The Database of Integer Sequences, Part 13 

     Part of the On-Line Encyclopedia of Integer Sequences


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

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

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    home page:          www.research.att.com/~njas/
    


(start)



%I A092670
%S A092670 1,1,1,1,1,2,2,2,2,2,2,3,3,3,6,6,6,11,11,22,22,22,22,41,41,41,41,114,
%T A092670 114,200,200,200,363,363,566,852,852,852,852,1655,1655,3054,3054,3054,
%U A092670 5777,5777,5777,10647,10647,10647,10647,19436,19436,33373,48360,90441
%N A092670 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0<x_1<...<x_k<=n.
%H A092670 Toshitaka Suzuki, <a href="http://www.research.att.com/~njas/sequences/b092670.txt">Table of n, a(n) for n = 1..610</a>
%H A092670 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ed.html#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A092670 a(n)=Sum(A092669(i), i=1..n)
%e A092670 a(6)=2 since there are two fractions 1=1/1 and 1=1/2+1/3+1/6.
%Y A092670 Cf. A020473, A038034, A092669, A006585.
%Y A092670 Adjacent sequences: A092667 A092668 A092669 this_sequence A092671 A092672 A092673
%Y A092670 Sequence in context: A090618 A072748 A030603 this_sequence A120450 A127238 A072114
%K A092670 nonn
%O A092670 1,6
%A A092670 Max Alekseyev (maxal(AT)cs.ucsd.edu), Mar 02 2004
%E A092670 More terms from T. Suzuki (suzuki(AT)scio.co.jp), Nov 24 2006

%I A120450
%S A120450 0,0,1,1,1,2,2,2,2,2,2,3,3,4,3,3,3,5,4,4,4,4,4,4,6,3,5,4,4,4,6,5,5,5,5,
%T A120450 7,5,6,6,6,6,7,5,6,7,6,7,9,8,8,6,7,7,8,7,9,6,10
%N A120450 Number of ways to express a prime p as 2*p1 + 3*p2, where p1, p2 are primes or 1.
%C A120450 It seems that every prime p > 3 can be expressed as 2*p1 + 3*p2, where p1, p2 are primes or 1. I have tested it for the first 1500 primes (up to 12553) and it is true.
%e A120450 a(11)=2 because we can write 31 (the 11th prime) as 2* 5 + 3* 7 OR 2* 11 + 3* 3
%e A120450 a(12)=3 because we can write 37 (the 12th prime) as 2* 2 + 3* 11 OR 2* 11 + 3* 5 OR 2* 17 + 3* 1
%Y A120450 Adjacent sequences: A120447 A120448 A120449 this_sequence A120451 A120452 A120453
%Y A120450 Sequence in context: A072748 A030603 A092670 this_sequence A127238 A072114 A090621
%K A120450 nonn
%O A120450 1,6
%A A120450 Vassilis Papadimitriou (bpapa(AT)sch.gr), Jul 20 2006

%I A127238
%S A127238 1,1,2,2,2,2,2,2,3,3,4,3,6,4,4,5,4,3,4,3,6,8,6,6,6,9,10,8,8,12,7,7,7,7,
%T A127238 7,6,8,13,12,8,7,12,8,13,14,12,15,13,12,12,13,14,13,12,15,12,12,12,15,
%U A127238 14,17,17,22,17,22,14,15
%N A127238 Diagonal sums of Thue-Morse binomial triangle A127236.
%F A127238 a(n)=sum{k=0..floor(n/2), A010060(binomial(n-k,k))}
%Y A127238 Adjacent sequences: A127235 A127236 A127237 this_sequence A127239 A127240 A127241
%Y A127238 Sequence in context: A030603 A092670 A120450 this_sequence A072114 A090621 A029378
%K A127238 easy,nonn
%O A127238 0,3
%A A127238 Paul Barry (pbarry(AT)wit.ie), Jan 10 2007

%I A072114
%S A072114 0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7,
%T A072114 7,7,7,7,7,7,8,8,9,10,10,10,10,10,11,11,12,12,12,12,12,12,12,12,12,12,
%U A072114 12,13,13,13,14,14,15,15,16,16,16,16,16,17,18,18,19,19,19,19,19,19,19
%N A072114 Number of 3-almost primes (A014612) <= n.
%C A072114 Number of k <= n such that bigomega(k) = 3.
%C A072114 Let A be a positive integer then card{ x <= n : bigomega(x) = A } ~ (n/Log(n))*Log(Log(n))^(A-1)/(A-1)!. For which n, card{ x <= n : bigomega(x) = 3 } >= card{ x <= n : bigomega(x) = 2 } ?
%D A072114 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
%D A072114 G. Tenenbaum, Introduction \`a la th\'eorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
%F A072114 a(n) = card{ x <= n : bigomega(x) = 3 }, asymptotically : a(n) ~ (n/log(n))*log(log(n))^2/2 [Landau, p. 211]
%o A072114 (PARI) for(n=1,100,print1(sum(i=1,n,if(bigomega(i)-3,0,1)),","))
%Y A072114 Cf. A014612, A109251, A001358, A072000.
%Y A072114 Adjacent sequences: A072111 A072112 A072113 this_sequence A072115 A072116 A072117
%Y A072114 Sequence in context: A092670 A120450 A127238 this_sequence A090621 A029378 A053278
%K A072114 easy,nonn
%O A072114 0,13
%A A072114 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 19 2002

%I A090621
%S A090621 0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,
%T A090621 8,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,
%U A090621 14,14,14,15,15,16,16,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19
%N A090621 Highest power of 16 dividing n!.
%F A090621 a(n) =A090622(n, 16) =[A011371(n)/4] =[A090616(n)/2] =[([n/2]+[n/4]+[n/8]+[n/16]+...)/4]. Almost n/4.
%e A090621 a(10)=2 since 10!=3628800=16^2*14175
%Y A090621 Cf. A011371, A054861, A090616, A027868, A054896, A090617, A090618, A090619, A090620.
%Y A090621 Adjacent sequences: A090618 A090619 A090620 this_sequence A090622 A090623 A090624
%Y A090621 Sequence in context: A120450 A127238 A072114 this_sequence A029378 A053278 A035466
%K A090621 nonn
%O A090621 0,11
%A A090621 Henry Bottomley (se16(AT)btinternet.com), Dec 06 2003

%I A029378
%S A029378 1,0,0,0,0,1,1,1,1,0,1,1,2,2,2,2,2,2,3,3,4,4,4,4,5,5,6,
%T A029378 6,7,7,8,8,9,9,10,11,12,12,13,13,15,15,17,17,18,19,20,21,
%U A029378 23,23,25,25,27,28,30,31,33,33,35,36,39,40,42,43,45,46
%N A029378 Expansion of 1/((1-x^5)(1-x^6)(1-x^7)(1-x^8)).
%Y A029378 Adjacent sequences: A029375 A029376 A029377 this_sequence A029379 A029380 A029381
%Y A029378 Sequence in context: A127238 A072114 A090621 this_sequence A053278 A035466 A122521
%K A029378 nonn
%O A029378 0,13
%A A029378 njas

%I A053278
%S A053278 1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,5,5,6,6,7,7,8,8,9,10,11,11,13,13,
%T A053278 14,15,16,17,19,19,21,22,24,25,28,29,31,32,35,36,40,41,44,46,49,51,56,
%U A053278 58,62,65,69,72,77,80,86,90,95,99,106,110,117,122,130,135,144,149,158
%N A053278 A '7th order' mock theta functions
%D A053278 Dean Hickerson, On the seventh order mock theta functions, Inventiones Mathematicae, 94 (1988) 661-677
%F A053278 G.f.: g(q, q^7), where g(x, q) = sum for n >= 1 of q^(n(n-1))/((1-x)(1-q/x)(1-q x)(1-q^2/x)...(1-q^(n-1) x)(1-q^n/x))
%t A053278 Series[Sum[q^(7n(n-1))/Product[1-q^Abs[7k+1], {k, -n, n-1}], {n, 1, 4}], {q, 0, 100}]
%Y A053278 Other '7th order' mock theta functions are at A053275, A053276, A053277, A053279, A053280.
%Y A053278 Adjacent sequences: A053275 A053276 A053277 this_sequence A053279 A053280 A053281
%Y A053278 Sequence in context: A072114 A090621 A029378 this_sequence A035466 A122521 A086394
%K A053278 nonn,easy
%O A053278 0,7
%A A053278 Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 19 1999

