+1)=q, d(i+1)=d(i)(q-1)/2 where q is least odd prime not dividing d(i);
%F A132181 n(i)=odd prime, d(i)=1 ==> a(i+1)=c*n(i), n(i+1)=c*n(i)+1, d(i+1)=c where c is least even integer such that c*n(i)+1 is prime;
%F A132181 n(i)=odd prime, d(i)=even ==> a(i+1)=1, n(i+1)=n(i), d(i+1)=d(i)/2;
%F A132181 n(i)=odd prime, d(i)= odd>=3 ==> a(i+1)=p-1, n(i+1)=n(i), d(i+1)=d(i)(p-1)/p where p is least prime divisor of d(i). (End)
%K A132181 nonn,new
%O A132181 1,2
%A A132181 Leroy Quet (qq-quet(AT)mindspring.com), Nov 04 2007
%E A132181 a(10) to a(59) and list of 200 terms added by Owen Whitby (whitbyo(AT)acm.org), May 07 2008

%I A139827
%S A139827 2,17,29,41,101,149,173,197,233,281,293,461,557,569,593,677,701,761,809,
%T A139827 821,857,941,953,1097,1217,1229,1289,1361,1481,1493,1553,1601,1613,1733,
%U A139827 1877,1889,1913,1949,1997,2081,2129,2141,2153,2213,2273
%N A139827 Primes of the form 2x^2+2xy+17y^2.
%C A139827 Discriminant=-132. Consider the quadratic form f(x,y)=ax^2+bxy+cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in OEIS.
%C A139827 When a=1 and b=0, f(x,y) is a principle quadratic form, whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such the the Jacobi symbol (-k/4N)=1.
%D A139827 David A. Cox, Primes of Form x^2 + n y^2, Wiley, 1989.
%F A139827 The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).
%Y A139827 Cf. A002313 (d=-4),
%Y A139827 A033200 (d=-8),
%Y A139827 A007645 (d=-12),
%Y A139827 A002144 (d=-16),
%Y A139827 A033205, A106865 (d=-20),
%Y A139827 A033199, A084865 (d=-24),
%Y A139827 A033207 (d=-28),
%Y A139827 A007519, A007520 (d=-32),
%Y A139827 A068228, A040117 (d=-36),
%Y A139827 A033201, A106889 (d=-40),
%Y A139827 A068228, A068229 (d=-48),
%Y A139827 A033210, A106906 (d=-52),
%Y A139827 A033212, A106859 (d=-60),
%Y A139827 A007519, A007521 (d=-64),
%Y A139827 A106950, A106949 (d=-72),
%Y A139827 A033215, A102271, A102273, A106972 (d=-84),
%Y A139827 A033216, A106984 (d=-88),
%Y A139827 A107008, A107003, A107006, A107007 (d=-96),
%Y A139827 A033205, A122487 (d=-100),
%Y A139827 A107134, A107133 (d=-112),
%Y A139827 A033220, A107135, A107136, A107137 (d=-120),
%Y A139827 A033222, A107138, A139827, A139828 (d=-132),
%Y A139827 A033225, A007639 (d=-148),
%Y A139827 A107145, A107144, A139829, A139830 (d=-160),
%Y A139827 A033229, A107146, A107147, A107148 (d=-168),
%Y A139827 A107152, A107151, A139831, A139832 (d=-180),
%Y A139827 A107008, A107006, A107154, A139530 (d=-192),
%Y A139827 A033236, A107165, A139833, A139834 (d=-228),
%Y A139827 A033237, A107166 (d=-232),
%Y A139827 A107152, A107167, A107168, A107169 (d=-240),
%Y A139827 A033245, A107178, A107179, A107180 (d=-280),
%Y A139827 A107008, A107007, A107154, A107181 (d=-288),
%Y A139827 A033250, A107188, A107189, A107190 (d=-312),
%Y A139827 A033254, A107199, A139835, A139836 (d=-340),
%Y A139827 A107202, A107201, A139837, A139838 (d=-352),
%Y A139827 A033202, A107210, A139839, A139840 (d=-372),
%Y A139827 A139643, A139841-A139843 (d=-408),
%Y A139827 A139644, A139844-A139850 (d=-420),
%Y A139827 A139645, A139851-A139853 (d=-448),
%Y A139827 A139502, A139854-A139860 (d=-480),
%Y A139827 A139646, A139861-A139863 (d=-520),
%Y A139827 A139647, A139864-A139866 (d=-532),
%Y A139827 A139648, A139867-A139873 (d=-660),
%Y A139827 A139506, A139874-A139880 (d=-672),
%Y A139827 A139649, A139881-A139883 (d=-708),
%Y A139827 A139650, A139884-A139886 (d=-760),
%Y A139827 A139651, A139887-A139893 (d=-840),
%Y A139827 A139652, A139894-A139896 (d=-928),
%Y A139827 A139502, A139897-A139903 (d=-960),
%Y A139827 A139653, A139904-A139906 (d=-1012),
%Y A139827 A139654, A139907-A139913 (d=-1092),
%Y A139827 A139655, A139914-A139920 (d=-1120),
%Y A139827 A139656, A139921-A139927 (d=-1248),
%Y A139827 A139657, A139928-A139934 (d=-1320),
%Y A139827 A139658, A139935-A139941 (d=-1380),
%Y A139827 A139659, A139942-A139948 (d=-1428),
%Y A139827 A139660, A139949-A139955 (d=-1540),
%Y A139827 A139661, A139956-A139962 (d=-1632),
%Y A139827 A139662, A139963-A139969 (d=-1848),
%Y A139827 A139663, A139970-A139976 (d=-2080),
%Y A139827 A139664, A139977-A139983 (d=-3040),
%Y A139827 A139665, A139984-A139998 (d=-3360),
%Y A139827 A139666, A139999-A140013 (d=-5280),
%Y A139827 A139667, A140014-A140028 (d=-5460),
%Y A139827 A139668, A140029-A140043 (d=-7392).
%K A139827 nonn,easy,new
%O A139827 1,1
%A A139827 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140043
%S A140043 47,311,383,647,719,839,983,1511,2399,2663,2687,3023,3191,3359,4007,
%T A140043 4079,4679,4871,5039,5087,5351,5591,5879,5927,6263,6359,6719,7703,7727,
%U A140043 8039,8111,8231,8783,9551,9623,9791,9887,10079,10223,10631,11399
%N A140043 Primes of the form 47x^2+38xy+47y^2.
%C A140043 Discriminant=-7392. See A139827 for more information.
%F A140043 The primes are congruent to {47, 311, 335, 383, 551, 647, 719, 815, 839, 983, 1175, 1343, 1391, 1511, 1655} (mod 1848).
%K A140043 nonn,easy,new
%O A140043 1,1
%A A140043 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140042
%S A140042 53,317,389,653,1061,1373,1709,1901,2069,2237,2333,2381,2909,3221,3389,
%T A140042 3413,3557,3677,3917,4013,4229,4349,4733,5021,5237,5261,5861,6029,6197,
%U A140042 6581,6869,6917,7109,7253,7877,8429,8933,9221,9293,9461
%N A140042 Primes of the form 44x^2+44xy+53y^2.
%C A140042 Discriminant=-7392. See A139827 for more information.
%F A140042 The primes are congruent to {53, 221, 317, 389, 485, 533, 653, 1037, 1061, 1325, 1373, 1541, 1565, 1709, 1829} (mod 1848).
%K A140042 nonn,easy,new
%O A140042 1,1
%A A140042 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140041
%S A140041 43,211,547,571,739,1051,1579,1723,2251,2683,2851,3187,3571,3691,3739,
%T A140041 3907,4099,4243,4363,5419,6091,6211,6379,6547,6619,6883,7603,7963,8059,
%U A140041 8443,8467,8731,8971,9283,9643,9787,9811,9907,10243,11083,11131
%N A140041 Primes of the form 43x^2+2xy+43y^2.
%C A140041 Discriminant=-7392. See A139827 for more information.
%F A140041 The primes are congruent to {43, 211, 403, 547, 571, 667, 739, 835, 1003, 1051, 1075, 1339, 1579, 1723, 1843} (mod 1848).