%I A035466
%S A035466 0,0,0,0,1,1,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,
%T A035466 8,8,9,10,12,12,14,14,15,16,17,19,22,22,24,25,27,29,32,34,38,39,42,44,
%U A035466 48,51,55,59,64,67,72,75,81,87,94,99,107,111,119,126,135,144,154
%N A035466 Number of partitions of n into parts 8k+5 or 8k+6.
%Y A035466 Adjacent sequences: A035463 A035464 A035465 this_sequence A035467 A035468 A035469
%Y A035466 Sequence in context: A090621 A029378 A053278 this_sequence A122521 A086394 A029226
%K A035466 nonn
%O A035466 1,18
%A A035466 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A122521
%S A122521 1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,5,5,7,7,8,8,9,9,12,12,15,15,17,
%T A122521 17,21,21,27,27,32,32,38,38,48,48,59,59,70,70,86,86,107,107,129,129,156,
%U A122521 156,193,193,236,236,285,285,349,349,429,429,521,521,634,634,778,778
%N A122521 Recursion: a(n) = a(n - 6) + a(n - 8) characteristic Polynomial:x^8-x^2-1.
%C A122521 Inspired by the recursion: a(n)=a(n-3)+a(n-4) as power doubled: x^4-x-1-->x^8-x^2-1 Root sum is zero and the ratio of the sequence is very low.
%F A122521 a(n) = a(n - 6) + a(n - 8)
%t A122521 a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; a[5] = 1; a[6] = 1; a[7] = 1; a[n_] := a[n] = a[n - 6] + a[n - 8] Table[a[n], {n, 0, 100}] (*vector Matrix Markov*) M = {{0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0, 0, 0}}; v[1] = Table[1, {n, 1, 8}] v[n_] := v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 1, 100}]
%Y A122521 Adjacent sequences: A122518 A122519 A122520 this_sequence A122522 A122523 A122524
%Y A122521 Sequence in context: A029378 A053278 A035466 this_sequence A086394 A029226 A093354
%K A122521 nonn,uned
%O A122521 1,9
%A A122521 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2006

%I A086394
%S A086394 1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4,5,5,7,7,8,10,11,12,16,19,21,23,29,34,
%T A086394 41,46,56,68,80,92,114,135,158,182,225,269,320,369,455,544,644,753,921,
%U A086394 1111,1321,1543,1891,2274,2711,3183,3895,4694,5591,6592,8051,9729,11624
%N A086394 (-1) times minimal coefficient of the polynomial (1-x)*(1-x^2)*...*(1-x^n).
%o A086394 (PARI) a(n)=-vecmin(vector(n*(n+1)/2,i,polcoeff(prod(k=1,n,1-x^k),i))) (from Benoit Cloitre)
%Y A086394 Cf. A086376.
%Y A086394 Cf. A025591.
%Y A086394 Adjacent sequences: A086391 A086392 A086393 this_sequence A086395 A086396 A086397
%Y A086394 Sequence in context: A053278 A035466 A122521 this_sequence A029226 A093354 A109701
%K A086394 nonn
%O A086394 1,7
%A A086394 Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003
%E A086394 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 12 2003
%E A086394 Further terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Sep 22 2003

%I A029226
%S A029226 1,0,1,0,1,0,1,1,2,2,2,2,2,2,3,3,5,4,6,4,6,5,7,7,9,9,10,
%T A029226 10,11,11,13,13,16,15,18,17,20,19,22,22,25,25,28,28,31,
%U A029226 31,34,34,38,38,42,42,46,46,50,50,55,55,60,60,65,65,70
%N A029226 Expansion of 1/((1-x^2)(1-x^7)(1-x^8)(1-x^9)).
%Y A029226 Adjacent sequences: A029223 A029224 A029225 this_sequence A029227 A029228 A029229
%Y A029226 Sequence in context: A035466 A122521 A086394 this_sequence A093354 A109701 A124751
%K A029226 nonn
%O A029226 0,9
%A A029226 njas

%I A093354
%S A093354 0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3,5,5,5,5,7,7,7,7,7,7,9,9,11,11,11,
%T A093354 11,11,11,11,11,11,11,13,13,16,16,16,16,17,17,17,17,17,17,20,20,20,20,
%U A093354 20,20,23,23,24,24,24,24,24,24,26,26,26,26,30,30,32,32,32,32,32,32,36
%N A093354 Number of prime triples (p,q,r) with p<q<r<=n and q-p=r-q.
%e A093354 a(20)=5: (3,5,7)_5-3=7-5, (3,7,11)_7-3=11-7,
%e A093354 (5,11,17)_11-5=17-11, (3,11,19)_11-3=19-11, and (7,13,19)_13-7=19-13.
%Y A093354 Adjacent sequences: A093351 A093352 A093353 this_sequence A093355 A093356 A093357
%Y A093354 Sequence in context: A122521 A086394 A029226 this_sequence A109701 A124751 A103374
%K A093354 nonn
%O A093354 1,11
%A A093354 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 27 2004

%I A109701
%S A109701 1,1,1,1,1,1,2,2,2,2,2,2,3,4,4,4,4,4,5,6,7,7,7,7,8,10,11,12,12,12,13,15,
%T A109701 17,18,19,19,20,23,26,28,29,30,31,34,38,41,43,44,46,50,55,60,63,65,67,
%U A109701 72,79,85,90,93,96,102,111,120,127,132,136,143,154,166,176,183,189,198
%N A109701 Number of partitions of n into parts each equal to 1 mod 6.
%F A109701 G.f.=1/product(1-x^(1+6j), j=0..infinity)-1; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 14 2006
%e A109701 a(10)=2 since 10 = 7+1+1+1 = 1+1+1+1+1+1+1+1+1+1
%p A109701 g:=1/product(1-x^(1+6*j),j=0..20)-1: gser:=series(g,x=0,77): seq(coeff(gser,x,n),n=1..74); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 14 2006
%Y A109701 Adjacent sequences: A109698 A109699 A109700 this_sequence A109702 A109703 A109704
%Y A109701 Sequence in context: A086394 A029226 A093354 this_sequence A124751 A103374 A137722
%K A109701 nonn
%O A109701 1,7
%A A109701 Erich Friedman (efriedma(AT)stetson.edu), Aug 07 2005

%I A124751
%S A124751 1,0,1,0,1,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,4,4,4,4,4,5,7,8,8,8,8,9,12,15,
%T A124751 16,16,16,17,21,27,31,32,32,33,38,48,58,63,64,65,71,86,106,121,127,129,
%U A124751 136,157,192,227,248
%V A124751 1,0,1,0,1,0,1,-1,1,-1,1,-1,1,-2,2,-2,2,-2,2,-3,4,-4,4,-4,4,-5,7,-8,8,-8,8,-9,12,-15,
%W A124751 16,-16,16,-17,21,-27,31,-32,32,-33,38,-48,58,-63,64,-65,71,-86,106,-121,127,-129,136,
%X A124751 -157,192,-227,248
%N A124751 Expansion of (1+x^2+x^4)/(1-x^6+x^7).
%C A124751 Diagonal sums of number triangle A124749.
%F A124751 a(n)=sum{k=0..floor(n/2), C(floor(k/3),n-2k)*(-1)^n}
%Y A124751 Adjacent sequences: A124748 A124749 A124750 this_sequence A124752 A124753 A124754
%Y A124751 Sequence in context: A029226 A093354 A109701 this_sequence A103374 A137722 A081305
%K A124751 easy,sign
%O A124751 0,14
%A A124751 Paul Barry (pbarry(AT)wit.ie), Nov 06 2006

%I A103374
%S A103374 1,1,1,1,1,1,1,2,2,2,2,2,2,3,4,4,4,4,4,5,7,8,8,8,8,9,12,15,16,16,16,17,
%T A103374 21,27,31,32,32,33,38,48,58,63,64,65,71,86,106,121,127,129,136,157,192,
%U A103374 227,248,256,265,293,349,419,475,504,521,558,642,768,894,979,1025,1079
%N A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).
%C A103374 k=6 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, and k=5 case is A103373.
%C A103374 The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
%C A103374 For this k=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.112775684278705470629704020571092935606859271855283681485701628007166332579528443459272836948847449
%C A103374 The sequence of prime values in this k=6 case is A103384; The sequence of semiprime values in this k=6 case is A103394.
%D A103374 Selmer, E.S., "On the irreducibility of certain trinomials", Math. Scand., 4 (1956) 287-302
%D A103374 Shallit, J., "A generalization of automatic sequences", Theoretical Computer Science, 61(1988)1-16.
%D A103374 Zanten, A. J. van, "The golden ratio in the arts of painting, building, and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.
%H A103374 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103374 J.-P. Allouche and T. Johnson, <a href="http://www.lri.fr/~allouche/johnson2.pdf">Narayana's Cows and Delayed Morphisms</a>
%e A103374 a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.
%t A103374 k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
%Y A103374 Cf. A000045, A000931, A079398, A103372-A103381, A103384, A103394.
%Y A103374 Adjacent sequences: A103371 A103372 A103373 this_sequence A103375 A103376 A103377
%Y A103374 Sequence in context: A093354 A109701 A124751 this_sequence A137722 A081305 A035665
%K A103374 nonn,easy
%O A103374 1,8
%A A103374 Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 03 2005
%E A103374 Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2005