%K A140041 nonn,easy,new
%O A140041 1,1
%A A140041 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140040
%S A140040 89,257,353,521,881,929,1049,1193,1433,1697,1721,2729,2777,2897,3041,
%T A140040 3617,3881,4049,4073,4217,4409,4889,5393,5417,5801,5897,6257,6473,6737,
%U A140040 6977,7481,7577,7649,8273,9161,9497,10169,10289,10433,10937
%N A140040 Primes of the form 33x^2+56y^2.
%C A140040 Discriminant=-7392. See A139827 for more information.
%F A140040 The primes are congruent to {89, 185, 257, 353, 377, 521, 713, 881, 929, 1049, 1193, 1433, 1697, 1721, 1769} (mod 1848).
%K A140040 nonn,easy,new
%O A140040 1,1
%A A140040 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140039
%S A140039 73,241,409,601,769,937,1249,1657,1777,1993,2089,2593,2617,2833,3121,
%T A140039 3673,3769,4177,4297,4441,4969,5521,5689,5953,6481,6529,6793,7321,7369,
%U A140039 7537,7873,7993,8161,8329,8377,8641,9049,9649,9721,10009,10177
%N A140039 Primes of the form 28x^2+28xy+73y^2.
%C A140039 Discriminant=-7392. See A139827 for more information.
%F A140039 The primes are congruent to {73, 145, 241, 409, 481, 601, 745, 769, 937, 985, 1249, 1273, 1657, 1777, 1825} (mod 1848).
%K A140039 nonn,easy,new
%O A140039 1,1
%A A140039 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140038
%S A140038 83,131,227,563,1091,1427,1811,1931,1979,2243,2411,2939,3251,3659,3779,
%T A140038 3923,4091,4259,4451,4787,5099,5507,5843,5939,6299,6947,6971,7523,7691,
%U A140038 8147,8291,8819,9203,9323,9371,9467,9539,9803,10139,10163
%N A140038 Primes of the form 24x^2+24xy+83y^2.
%C A140038 Discriminant=-7392. See A139827 for more information.
%F A140038 The primes are congruent to {83, 131, 227, 299, 395, 563, 635, 755, 899, 923, 1091, 1139, 1403, 1427, 1811} (mod 1848).
%K A140038 nonn,easy,new
%O A140038 1,1
%A A140038 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140037
%S A140037 101,173,293,461,677,941,1613,1949,2141,2309,2477,2789,3461,3533,3797,
%T A140037 3989,4133,4157,4373,4637,5309,5381,5477,5717,5981,6173,6221,7013,7229,
%U A140037 7829,7853,8069,8741,8861,9173,9341,9413,9533,9677,10181,10589
%N A140037 Primes of the form 24x^2+77y^2.
%C A140037 Discriminant=-7392. See A139827 for more information.
%F A140037 The primes are congruent to {101, 173, 293, 437, 461, 629, 677, 941, 965, 1349, 1469, 1517, 1613, 1685, 1781} (mod 1848).
%K A140037 nonn,easy,new
%O A140037 1,1
%A A140037 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140036
%S A140036 109,277,373,541,613,877,1117,1381,1429,1597,1789,1933,2053,2221,2389,
%T A140036 2437,3109,3229,3637,4813,4957,5077,5653,5749,5821,6133,6421,6661,7333,
%U A140036 7477,7669,7933,8269,8821,9181,9349,9613,9781,9829,10357,10501
%N A140036 Primes of the form 21x^2+88y^2.
%C A140036 Discriminant=-7392. See A139827 for more information.
%F A140036 The primes are congruent to {85, 109, 205, 277, 373, 541, 589, 613, 877, 1117, 1261, 1381, 1429, 1597, 1789} (mod 1848).
%K A140036 nonn,easy,new
%O A140036 1,1
%A A140036 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140035
%S A140035 157,181,229,397,661,829,1021,1237,1741,2029,2341,2677,3037,3853,3877,
%T A140035 4093,4261,4357,4933,5197,5437,5701,6037,6229,6373,6733,6781,7549,7573,
%U A140035 7621,7789,8053,8221,8581,8629,8893,9133,9397,9421,9733,9901
%N A140035 Primes of the form 12x^2+12xy+157y^2.
%C A140035 Discriminant=-7392. See A139827 for more information.
%F A140035 The primes are congruent to {157, 181, 229, 397, 493, 565, 661, 685, 829, 1021, 1189, 1237, 1357, 1501, 1741} (mod 1848).
%K A140035 nonn,easy,new
%O A140035 1,1
%A A140035 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140034
%S A140034 11,179,443,683,947,1499,1523,1787,2003,2027,2531,2699,2843,2963,3347,
%T A140034 3371,3851,4019,4139,4211,4547,4643,4691,5483,5531,5867,5987,6323,6491,
%U A140034 6659,7043,7331,7547,7907,8171,8243,8387,9227,9419,9923,10091
%N A140034 Primes of the form 11x^2+168y^2.
%C A140034 Discriminant=-7392. See A139827 for more information.
%F A140034 Except for 11,the primes are congruent to {155, 179, 323, 443, 515, 683, 779, 851, 947, 995, 1115, 1499, 1523, 1787, 1835} (mod 1848).
%K A140034 nonn,easy,new
%O A140034 1,1
%A A140034 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140033
%S A140033 233,281,569,809,953,1289,1481,1913,2081,2129,2153,2417,2657,2801,2969,
%T A140033 3137,3329,3593,3761,3929,4001,4649,4817,5441,5849,6113,6353,6833,7193,
%U A140033 7457,7673,8513,8681,9041,9137,9209,9473,9521,10193,10313
%N A140033 Primes of the form 8x^2+8xy+233y^2.
%C A140033 Discriminant=-7392. See A139827 for more information.
%F A140033 The primes are congruent to {65, 233, 281, 305, 569, 809, 953, 1073, 1121, 1289, 1481, 1625, 1649, 1745, 1817} (mod 1848).
%K A140033 nonn,easy,new
%O A140033 1,1
%A A140033 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140032
%S A140032 239,263,359,431,743,1031,1583,2087,2111,2207,2543,2591,2879,3119,4127,
%T A140032 4391,4463,4967,5231,5279,5639,5783,5807,5903,6287,6311,6959,7079,7127,
%U A140032 7487,7823,7919,8087,8423,8663,8807,9479,9767,10007,10271
%N A140032 Primes of the form 8x^2+231y^2.
%C A140032 Discriminant=-7392. See A139827 for more information.
%F A140032 The primes are congruent to {95, 239, 263, 359, 431, 527, 695, 743, 767, 1031, 1271, 1415, 1535, 1583, 1751} (mod 1848).
%K A140032 nonn,easy,new
%O A140032 1,1
%A A140032 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140031
%S A140031 7,271,439,607,1063,1231,1399,1447,2239,2287,2383,2551,2719,2791,3079,
%T A140031 3559,3583,3967,4231,4567,4639,4759,4903,5407,5431,6079,6151,6247,6607,
%U A140031 6991,7927,8263,8599,8623,8839,9103,9127,9511,9631,9679,10111
%N A140031 Primes of the form 7x^2+264y^2.
%C A140031 Discriminant=-7392. See A139827 for more information.
%F A140031 Except for 7, the primes are congruent to {271, 391, 439, 535, 607, 703, 871, 943, 1063, 1207, 1231, 1399, 1447, 1711, 1735} (mod 1848).
%K A140031 nonn,easy,new
%O A140031 1,1
%A A140031 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140030
%S A140030 463,487,631,751,823,991,1087,1303,1423,1831,2143,2311,2671,3271,3943,
%T A140030 4159,4327,4447,4519,4783,4951,4999,5119,5503,5527,5791,5839,6007,6367,
%U A140030 6703,6967,7351,7639,7687,7879,8647,9199,9871,10399,10663,11047
%N A140030 Primes of the form 4x^2+4xy+463y^2.
%C A140030 Discriminant=-7392. See A139827 for more information.
%F A140030 The primes are congruent to {247, 295, 463, 487, 631, 751, 823, 991, 1087, 1159, 1255, 1303, 1423, 1807, 1831} (mod 1848).
%K A140030 nonn,easy,new
%O A140030 1,1
%A A140030 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140029
%S A140029 3,619,643,691,859,1123,1291,1483,1699,2203,2467,2539,2707,2803,2971,
%T A140029 3331,3499,3547,4051,4339,4651,4723,4987,5179,5347,5659,6163,6571,6691,
%U A140029 7027,7243,7507,8011,8419,8539,9043,9091,9859,9883,9931,10099
%N A140029 Primes of the form 3x^2+616y^2.