%I A137722
%S A137722 0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4,5,5,5,5,6,6,7,7,7,7,7,8,9,9,
%T A137722 9,9,10,11,12,12,13,13,14,14,15,15,15,15,16,17,18,18,18,19,20,21,22,22,
%U A137722 22,22,23,24,24,25,26,26,27,28,29,29,29,29,30,30,31,31,32,32,33,33,34
%N A137722 Number of numbers not greater than n with exactly one prime gap in their factorization.
%C A137722 a(n) > a(n-1) iff A073490(n) = 1;
%C A137722 A137721(n) > a(n) for n < 134;
%C A137722 A137721(n) < a(n) for n > 140.
%H A137722 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/b137722.txt">Table of n, a(n) for n = 1..1000</a>
%Y A137722 Cf. A073493.
%Y A137722 Adjacent sequences: A137719 A137720 A137721 this_sequence A137723 A137724 A137725
%Y A137722 Sequence in context: A109701 A124751 A103374 this_sequence A081305 A035665 A027388
%K A137722 nonn
%O A137722 1,14
%A A137722 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 09 2008

%I A081305
%S A081305 0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4,5,5,5,5,6,6,7,7,8,8,8,9,10,
%T A081305 10,10,10,11,12,13,13,14,14,15,15,16,16,16,16,17,18,19,19,19,20,21,22,
%U A081305 23,23,24,24,25,26,26,27,28,28,29,30,31,31,31,31,32,32,33,33,34,34,35
%N A081305 Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).
%C A081305 a(n)+A081304(n)=n; a(114)=A081304(114)=57;
%C A081305 a(n)<=n/2 for n<=114, a(n)>n/2 for n>114.
%Y A081305 Cf. A069900, A081303.
%Y A081305 Adjacent sequences: A081302 A081303 A081304 this_sequence A081306 A081307 A081308
%Y A081305 Sequence in context: A124751 A103374 A137722 this_sequence A035665 A027388 A083261
%K A081305 nonn
%O A081305 1,14
%A A081305 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 17 2003

%I A035665
%S A035665 0,0,0,0,0,0,0,1,0,1,0,1,0,2,2,2,2,2,2,3,4,6,4,7,4,8,6,12,10,13,12,14,
%T A035665 15,18,21,24,23,29,26,35,33,46,41,52,50,58,61,71,77,84,89,99,101,120,
%U A035665 121,146,142,168,166,192,199,226,239,259,275,300,316,354,370,416,422
%N A035665 Number of partitions of n into parts 7k+2 and 7k+6 with at least one part of each type.
%Y A035665 Adjacent sequences: A035662 A035663 A035664 this_sequence A035666 A035667 A035668
%Y A035665 Sequence in context: A103374 A137722 A081305 this_sequence A027388 A083261 A003648
%K A035665 nonn
%O A035665 1,14
%A A035665 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A027388
%S A027388 0,2,2,2,2,2,2,4,0,1,2,4,4,4,4,4,3,6,2,3,2,4,4,4,4,4,3
%N A027388 Write digits for n, count endpoints (version 3).
%Y A027388 Adjacent sequences: A027385 A027386 A027387 this_sequence A027389 A027390 A027391
%Y A027388 Sequence in context: A137722 A081305 A035665 this_sequence A083261 A003648 A003642
%K A027388 nonn
%O A027388 0,2
%A A027388 njas

%I A083261
%S A083261 1,2,2,2,2,2,2,4,2,2,2,2,2,6,2,2,2,2,2,6,6,2,2,4,2,2,4,2,2,2,2,2,6,6,6,
%T A083261 2,2,6,6,2,2,2,2,12,6,2,2,4,4,6,6,2,2,6,6,6,6,2,2,2,2,6,4,2,6,2,2,6,6,2,
%U A083261 2,2,2,6,12,6,6,2,2,16,2,2,2,6,6,6,6,2,2,6,6,6,6,6,6,2,2,12,12,2,2,2,2
%N A083261 a(n)=GCD[A046523(n+1), A046523(n)].
%Y A083261 Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083255-A083260.
%Y A083261 Adjacent sequences: A083258 A083259 A083260 this_sequence A083262 A083263 A083264
%Y A083261 Sequence in context: A081305 A035665 A027388 this_sequence A003648 A003642 A100007
%K A083261 nonn
%O A083261 1,2
%A A083261 Labos E. (labos(AT)ana.sote.hu), May 09 2003

%I A003648 M0210
%S A003648 1,2,2,2,2,2,2,4,2,2,2,2,4,2,4,4,2,4,4,2,2,2,4,4,2,2,2,4,2,4,2,2,4,6,4,
%T A003648 2,2,4,4,2,4,4,2,2,2,4,4,4,4,2,4,2,4,2,4,2,2,4,2,6,4,4,2,4,4,2,4,2,2,2,
%U A003648 4,4,4,2,4,4,4,4,4,2,4,8,2,2,4,2,8,2,2,4,2,8,4,6,8,2,4,8,12,4,2,4,4,2
%N A003648 Class number of quadratic forms with discriminant 4n, n == 2,3^( mod 4).
%D A003648 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%Y A003648 Cf. A003647.
%Y A003648 Adjacent sequences: A003645 A003646 A003647 this_sequence A003649 A003650 A003651
%Y A003648 Sequence in context: A035665 A027388 A083261 this_sequence A003642 A100007 A104369
%K A003648 nonn
%O A003648 2,2
%A A003648 njas, Mira Bernstein

%I A003642 M0211
%S A003642 1,1,2,2,2,2,2,2,4,2,2,2,4,4,2,2,2,2,4,2,2,4,2,2,2,4,4,4,4,2,2,4,4,2,4,
%T A003642 2,2,4,2,2,2,4,8,2,2,4,2,4,2,2,4,4,4,2,2,4,4,2,4,2,2,4,2,2,4,8,2,4,2,4,
%U A003642 4,2,2,4,4,4,4,2,2,2,4,2,4,2,4,8,4,2,4,2,4,4,2,2,4,2,4,4,2,4,4,4,2,2,4
%N A003642 Number of genera of quadratic field with discriminant -4n, where -n == 2 or 3 (mod 4).
%D A003642 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%H A003642 <a href="http://www.research.att.com/~njas/sequences/Sindx_Qua.html#quadfield">Index entries for sequences related to quadratic fields</a>
%Y A003642 Cf. A003641.
%Y A003642 Adjacent sequences: A003639 A003640 A003641 this_sequence A003643 A003644 A003645
%Y A003642 Sequence in context: A027388 A083261 A003648 this_sequence A100007 A104369 A051702
%K A003642 nonn
%O A003642 1,3
%A A003642 njas, Mira Bernstein

%I A100007
%S A100007 1,2,2,2,2,2,2,4,2,2,4,2,2,2,2,2,4,4,2,4,2,2,4,2,2,4,2,4,4,2,2,4,4,2,4,
%T A100007 2,2,4,4,2,2,2,4,4,2,4,4,4,2,4,2,2,8,2,2,4,2,4,4,4,2,4,2,2,4,2,4,4,2,2,
%U A100007 4,4,4,4,2,2,4,4,2,4,4,2,8,2,2,4,2,4,4,2,2,4,4,4,4,2,2,8,2,2,4,4,4,4,4
%N A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.
%e A100007 a(13)=2 because among the three divisors of 25 only 1 and 25 are unitary.
%p A100007 with(numtheory): for n from 1 to 120 do printf(`%d,`,2^nops(ifactors(2*n-1)[2])) od: (Deutsch)
%Y A100007 Adjacent sequences: A100004 A100005 A100006 this_sequence A100008 A100009 A100010
%Y A100007 Sequence in context: A083261 A003648 A003642 this_sequence A104369 A051702 A073130
%K A100007 nonn,easy
%O A100007 1,2
%A A100007 njas, Nov 20 2004
%E A100007 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 24 2004

%I A104369
%S A104369 2,2,2,2,2,2,4,2,2,4,2,2,4,4,4,16,16,4,2,16,4,4,8,4,2,16,8,8,2,8,16,32,
%T A104369 16,32,16,8,8,16,128,4,32,16,16,4,16,8,16,4,8,128
%N A104369 Number of divisors of A104350(n) + 1.
%C A104369 a(n) = A000005(A104365(n)).
%H A104369 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a104350.txt">Products of largest prime factors of numbers <= n</a>
%Y A104369 Cf. A104370, A104361, A064144.
%Y A104369 Adjacent sequences: A104366 A104367 A104368 this_sequence A104370 A104371 A104372
%Y A104369 Sequence in context: A003648 A003642 A100007 this_sequence A051702 A073130 A064136
%K A104369 nonn
%O A104369 1,1
%A A104369 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 06 2005

%I A051702
%S A051702 1,1,2,2,2,2,2,2,4,2,2,4,2,2,4,6,2,2,4,2,2,4,4,6,4,2,2,2,2,4,4,4,2,2,2,
%T A051702 2,6,4,4,6,2,2,2,2,2,2,12,4,2,2,4,2,2,6,6,6,2,2,4,2,2,10,4,2,2,4,6,6,2,
%U A051702 2,4,6,6,6,4,4,6,4,4,8,2,2,2,2,4,4,6,4,2,2,4,8,4,4,4,4,6,2,2,6,6,6,6,2
%N A051702 Distance from n-th prime to closest prime.
%H A051702 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b051702.txt">Table of n, a(n) for n=1..10000</a>
%e A051702 Closest primes to 2,3,5,7,11 are 3,2,3,5,13.
%Y A051702 Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
%Y A051702 A046929(n+1) + 1.
%Y A051702 Adjacent sequences: A051699 A051700 A051701 this_sequence A051703 A051704 A051705
%Y A051702 Sequence in context: A003642 A100007 A104369 this_sequence A073130 A064136 A072924
%K A051702 nonn,easy,nice
%O A051702 1,3
%A A051702 njas
%E A051702 More terms from James A. Sellers (sellersj(AT)math.psu.edu)

%I A073130
%S A073130 1,2,2,2,2,2,2,4,2,2,6,2,2,4,6,6,2,6,2,2,2,2,6,8,2,2,4,2,2,2,2,2,2,2,2,
%T A073130 6,6,4,2,2,2,2,2,2,2,4,4,4,2,4,2,2,2,6,6,6,2,2,4,2,2,2,4,2,2,2,6,2,2,2,
%U A073130 6,4,6,6,2,6,4,2,2,2,2,2,2,6,2,6,4,2,2,4,4,2,4,2,2,2,12,2,6,2,2,2,6,2
%N A073130 a(n)=GCD[p[w+1]-p[w],p[p[w+1]]-p[p[w]]],here p(j) is the j-th prime.
%t A073130 Table[GCD[Prime[w+1]-Prime[w], Prime[Prime[w+1]]-Prime[Prime[w]]], {w, 1, 256}]
%Y A073130 Cf. A073131, A073132.
%Y A073130 Adjacent sequences: A073127 A073128 A073129 this_sequence A073131 A073132 A073133
%Y A073130 Sequence in context: A100007 A104369 A051702 this_sequence A064136 A072924 A036263
%K A073130 nonn
%O A073130 1,2
%A A073130 Labos E. (labos(AT)ana.sote.hu), Jul 16 2002

%I A064136
%S A064136 2,2,2,2,2,2,4,2,4,2,2,16,4,8,8,4,4,8,2,16,4,2,4,16,16,4,2,4,16,8,2,16,
%T A064136 8,16,16,48,8,4,4,8,4,8,4,4,16,8,16,16,4,16,8,8,2,16,16,32
%N A064136 Number of divisors of 11^n + 1 that are relatively prime to 11^m + 1 for all 0 < m < n.
%H A064136 Sam Wagstaff, Cunningham Project, <a href="http://www.cerias.purdue.edu/homes/ssw/cun/pmain501">Factorizations of 11^n-1, n<=240</a>
%t A064136 a = {1}; Do[ d = Divisors[ 11^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 11^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 55} ]
%Y A064136 Adjacent sequences: A064133 A064134 A064135 this_sequence A064137 A064138 A064139
%Y A064136 Sequence in context: A104369 A051702 A073130 this_sequence A072924 A036263 A060447
%K A064136 nonn
%O A064136 0,1
%A A064136 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2001

%I A072924
%S A072924 1,2,2,2,2,2,2,4,3,3,3,3,6,8,7,6,6,7,5,11,2,2,9,4,6,10,5,9,5,6,4,7,10,
%T A072924 11,7,6,4,3,10,4,4,3,5,4,17,6,11,7,5,14,12,8,6,11,4,14,8,7,3,16,4,21,8,
%U A072924 12,7,8,7,7,18,12,8,17,10,12,28,6,24,16,12,16,18,7,6,6,7,11,8,14,24,8
%N A072924 Least k such that floor((1+1/k)^n) is prime.
%F A072924 It seems that a(n)/sqrt(n) is bounded. More precisely for n large enough it seems that (1/2)*sqrt(n) < a(n) < 3*sqrt(n)
%o A072924 (PARI) a(n)=if(n<0,0,s=1; while(isprime(floor((1+1/s)^n)) == 0,s++); s)
%Y A072924 Adjacent sequences: A072921 A072922 A072923 this_sequence A072925 A072926 A072927
%Y A072924 Sequence in context: A051702 A073130 A064136 this_sequence A036263 A060447 A118177
%K A072924 easy,nonn
%O A072924 1,2
%A A072924 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 11 2002

%I A036263
%S A036263 1,0,2,2,2,2,2,2,4,4,2,2,2,2,0,4,4,2,2,4,2,2,2,4,2,2,2,2,10,10,
%T A036263 2,4,8,8,4,0,2,2,0,4,8,8,2,2,10,0,8,2,2,2,4,8,4,0,0,4,4,2,2,8,4,
%U A036263 10,2,2,10,8,4,8,2,2,2,2,0,2,2,2,4,4,2,8,8,8,4,2,2,2,4,2,2,8,4
%V A036263 1,0,2,-2,2,-2,2,2,-4,4,-2,-2,2,2,0,-4,4,-2,-2,4,-2,2,2,-4,-2,2,-2,2,10,-10,
%W A036263 2,-4,8,-8,4,0,-2,2,0,-4,8,-8,2,-2,10,0,-8,-2,2,2,-4,8,-4,0,0,-4,4,-2,-2,8,4,
%X A036263 -10,-2,2,10,-8,4,-8,2,2,2,-2,0,-2,2,2,-4,4,2,-8,8,-8,4,-2,2,2,-4,-2,2,8,-4
%N A036263 Second differences of primes.
%H A036263 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b036263.txt">Table of n, a(n) for n=1..10000</a>
%t A036263 Table[ Prime[n - 1] + Prime[n + 1] - 2*Prime[n], {n, 2, 105}]
%o A036263 (PARI) for(n=2,100,print1(prime(n+2)-2*prime(n+1)+prime(n)","))
%Y A036263 Cf. A001223, A036262, A051635, A006562 & A051634.
%Y A036263 Adjacent sequences: A036260 A036261 A036262 this_sequence A036264 A036265 A036266
%Y A036263 Sequence in context: A073130 A064136 A072924 this_sequence A060447 A118177 A105069
%K A036263 sign,easy,nice
%O A036263 1,3
%A A036263 njas

%I A060447
%S A060447 1,1,1,2,2,2,2,2,2,4,4,2,5,5,4,4,4,4,4,4,8,5,8,8,5,5,8,7,7,7,11,11,11,
%T A060447 11,11,11,11,11,11,17,11,14,7,7,11,11,11,11,19,20,11,11,11,11,14,14,22,
%U A060447 17,17,17,16,14,14,16,20
%N A060447 Cyclic token-passing numbers with pattern 121: players 1, 2, ..., n are seated around a table. Each has a penny. Player 1 passes a penny to player 2, who passes two pennies to player 3, who passes a penny to player 4. Player 4 passes a penny to player 5, who passes two pennies to player 6, who passes a penny to player 7, and so on, players passing 1,2,1,1,2,1,... pennies to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Sequence gives number of players remaining when game reaches periodic state.
%D A060447 Suggested by 58th William Lowell Putnam Mathematical Competition, 1997, Problem A-2.
%H A060447 <a href="http://math.ucsd.edu/~pfitz/pastputnam.html">Putnam Mathematical Competitions</a>
%e A060447 a(10)=4 because 4 players (numbers 4, 6, 9, 10) remain.
%Y A060447 Adjacent sequences: A060444 A060445 A060446 this_sequence A060448 A060449 A060450
%Y A060447 Sequence in context: A064136 A072924 A036263 this_sequence A118177 A105069 A100825
%K A060447 easy,nonn,nice
%O A060447 1,4
%A A060447 Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 08 2001