%C A140029 Discriminant=-7392. See A139827 for more information.
%F A140029 Except for 3, the primes are congruent to {115, 355, 619, 643, 691, 859, 955, 1027, 1123, 1147, 1291, 1483, 1651, 1699, 1819} (mod 1848).
%K A140029 nonn,easy,new
%O A140029 1,1
%A A140029 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140028
%S A140028 43,127,547,823,883,907,1303,1327,1663,2083,3067,3823,3847,3943,4027,
%T A140028 4447,4603,4663,4783,5443,5503,6007,6343,6367,6763,6967,7687,7723,8467,
%U A140028 8527,8563,8863,9283,9403,9643,9907,10243,10663,10903,11047
%N A140028 Primes of the form 42x^2+42xy+43y^2.
%C A140028 Discriminant=-5460. See A139827 for more information.
%F A140028 The primes are congruent to {43, 127, 547, 667, 823, 883, 907, 1303, 1327, 1387, 1507, 1663, 1843, 1927, 2083, 2167, 2227, 2263, 3007, 3067, 3103, 3403, 3487, 3823, 3847, 3943, 4027, 4183, 4267, 4447, 4603, 4663, 4783, 5203, 5287, 5443} (mod 5460).
%K A140028 nonn,easy,new
%O A140028 1,1
%A A140028 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140027
%S A140027 37,193,457,613,877,1033,1597,2017,2137,2293,2377,2437,2797,3217,3313,
%T A140027 3697,3733,4153,5077,5233,5413,5653,6073,6337,6637,7057,7417,7477,7753,
%U A140027 8353,8677,9157,9277,9433,9613,10333,10753,10837,10957,11113
%N A140027 Primes of the form 37x^2+4xy+37y^2.
%C A140027 Discriminant=-5460. See A139827 for more information.
%F A140027 The primes are congruent to {37, 193, 253, 457, 613, 697, 877, 1033, 1177, 1513, 1597, 1633, 1957, 2017, 2137, 2293, 2377, 2437, 2797, 2893, 3193, 3217, 3313, 3697, 3733, 3817, 3973, 4153, 4453, 4477, 4873, 5077, 5233, 5293, 5377, 5413} (mod 5460).
%K A140027 nonn,easy,new
%O A140027 1,1
%A A140027 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140026
%S A140026 179,191,491,599,659,911,1031,1439,1499,1871,2339,2531,2591,3119,3299,
%T A140026 3371,3539,3719,3851,4211,4391,5279,5399,5639,5651,6491,6659,6899,6959,
%U A140026 7151,7211,7331,8219,8831,8999,9311,9851,10091,10271,10739,10859
%N A140026 Primes of the form 35x^2+39y^2.
%C A140026 Discriminant=-5460. See A139827 for more information.
%F A140026 The primes are congruent to {179, 191, 491, 599, 659, 779, 911, 1031, 1199, 1271, 1439, 1499, 1691, 1751, 1871, 2279, 2291, 2339, 2531, 2591, 2759, 3119, 3299, 3371, 3431, 3539, 3719, 3851, 4211, 4391, 4559, 4631, 4811, 4859, 5279, 5399} (mod 5460).
%K A140026 nonn,easy,new
%O A140026 1,1
%A A140026 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140025
%S A140025 53,113,233,653,953,1733,2213,2237,2297,2417,2753,2837,3137,3917,4013,
%T A140025 4397,4733,4937,5573,5693,6113,6197,6353,6917,7193,7253,7673,7757,7877,
%U A140025 8297,8537,8597,9377,9437,9473,9857,10193,10313,10973,11657
%N A140025 Primes of the form 30x^2+30xy+53y^2.
%C A140025 Discriminant=-5460. See A139827 for more information.
%F A140025 The primes are congruent to {53, 113, 233, 653, 737, 893, 953, 1037, 1457, 1577, 1733, 1793, 1817, 2213, 2237, 2297, 2417, 2573, 2753, 2837, 2993, 3077, 3137, 3173, 3917, 3977, 4013, 4313, 4397, 4733, 4757, 4853, 4937, 5093, 5177, 5357} (mod 5460).