%I A118177
%S A118177 1,1,2,2,2,2,2,2,4,4,3,2,2,2,6,4,2,5,4,6,5,2,2,2,2,4,4,3,4,2,4,2,8,3,4,
%T A118177 4,6,4,4,4,8,4,4,8,2,8,2,2,6,6,4,4,4,3,9,4,2,4,4,2,4,4,4,6,8,12,6,12,6,
%U A118177 7,3,4,4,2,8,2,6,2,8,2,8,4,4,5,8,4,4,8,6,4,2,5,2,4,4,4,4,4,8,6
%N A118177 a(n) = number of positive divisors of A118176(n).
%F A118177 a(n) = A000005(A118176(n)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 06 2007
%Y A118177 Cf. A118176.
%Y A118177 Adjacent sequences: A118174 A118175 A118176 this_sequence A118178 A118179 A118180
%Y A118177 Sequence in context: A072924 A036263 A060447 this_sequence A105069 A100825 A008767
%K A118177 nonn
%O A118177 1,3
%A A118177 Leroy Quet (qq-quet(AT)mindspring.com), Apr 13 2006
%E A118177 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 06 2007

%I A105069
%S A105069 1,2,2,2,2,2,2,4,4,4,2,2,3,3,3,4,2,4,3,5
%N A105069 Number of distinct prime divisors of 100^n - 3.
%e A105069 If n=1, then 100^1 - 3 = 97 which is prime so the first term is 1.
%Y A105069 Adjacent sequences: A105066 A105067 A105068 this_sequence A105070 A105071 A105072
%Y A105069 Sequence in context: A036263 A060447 A118177 this_sequence A100825 A008767 A105255
%K A105069 nonn
%O A105069 1,2
%A A105069 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 05 2005

%I A100825
%S A100825 0,0,0,0,2,2,2,2,2,2,4,4,4,4,4,4,2,6,6,4,6,4,4,4,4,6,6,8,6,8,8,8,8,8,8,
%T A100825 8,6,6,10,12,12,8,10,10,12,10,10,14,14,14,12,12,14,12,10,14,14,16,16,16,
%U A100825 18,14,16,16,16,14,18,16,18,18,18,18,18,16,20,20,18,22,22,22,20,18,20
%N A100825 In decimal representation: minimal number of editing steps (delete, insert, or substitute) to transform 2^n into its reversal.
%H A100825 Michael Gilleland, <a href="http://www.merriampark.com/ld.htm">Levenshtein Distance</a> [It has been suggested that this algorithm gives incorrect results sometimes. - njas]
%F A100825 a(n) = LevenshteinDistance(A000079(n), A004094(n)).
%e A100825 n=19: 2^19 = 524288=[5]24288 -> 824288=[]824288 ->
%e A100825 8824288=882428[8] -> 882428=88242[8] -> 882425=A004094(19):
%e A100825 a(19) = #{subst[5->8], ins[8], del[8], subst[8->5]} = 4.
%Y A100825 Adjacent sequences: A100822 A100823 A100824 this_sequence A100826 A100827 A100828
%Y A100825 Sequence in context: A060447 A118177 A105069 this_sequence A008767 A105255 A129381
%K A100825 nonn,base
%O A100825 1,5
%A A100825 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 06 2005

%I A008767
%S A008767 0,0,0,0,0,0,0,1,2,2,2,2,2,2,4,6,6,6,6,6,6,9,12,12,12,12,
%T A008767 12,12,16,20,20,20,20,20,20,25,30,30,30,30,30,30,36,42,
%U A008767 42,42,42,42,42,49,56,56,56,56,56,56,64,72,72,72,72,72
%N A008767 floor(n/7)*ceil(n/7).
%Y A008767 Adjacent sequences: A008764 A008765 A008766 this_sequence A008768 A008769 A008770
%Y A008767 Sequence in context: A118177 A105069 A100825 this_sequence A105255 A129381 A139514
%K A008767 nonn
%O A008767 0,9
%A A008767 njas

%I A105255
%S A105255 1,2,2,2,2,2,2,5,3,2,5,3,1,6,4
%N A105255 Number of distinct prime divisors of 88...889 (with n 8s).
%e A105255 The number of distinct prime divisors of 89 is 1 (prime).
%e A105255 The number of distinct prime divisors of 889 is 2.
%e A105255 The number of distinct prime divisors of 8889 is 2.
%Y A105255 Cf. A104543.
%Y A105255 Adjacent sequences: A105252 A105253 A105254 this_sequence A105256 A105257 A105258
%Y A105255 Sequence in context: A105069 A100825 A008767 this_sequence A129381 A139514 A068323
%K A105255 nonn
%O A105255 0,2
%A A105255 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 29 2005

%I A129381
%S A129381 1,1,2,2,2,2,2,2,8,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,24,25,25,25,25,25,25,
%T A129381 25,25,25,25,12,12,12,12,12,12,12,12,12,12,12,12,12,48,49,49,49,49,49,
%U A129381 49,49,49,49,49,49,49,49,49,32,32,32,32,32,32,32,32,32,32,32,32,32,32
%N A129381 a(1)=1. a(n) = the number of earlier terms which are coprime to floor(sqrt(n)).
%e A129381 Floor(sqrt(11)) = 3. So a(11) is the number of terms from among a(1),a(2),...a(10) which are coprime to 3. Terms a(1) through a(9) are each coprime to 3, so a(11) = 9.
%p A129381 a[1]:=1: for n from 2 to 100 do ct:=0: for j from 1 to n-1 do if igcd(a[j],floor(sqrt(n)))=1 then ct:=ct+1 else fi od: a[n]:=ct: od: seq(a[n],n=1..100); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 2007
%Y A129381 Cf. A129382.
%Y A129381 Adjacent sequences: A129378 A129379 A129380 this_sequence A129382 A129383 A129384
%Y A129381 Sequence in context: A100825 A008767 A105255 this_sequence A139514 A068323 A054990
%K A129381 nonn
%O A129381 1,3
%A A129381 Leroy Quet (qq-quet(AT)mindspring.com), Apr 12 2007
%E A129381 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 2007

%I A139514
%S A139514 2,2,2,2,2,2,11,23,139,90439,33156439637,87550414616253068989,
%T A139514 751473085670398285260591818545427587609,
%U A139514 9405222481347574254746223047204588161218024092399608112777273401749812628709
%N A139514 Denominators of an Egyptian fraction for Pi, using only prime numbers and allowing repetitions.
%e A139514 1/2+1/2+1/2+1/2+1/2+1/2 = 3 (integer part of Pi)
%e A139514 3+1/11+1/23+1/139+1/90439 gives an approximation which is good to 10 decimal digits.
%e A139514 Adding the other fractions we reach a good approximation to 19, 38, 75, 149, 296, 591 decimal digits.
%p A139514 P:=proc(n) local a,i; a:=evalf(Pi-3,100); for i from 1 by 1 to 6 do print(2); od; for i from 1 by 1 to n do if 1/ithprime(i)<a then a:=a-1/ithprime(i); print(a); print(ithprime(i)); fi; od; end: P(100000);
%Y A139514 Cf. A139515-A139523.
%Y A139514 Adjacent sequences: A139511 A139512 A139513 this_sequence A139515 A139516 A139517
%Y A139514 Sequence in context: A008767 A105255 A129381 this_sequence A068323 A054990 A046921
%K A139514 easy,nonn,new
%O A139514 0,1
%A A139514 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), Apr 24 2008

%I A068323
%S A068323 0,1,1,1,2,2,2,2,2,3,1,3,2,3,4,3,1,4,2,4,4,4,1,5,2,4,1,4,1,6,2,3,5,5,2,
%T A068323 6,1,3,5,5,1,7,2,5,3,5,1,7,2,6,5,5,1,7
%N A068323 a(n)=number of arithmetic progressions of primes, nondecreasing with sum n.
%Y A068323 Adjacent sequences: A068320 A068321 A068322 this_sequence A068324 A068325 A068326
%Y A068323 Sequence in context: A105255 A129381 A139514 this_sequence A054990 A046921 A078178
%K A068323 easy,nonn
%O A068323 1,5
%A A068323 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 27 2002

%I A054990
%S A054990 1,1,1,2,2,2,2,2,3,2,1,3,2,2,3,5,3,6,2,2,3,3,4,2,2,2,1,2,3,5,4,4,5,2,5,
%T A054990 6,1,2,4,7,1,3,4,3,3,3,4,2,5,5,6,4,4,2,2,4,3,4,2,4,4,3,5,3,4,5,4,5,6,5,
%U A054990 2,7,1,4,2,3,1,6,3,4,7,3,3,3,5,5,4,3,8,3,6,2,4,3,4,5,6,6,5,5,4,5
%N A054990 Number of prime divisors of n! + 1 (counted with multiplicity).
%C A054990 The smallest k! with n prime factors occurs for n in A060250.
%C A054990 103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 10 2003
%H A054990 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A054990 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha104.htm">Factorizations of many number sequences</a>
%H A054990 R. G. Wilson v, <a href="http://www.research.att.com/~njas/sequences/a38507.txt">Explicit factorizations</a>
%H A054990 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/factors/factorial+">Factors of n!+1</a>.
%e A054990 a(2)=2 because 4! + 1 = 25 = 5*5
%t A054990 a[q_] := Module[{x, n}, x=FactorInteger[q!+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
%o A054990 (PARI) for(n=1,64,print1(bigomega(n!+1),","))
%Y A054990 Cf. A000040 (prime numbers), A001359 (twin primes). Also A054988, A054989, A054991, A054992.
%Y A054990 Cf. A066856 (number of distinct prime divisors of n!+1), A084846 (mu(n!+1)).
%Y A054990 Adjacent sequences: A054987 A054988 A054989 this_sequence A054991 A054992 A054993
%Y A054990 Sequence in context: A129381 A139514 A068323 this_sequence A046921 A078178 A105068
%K A054990 nonn,hard
%O A054990 1,4
%A A054990 Arne Ring (arne.ring(AT)epost.de), May 30 2000
%E A054990 More terms from Robert G. Wilson V (rgwv(AT)rgwv.com), Mar 23 2001
%E A054990 More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 10 2003

%I A046921
%S A046921 1,2,2,2,2,2,3,2,1,4,3,2,3,1,2,4,2,2,4,3,2,3,3,2,4,3,2,5,1,2,6,3,1,
%T A046921 3,4,2,5,4,2,6,3,2,4,2,3,6,2,1,4,3,4,6,4,2,6,5,2,6,3,2,5,1,2,3,5,4,
%U A046921 5,4,1,8,4,1,6,3,2,6,2,2,6,6,1,4,5,3,7,4,3,6,2,3,10,2,3,4,4,3,3,4,2
%N A046921 Number of ways to express 2n+1 as p+2a^2; p = 1 or prime, a >= 0.
%C A046921 Goldbach conjectured this sequence is never zero.
%H A046921 L. Hodges, <a href="http://learning.physics.iastate.edu/hodges/mm-1.pdf">A lesser-known Goldbach conjecture</a>, Math. Mag., 66 (1993), 45-47.
%H A046921 <a href="http://www.research.att.com/~njas/sequences/Sindx_Go.html#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%Y A046921 Adjacent sequences: A046918 A046919 A046920 this_sequence A046922 A046923 A046924
%Y A046921 Sequence in context: A139514 A068323 A054990 this_sequence A078178 A105068 A120676
%K A046921 nonn
%O A046921 0,2
%A A046921 David W. Wilson (davidwwilson(AT)comcast.net)

%I A078178
%S A078178 2,2,2,2,2,3,2,2,2,2,4,2,16,2,2,4,3,2,2,2,7,4,2,3,2,3,2,10,2,2,108,3,6,
%T A078178 2,3,7,2,2,4,2,16,3,2,2,2,20,2,7,2,3,3,2,2,2,2,9,4,2,2,7,8,3,2,2,2,24,
%U A078178 2,6,2,12,4,3,8,6,2,4,3,9,194,3,13,2,8,2,2,3,8,2,10,6,4,2,2,54,2,132,4
%N A078178 Least k>=2 such that n^k + n - 1 is prime.
%C A078178 n^a(n) + n - 1 = A078179(n).
%e A078178 7^2+7-1=5*11, but 7^3+7-1=349=A000040(70), therefore a(7)=3.
%Y A078178 Cf. A076845, A078179.
%Y A078178 Adjacent sequences: A078175 A078176 A078177 this_sequence A078179 A078180 A078181
%Y A078178 Sequence in context: A068323 A054990 A046921 this_sequence A105068 A120676 A125973
%K A078178 nonn
%O A078178 2,1
%A A078178 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 20 2002
%E A078178 More terms from Benoit Cloitre, Nov 20 2002

%I A105068
%S A105068 1,1,1,2,2,2,2,2,3,2,2,2,3,2,4,4,1,4,5,4,2,2,4,2,2,3,4,3,4,3,6,4,3,2,4,
%T A105068 4,5,3,5,5,4,5,7,4,5,4,7,5,4,4
%N A105068 Number of distinct prime divisors of 10^n - 3.
%e A105068 If n=1, 2 or 3, then 10^n - 3 = prime and thus the first three terms are 1.
%t A105068 Table[Length[FactorInteger[10^n - 3]], {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 21 2006
%Y A105068 Adjacent sequences: A105065 A105066 A105067 this_sequence A105069 A105070 A105071
%Y A105068 Sequence in context: A054990 A046921 A078178 this_sequence A120676 A125973 A001031
%K A105068 nonn
%O A105068 1,4
%A A105068 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 05 2005
%E A105068 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 21 2006

%I A120676
%S A120676 1,2,2,2,2,2,3,2,2,3,2,2,2,3,3,2,3,2,2,2,3,2,3,3,2,2,3,2,3,2,2,3,2,2,3,
%T A120676 3,2,3,3,3,2,2,2,4,2,2,3,2,3,3,3,2,3,2,3,2,2,3,3,3,2,2,3,2,3,3,2,4,2,2,
%U A120676 3,2,2,3,3,3,2,2,4,2,2,3,3,3,3,2,3,3,3,3,3,2,2,2,4,2,3,3,2,2,3,3,2,3,4
%N A120676 Number of prime factors of even square-free numbers A039956.
%F A120676 a(n)=A001221(A039956(n))=A001222(A039956(n))=A120675(n)+1.
%p A120676 issquarefree := proc(n::integer) local nf, ifa ; nf := op(2,ifactors(n)) ; for ifa from 1 to nops(nf) do if op(2,op(ifa,nf)) >= 2 then RETURN(false) ; fi ; od : RETURN(true) ; end: A001221 := proc(n::integer) RETURN(nops(numtheory[factorset](n))) ; end: A039956 := proc(maxn) local n,a ; a := [2] ; for n from 4 to maxn by 2 do if issquarefree(n) then a := [op(a),n] ; fi ; od : RETURN(a) ; end: A120676 := proc(maxn) local a,n; a := A039956(maxn) ; for n from 1 to nops(a) do a := subsop(n=A001221(a[n]),a) ; od ; RETURN(a) ; end: nmax := 600 : a := A120676(nmax) : for n from 1 to nops(a) do printf("%d,",a[n]) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 17 2006
%Y A120676 Adjacent sequences: A120673 A120674 A120675 this_sequence A120677 A120678 A120679
%Y A120676 Sequence in context: A046921 A078178 A105068 this_sequence A125973 A001031 A035250
%K A120676 nonn
%O A120676 1,2
%A A120676 Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 24 2006
%E A120676 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 17 2006

%I A125973
%S A125973 2,2,2,2,2,3,2,2,14,4,7,2,38,6,7,3,4,10,2,9,74,6,10,7,4,61,20,4,5,9,6,
%T A125973 16,6,8,2,9,4,10,2,48,44,163,9,2,95,3,27,70,6,26,57,9,6,8,207,2,27,15,
%U A125973 45,7,69,199,55,16,2,5,12,43,137,39,9,57,5,20,4,115,2,103,45,15,20,109
%N A125973 Smallest k such that k^n + k^(n-1) - 1 is prime.
%e A125973 Consider n = 6. k^6 + k^5 - 1 evaluates to 1, 95, 971 for k = 1, 2, 3. Only the last of these numbers is prime, hence a(6) = 3.
%o A125973 (PARI) {m=82;for(n=1,m,k=1;while(!isprime(k^n+k^(n-1)-1),k++);print1(k,","))} - Klaus Brockhaus, Dec 17 2006
%Y A125973 Cf. A000040, A045546, A125881-A125885, A125965-A125972, A126017.
%Y A125973 Adjacent sequences: A125970 A125971 A125972 this_sequence A125974 A125975 A125976
%Y A125973 Sequence in context: A078178 A105068 A120676 this_sequence A001031 A035250 A067743
%K A125973 nonn
%O A125973 1,1
%A A125973 Artur Jasinski (grafix(AT)csl.pl), Dec 14 2006
%E A125973 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 17 2006