%K A140025 nonn,easy,new
%O A140025 1,1
%A A140025 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140024
%S A140024 59,479,839,1151,1259,1319,1571,1931,2099,2351,2399,2411,2579,2819,3191,
%T A140024 3359,3491,3659,3671,3911,4259,4451,4751,4919,5039,5351,5519,5939,6011,
%U A140024 6131,6299,6359,6599,6719,6779,7451,7559,7691,8039,8231
%N A140024 Primes of the form 26x^2+26xy+59y^2.
%C A140024 Discriminant=-5460. See A139827 for more information.
%F A140024 The primes are congruent to {59, 479, 551, 671, 839, 899, 1139, 1151, 1259, 1319, 1571, 1679, 1931, 1991, 2099, 2231, 2351, 2399, 2411, 2579, 2771, 2819, 3191, 3239, 3359, 3491, 3659, 3671, 3911, 4259, 4331, 4451, 4751, 4919, 5039, 5351} (mod 5460).
%K A140024 nonn,easy,new
%O A140024 1,1
%A A140024 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140023
%S A140023 149,281,401,449,821,1061,1229,1289,1409,1709,2069,2381,2741,2801,2969,
%T A140023 3089,3581,4349,4649,5189,5441,5741,5861,6329,6521,6689,6701,6869,7001,
%U A140023 7109,7529,7841,7949,8429,9041,9209,9521,10301,10889,11069
%N A140023 Primes of the form 21x^2+65y^2.
%C A140023 Discriminant=-5460. See A139827 for more information.
%F A140023 The primes are congruent to {149, 281, 401, 449, 821, 869, 1061, 1229, 1241, 1289, 1409, 1541, 1649, 1709, 1961, 2069, 2321, 2381, 2489, 2501, 2741, 2801, 2969, 3089, 3161, 3581, 3749, 4061, 4181, 4349, 4649, 4769, 4841, 5189, 5429, 5441} (mod 5460).
%K A140023 nonn,easy,new
%O A140023 1,1
%A A140023 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140022
%S A140022 151,331,379,499,631,739,1051,1471,1579,1831,2179,2251,2671,3271,3739,
%T A140022 3931,4519,4831,4951,4999,5419,5779,5791,5839,6091,6199,6619,6871,7039,
%U A140022 7351,7639,8731,9199,9319,9391,9739,10159,10459,10831,11071
%N A140022 Primes of the form 15x^2+91y^2.
%C A140022 Discriminant=-5460. See A139827 for more information.
%F A140022 The primes are congruent to {151, 319, 331, 379, 499, 631, 739, 799, 1051, 1159, 1411, 1471, 1579, 1591, 1831, 1891, 2059, 2179, 2251, 2671, 2839, 3151, 3271, 3439, 3739, 3859, 3931, 4279, 4519, 4531, 4699, 4831, 4951, 4999, 5371, 5419} (mod 5460).
%K A140022 nonn,easy,new
%O A140022 1,1
%A A140022 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140021
%S A140021 101,269,521,881,1049,1109,1301,1361,1889,1949,2141,2441,2609,2729,2861,
%T A140021 3041,3449,3461,3701,3821,4241,4289,4889,5381,5669,5801,5981,6569,6761,
%U A140021 7229,7349,7829,7901,8069,8501,8609,9161,9281,9749,9929
%N A140021 Primes of the form 14x^2+14xy+101y^2.
%C A140021 Discriminant=-5460. See A139827 for more information.
%F A140021 The primes are congruent to {101, 209, 269, 341, 521, 881, 1049, 1109, 1301, 1349, 1361, 1769, 1889, 1949, 2141, 2201, 2369, 2441, 2609, 2729, 2861, 2981, 3041, 3149, 3449, 3461, 3701, 3821, 4241, 4289, 4469, 4541, 4709, 4889, 5249, 5381} (mod 5460).
%K A140021 nonn,easy,new
%O A140021 1,1
%A A140021 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140020
%S A140020 13,157,313,433,937,997,1153,1693,1777,1993,2617,2677,2833,3253,3433,
%T A140020 3457,3793,3877,4177,4273,5113,5197,5437,5953,6373,6397,6733,7237,7297,
%U A140020 7537,8293,8713,8893,9337,9733,10477,10513,10657,11353,11437
%N A140020 Primes of the form 13x^2+105y^2.
%C A140020 Discriminant=-5460. See A139827 for more information.