%I A001031 M0213 N0077
%S A001031 1,2,2,2,2,2,3,2,3,3,3,4,3,2,4,3,4,4,3,3,5,4,4,6,4,3,6,3,4,7,4,5,6,3,5,
%T A001031 7,6,5,7,5,5,9,5,4,10,4,5,7,4,6,9,6,6,9,7,7,11,6,6,12,4,5,10,4,7,10,6,5,
%U A001031 9,8,8,11,6,5,13,5,8,11,6,8,10,6,6,14,9,6,12,7,7,15,7,8,13,5,8,12,8,9
%N A001031 Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).
%D A001031 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
%D A001031 Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture, Faber and Faber, 2001
%D A001031 R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
%D A001031 G. H. Hardy and J. E. Littlewood, Some problems of `partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
%D A001031 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 79.
%D A001031 J. Richstein, Verifying the Goldbach conjecture up to 4*10^14, Mathematics of Computation, Vol. 70, No. 236, pp. 1745-1749, 2000.
%D A001031 Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931-934, October 1993.
%D A001031 M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.
%H A001031 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b001031.txt">Table of n, a(n) for n=1..10000</a>
%H A001031 T. Oliveira e Silva, <a href="http://www.ieeta.pt/~tos/goldbach.html">Goldbach conjecture verification</a>
%H A001031 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A001031 <a href="http://www.research.att.com/~njas/sequences/Sindx_Go.html#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%Y A001031 Cf. A002372 (the main entry), A002373, A002374, A002375, A045917, A006307.
%Y A001031 Adjacent sequences: A001028 A001029 A001030 this_sequence A001032 A001033 A001034
%Y A001031 Sequence in context: A105068 A120676 A125973 this_sequence A035250 A067743 A029230
%K A001031 nonn,easy,nice
%O A001031 1,2
%A A001031 njas
%E A001031 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 19 2003

%I A035250
%S A035250 1,2,2,2,2,2,3,2,3,4,4,4,4,3,4,5,5,4,5,4,5,6,6,6,6,6,7,7,7,7,8,7,7,8,8,
%T A035250 9,10,9,9,10,10,10,10,9,10,10,10,9,10,10,11,12,12,12,13,13,14,14,14,13,
%U A035250 13,12,12,13,13,14,14,13,14,15,15,14,14,13,14,15,15,15,16,15,15,16,16
%N A035250 Number of primes between n and 2n (inclusive).
%C A035250 By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e. a(n) is positive for all n.
%C A035250 The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 01 2007
%D A035250 Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.
%H A035250 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b035250.txt">Table of n, a(n) for n=1..1000</a>
%H A035250 International Mathematics Olympiad, <a href="http://dmoz.org/Science/Math/Number_Theory/Prime_Numbers/">Proof of Bertrand's Postulate</a>
%e A035250 The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
%Y A035250 Adjacent sequences: A035247 A035248 A035249 this_sequence A035251 A035252 A035253
%Y A035250 Sequence in context: A120676 A125973 A001031 this_sequence A067743 A029230 A084294
%K A035250 nonn
%O A035250 1,2
%A A035250 Erich Friedman (erich.friedman(AT)stetson.edu)

%I A067743
%S A067743 0,1,2,2,2,2,2,3,2,4,2,4,2,4,2,4,2,5,2,4,4,4,2,6,2,4,4,4,2,6,2,5,4,4,2,
%T A067743 8,2,4,4,6,2,6,2,6,4,4,2,8,2,5,4,6,2,6,4,6,4,4,2,10,2,4,4,6,4,6,2,6,4,
%U A067743 6,2,9,2,4,6,6,2,8,2,8,4,4,2,10,4,4,4,6,2,10,2,6,4,4,4,10,2,5,4,8,2,8
%N A067743 Number of divisors of n not in the half-open interval [sqrt(n/2), sqrt(n*2)).
%D A067743 Problem 10847, Amer. Math. Monthly 109, (2002), p. 80.
%F A067743 a(n) = A000005(n) - A067742(n).
%F A067743 G.f.: sum(k=1,N, z^(2*k^2)*(1+z^k)/(1-z^k) ). - Joerg Arndt, May 12 2008
%F A067743 Direct proof of Joerg Arndt's g.f., from Max Alekseyev, May 13 2008 (Start):
%F A067743 We need to count divisors d|n such that d^2<=n/2 or d^2>2n. In the latter case, let's switch to co-divisor, replacing d by n/d.
%F A067743 Then we need to find the total count of: 1) divisors d|n such that 2d^2<=n; 2) divisors d|n such that 2d^2<n.
%F A067743 Let d|n and 2d^2<=n. Then n-2d^2 must be a multiple of d, i.e. n-2d^2=td for some integer t>=0.
%F A067743 Moreover it is easy to see that 1) is equivalent to n = 2d^2 + td for some integer t>=0. Therefore the answer for 1) is the coefficient of z^n in SUM[d=1..oo] SUM[t=0..oo] x^(2d^2 + td) = SUM[d=1..oo] x^(2d^2)/(1 - x^d).
%F A067743 Similarly, the answer for 2) is SUM[d=1..oo] x^(2d^2)/(1 - x^d) * x^d.
%F A067743 Therefore the g.f. for A067743 is SUM[d=1..oo] x^(2d^2)/(1 - x^d) + SUM[d=1..oo] x^(2d^2)/(1 - x^d) * x^d = SUM[d=1..oo] x^(2d^2)/(1 - x^d) * (1 + x^d), as proposed. (End)
%e A067743 a(6)=2 because 2 divisors of 6 (i.e. 1 and 6) fall outside sqrt(3) to sqrt(12).
%o A067743 (PARI from M. F. Hasler, May 12 2008) A067743(n)=sumdiv( n,d, d*d<n/2 || d*d >= 2*n )
%Y A067743 Cf. A067742, A000005.
%Y A067743 Adjacent sequences: A067740 A067741 A067742 this_sequence A067744 A067745 A067746
%Y A067743 Sequence in context: A125973 A001031 A035250 this_sequence A029230 A084294 A067752
%K A067743 easy,nonn,new
%O A067743 1,3
%A A067743 Marc LeBrun (mlb(AT)well.com), Jan 29 2002

%I A029230
%S A029230 1,0,1,0,1,0,1,1,1,2,2,2,2,2,3,2,4,3,5,4,6,5,6,6,7,7,8,
%T A029230 9,10,10,12,11,13,12,15,14,17,17,19,19,21,21,23,23,26,26,
%U A029230 29,29,32,32,35,35,38,38,42,42,46,46,50,50,54,54,58,59
%N A029230 Expansion of 1/((1-x^2)(1-x^7)(1-x^9)(1-x^10)).
%Y A029230 Adjacent sequences: A029227 A029228 A029229 this_sequence A029231 A029232 A029233
%Y A029230 Sequence in context: A001031 A035250 A067743 this_sequence A084294 A067752 A025422
%K A029230 nonn
%O A029230 0,10
%A A029230 njas

%I A084294
%S A084294 2,2,2,2,2,3,3,2,3,3,3,4,4,3,4,4,5,5,5,5,4,5,5,7,6,6,5,5,5,5,7,7,7,7,8,
%T A084294 7,8,9,8,8,7,7,9,8,9,8,9,11,10,10,11,10,10,9,10,11,10,9,9,9,8,10,11,11,
%U A084294 10,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,14,15,14,13
%N A084294 Number of primes in [p(n),n+p(n)] closed interval, where p(n) is the n-th prime.
%F A084294 a(n)=Pi[n+p(n)]=A000720(n+A000040(n))
%t A084294 t[x_] := Table[w, {w, Prime[x], x+Prime[x]}] Table[Count[PrimeQ[t[n]], True], {n, 1, 128}] or Table[PrimePi[n+Prime[n]]-n+1, {n, 1, 25}];
%Y A084294 Cf. A000040, A000720, A061067, A061068, A084295.
%Y A084294 Adjacent sequences: A084291 A084292 A084293 this_sequence A084295 A084296 A084297
%Y A084294 Sequence in context: A035250 A067743 A029230 this_sequence A067752 A025422 A078640
%K A084294 nonn
%O A084294 1,1
%A A084294 Labos E. (labos(AT)ana.sote.hu), May 27 2003