%F A140020 The primes are congruent to {13, 157, 313, 433, 493, 517, 913, 937, 997, 1153, 1273, 1693, 1777, 1837, 1993, 2077, 2497, 2617, 2677, 2833, 3013, 3097, 3253, 3337, 3433, 3457, 3793, 3877, 4177, 4213, 4273, 5017, 5053, 5113, 5197, 5353, 5437} (mod 5460).
%K A140020 nonn,easy,new
%O A140020 1,1
%A A140020 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140019
%S A140019 139,199,439,859,1039,1231,1291,1459,1531,1699,1951,2131,2239,2539,2551,
%T A140019 2791,3331,3559,3631,4339,4651,4759,5431,5659,5851,6691,6991,7159,7411,
%U A140019 7591,7699,8011,8839,9091,9439,9931,10111,10531,10891,11059
%N A140019 Primes of the form 10x^2+10xy+139y^2.
%C A140019 Discriminant=-5460. See A139827 for more information.
%F A140019 The primes are congruent to {139, 199, 391, 439, 451, 859, 979, 1039, 1231, 1291, 1459, 1531, 1699, 1819, 1951, 2071, 2131, 2239, 2539, 2551, 2791, 2911, 3331, 3379, 3559, 3631, 3799, 3979, 4339, 4471, 4651, 4759, 4819, 4891, 5071, 5431} (mod 5460).
%K A140019 nonn,easy,new
%O A140019 1,1
%A A140019 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140018
%S A140018 7,223,307,643,787,1123,1567,1627,1783,1867,1987,2203,2803,3307,3463,
%T A140018 3547,3583,3967,4483,4903,4987,5323,5647,5683,6247,6823,7027,7243,7507,
%U A140018 7867,8263,8287,8923,9007,9043,9547,10303,10723,11863,12043
%N A140018 Primes of the form 7x^2+195y^2.
%C A140018 Discriminant=-5460. See A139827 for more information.
%F A140018 The primes are congruent to {7, 187, 223, 307, 643, 787, 943, 1123, 1207, 1363, 1567, 1627, 1783, 1867, 1903, 1987, 2047, 2203, 2407, 2803, 2827, 3127, 3307, 3463, 3547, 3583, 3967, 4063, 4087, 4387, 4483, 4843, 4903, 4987, 5143, 5263, 5323} (mod 5460).
%K A140018 nonn,easy,new
%O A140018 1,1
%A A140018 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140017
%S A140017 229,241,349,409,661,769,1021,1321,1489,1669,1861,2281,2749,3001,3541,
%T A140017 4129,4441,5101,5449,5689,5701,5869,6121,6229,6481,6781,6949,6961,7129,
%U A140017 7321,7369,7741,7789,8209,8221,8461,9001,9901,10069,10909,11149
%N A140017 Primes of the form 6x^2+6xy+229y^2.
%C A140017 Discriminant=-5460. See A139827 for more information.
%F A140017 The primes are congruent to {229, 241, 349, 409, 661, 769, 1021, 1081, 1189, 1321, 1441, 1489, 1501, 1669, 1861, 1909, 2281, 2329, 2449, 2581, 2749, 2761, 3001, 3349, 3421, 3541, 3841, 4009, 4129, 4441, 4609, 5029, 5101, 5221, 5389, 5449} (mod 5460).
%K A140017 nonn,easy,new
%O A140017 1,1
%A A140017 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140016
%S A140016 5,293,353,593,773,1097,1217,1553,1697,2273,2477,2693,2777,2897,2957,
%T A140016 4217,4373,4457,4493,4877,4973,5297,5393,5813,5897,6053,6173,7013,7577,
%U A140016 7937,8237,8273,8573,9173,9497,9677,9833,10337,10433,10457,10853
%N A140016 Primes of the form 5x^2+273y^2.
%C A140016 Discriminant=-5460. See A139827 for more information.
%F A140016 The primes are congruent to {5, 293, 353, 437, 593, 713, 773, 1097, 1133, 1217, 1553, 1697, 1853, 2033, 2117, 2273, 2477, 2537, 2693, 2777, 2813, 2897, 2957, 3113, 3317, 3713, 3737, 4037, 4217, 4373, 4457, 4493, 4877, 4973, 4997, 5297, 5393} (mod 5460).
%K A140016 nonn,easy,new
%O A140016 1,1
%A A140016 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140015
%S A140015 3,467,503,563,647,887,1223,1427,1823,1847,1907,2063,2687,2903,3407,
%T A140015 3527,3923,4007,4703,4787,5087,5927,6263,6803,6863,7283,7307,7523,7643,
%U A140015 8147,8363,8447,8867,9203,9467,9623,9803,10163,10247,11423,11483
%N A140015 Primes of the form 3x^2+455y^2.
%C A140015 Discriminant=-5460. See A139827 for more information.
%F A140015 The primes are congruent to {3, 467, 503, 563, 647, 803, 887, 1067, 1223, 1343, 1403, 1427, 1823, 1847, 1907, 2063, 2183, 2603, 2687, 2747, 2903, 2987, 3407, 3527, 3587, 3743, 3923, 4007, 4163, 4247, 4343, 4367, 4703, 4787, 5087, 5123, 5183} (mod 5460).
%K A140015 nonn,easy,new
%O A140015 1,1
%A A140015 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140014
%S A140014 2,683,743,827,863,947,1103,1163,1367,1523,1607,1787,2087,2423,2543,
%T A140014 2927,3203,3347,3803,4127,4643,5387,5783,5987,6143,6203,6287,6323,6563,
%U A140014 6827,6983,7247,7547,7883,8387,8663,8747,8807,9587,10067,10103
%N A140014 Primes of the form 2x^2+2xy+683y^2.
%C A140014 Discriminant=-5460. See A139827 for more information.
%F A140014 The primes are congruent to {2, 323, 527, 683, 743, 827, 863, 947, 1103, 1163, 1367, 1523, 1607, 1787, 1943, 2087, 2423, 2507, 2543, 2867, 2927, 3047, 3203, 3287, 3347, 3707, 3803, 4103, 4127, 4223, 4607, 4643, 4727, 4883, 5063, 5363, 5387} (mod 5460).
%K A140014 nonn,easy,new
%O A140014 1,1
%A A140014 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140013
%S A140013 41,281,569,761,809,1289,1361,1481,1601,1889,2081,2129,2441,2609,2801,
%T A140013 2969,3209,3329,3449,3761,3929,4001,4241,4289,4649,4721,5081,5441,5849,
%U A140013 6089,6569,6761,8081,8609,8681,9041,9209,9281,9521,9929,10529
%N A140013 Primes of the form 41x^2+38xy+41y^2.
%C A140013 Discriminant=-5280. See A139827 for more information.
%F A140013 The primes are congruent to {41, 161, 281, 329, 569, 689, 761, 809, 1121, 1289} (mod 1320).
%K A140013 nonn,easy,new
%O A140013 1,1
%A A140013 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140012
%S A140012 43,283,307,523,547,787,1627,1723,1867,1987,2683,3163,3187,3307,3643,
%T A140012 3907,4003,4243,4363,4483,4507,5227,5323,5563,5683,5827,6067,6547,6883,
%U A140012 6907,7603,7867,7963,8443,8467,8707,8923,9187,9283,9547,9643
%N A140012 Primes of the form 40x^2+40xy+43y^2.
%C A140012 Discriminant=-5280. See A139827 for more information.
%F A140012 The primes are congruent to {43, 283, 307, 403, 523, 547, 667, 787, 1003, 1267} (mod 1320).
%K A140012 nonn,easy,new
%O A140012 1,1
%A A140012 T. D. Noe (noe(AT)sspectra.com), May 02 2008

%I A140011
%S A140011 37,157,397,757,1093,1213,1237,1453,2293,2557,2677,2797,3037,3613,3733,
%T A140011 3853,3877,4093,4357,4933,5197,5413,5437,6037,6373,6637,6733,6997,7573,
%U A140011 8053,8317,8677,8893,9013,9133,9157,9277,9397,9733,10333
%N A140011 Primes of the form 37x^2+14xy+37y^2.
%C A140011 Discriminant=-5280. See A139827 for more information.
%F A140011 The primes are congruent to {37, 133, 157, 397, 493, 757, 973, 1093, 1213, 1237} (mod 1320).
%K A140011 nonn,easy,new
%O A140011 1,1
%A A140