%I A067752
%S A067752 1,1,2,2,2,2,2,3,3,2,3,4,2,3,4,4,3,3,3,5,4,2,4,6,3,4,5,4,4,4,4,6,4,3,6,
%T A067752 7,2,4,6,6,5,4,3,7,6,3,6,8,4,5,6,5,4,6,6,9,4,2,7,8,4,5,8,7,6,6,3,8,6,4,
%U A067752 8,9,3,6,8,7,6,4,6,11,7,3,7,10,4,6,8,6,7
%N A067752 Number of unordered solutions of xy+xz+yz=n in nonnegative integers.
%C A067752 An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe (noe(AT)sspectra.com), Jun 14 2006
%H A067752 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b067752.txt">Table of n, a(n) for n = 1..10000</a>
%e A067752 a(12)=4 because of (0,1,12),(0,2,6),(0,3,4),(2,2,2).
%e A067752 a(20)=5 because of (0,1,20),(0,2,10),(0,4,5),(1,2,6),(2,2,4).
%t A067752 Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt++ ], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] - T. D. Noe (noe(AT)sspectra.com), Jun 14 2006
%Y A067752 Cf. A067751, A067753, A067754.
%Y A067752 Adjacent sequences: A067749 A067750 A067751 this_sequence A067753 A067754 A067755
%Y A067752 Sequence in context: A067743 A029230 A084294 this_sequence A025422 A078640 A006374
%K A067752 easy,nonn
%O A067752 1,3
%A A067752 Colin L. Mallows (colinm(AT)avaya.com), Jan 31 2002
%E A067752 Corrected, extended, and edited by John W. Layman (layman(AT)math.vt.edu, Dec 3 2004

%I A025422
%S A025422 1,1,1,1,2,2,2,2,2,3,3,2,3,4,3,3,4,4,5,4,5,6,6,4,5,7,6,6,7,8,8,7,6,8,9,7,
%T A025422 10,11,10,9,10,10,11,10,10,15,12,10,11,13,14,12,15,16,18,15,13,16,17,14,
%U A025422 16,20,17,18,16,18,21,18,19,23,24,18,21,21,22,22,23,26,25,24,21,27,27,23
%N A025422 Number of partitions of n into 7 squares.
%Y A025422 Adjacent sequences: A025419 A025420 A025421 this_sequence A025423 A025424 A025425
%Y A025422 Sequence in context: A029230 A084294 A067752 this_sequence A078640 A006374 A064876
%K A025422 nonn
%O A025422 0,5
%A A025422 David W. Wilson (davidwwilson(AT)comcast.net)

%I A078640
%S A078640 1,1,1,1,1,2,2,2,2,2,3,3,2,3,4,5,5,5,4,4,5,6,6,6,6,7,8,6,7,8,9,9,6,7,8,
%T A078640 11,10,8,9,9,11,11,10,10,11,14,13,12,11,12
%N A078640 Number of numbers between 1 and n-1 that are coprime to n(n+1)(n+2).
%C A078640 Does every integer appear?
%e A078640 a(10) is the number of numbers between 1 and 9 that are coprime to 10.11.12, which leaves 1 and 7, hence a(10)=2.
%o A078640 (PARI) newphi(v)=local(vl,fl,np); vl=length(v); np=0; for (s=1,v[1],fl=false; for (r=1,vl,if (gcd(s,v[r])>1,fl=true; break)); if (fl==false,np++)); np v=vector(3); for (i=1,50,v[1]=i; v[2]=i+1; v[3]=i+2; print1(newphi(v)","))
%Y A078640 Adjacent sequences: A078637 A078638 A078639 this_sequence A078641 A078642 A078643
%Y A078640 Sequence in context: A084294 A067752 A025422 this_sequence A006374 A064876 A105517
%K A078640 nonn
%O A078640 1,6
%A A078640 Jon Perry (perry(AT)globalnet.co.uk), Dec 12 2002

%I A006374 M0214
%S A006374 1,1,2,2,2,2,2,3,3,2,4,4,2,4,4,4,4,3,4,6,4,2,6,6,3,6,6,4,6,4,6,7,4,
%T A006374 4,8,8,2,6,8,6,8,4,4,10,6,4,10,8,5,7,8,6,6,8,8,12,4,2,12,8,6,8,10,8,
%U A006374 8,8,4,12,8,4,14,9,4,10,10,10,8,4,10,14,9,4,12,12,4,10,12,6,12,10,8
%N A006374 Number of reduced binary quadratic forms of determinant n.
%D A006374 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 360.
%D A006374 A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", Springer-Verlag, New York, 1986, p. 186.
%Y A006374 Cf. A006371, A006375, A096446, A096445.
%Y A006374 Adjacent sequences: A006371 A006372 A006373 this_sequence A006375 A006376 A006377
%Y A006374 Sequence in context: A067752 A025422 A078640 this_sequence A064876 A105517 A056813
%K A006374 nonn,nice,easy
%O A006374 1,3
%A A006374 njas

%I A064876
%S A064876 0,1,1,1,2,2,2,2,2,3,3,3,2,3,3,3,4,4,3,3,4,4,3,3,4,5,5,5,5,5,5,5,4,4,5,
%T A064876 5,6,6,6,6,6,5,5,5,6,6,6,6,4,7,7,7,6,7,7,7,6,7,7,7,7,6,6,6,8,8,8,7,8,8,
%U A064876 6,6,6,8,7,7,6,8,7,7,8,9,9,9,8,9,9,9,6,8,9,9,9,8,9,9,8,9,7,7,10,10,10
%N A064876 Last of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + A064874(n)^2 + A064875(n)^2 + a(n)^2.
%e A064876 a(18) = 3: 18 = A064873(18)^2 + A064874(18)^2 + A064875(18)^2 + a(18)^2 = 0 + 0 + 9 + 9 and the other decompositions (0, 1, 1, 4) and (1, 2, 2, 3) are greater than (0, 0, 3, 3).
%Y A064876 Cf. A064873, A064874, A064875, A064877.
%Y A064876 Adjacent sequences: A064873 A064874 A064875 this_sequence A064877 A064878 A064879
%Y A064876 Sequence in context: A025422 A078640 A006374 this_sequence A105517 A056813 A125269
%K A064876 nonn
%O A064876 0,5
%A A064876 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 10 2001

%I A105517
%S A105517 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,
%T A105517 1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,
%U A105517 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105517 a(n) = #{k: A008963(k) = 7 and 0<=k<=n};
%C A105517 a(A105507(n)) = a(A105507(n) - 1) + 1;
%C A105517 n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + a(n) + A105518(n) + A105519(n).
%Y A105517 Cf. A000030, A000045.
%Y A105517 Adjacent sequences: A105514 A105515 A105516 this_sequence A105518 A105519 A105520
%Y A105517 Sequence in context: A078640 A006374 A064876 this_sequence A056813 A125269 A077430
%K A105517 nonn,base
%O A105517 0,45
%A A105517 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005

%I A056813
%S A056813 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,
%T A056813 5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A056813 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A056813 Largest non-unitary prime factor of LCM[1,...,n]; i.e. the largest prime which occurs to power > 1 in prime factorization of LCM[1,..,n].
%C A056813 For n>0, p(n) appears {(p(n+1))^2 - (p(n))^2} times [from n=(p(n))^2 to n=(p(n+1))^2 - 1], i.e., A000040(n) appears A069482(n) times[from n=A001248(n) to n=A084920(n+1)] - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 31 2005
%F A056813 a(n)=p(w) if p(w)^2 <= n < p(w+1)^2.
%e A056813 The j-th prime appears at the position of its square, at n=P(j)^2.
%Y A056813 A056168, A056170.
%Y A056813 Adjacent sequences: A056810 A056811 A056812 this_sequence A056814 A056815 A056816
%Y A056813 Sequence in context: A006374 A064876 A105517 this_sequence A125269 A077430 A105513
%K A056813 nonn
%O A056813 0,4
%A A056813 Labos E. (labos(AT)ana.sote.hu), Aug 28 2000

%I A125269
%S A125269 1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A125269 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A125269 4,5,5,4,4,5,5,4,5,5,5,5,4,4,5,5,5,5,5,5,5,5,5,5,5,4,4,5,5,5,5,5,5,5,5
%N A125269 Minimal number of states required for a 2-symbol, 5-tuple Turing machine that takes n steps on an initially blank tape before halting.
%C A125269 If BB(n) = A060843(n), then a(BB(n)) = n, and that is the last occurrence of n in this sequence. a(n) will not become monotonic; if it did, we could compute BB(n), since a(n) is computable. a(n) diverges properly, but does so very slowly. The terms with values 1,2,3 were computed by exhaustive search. The terms with value 4 were inferred from knowing that they are greater than 3 and from the observation that for all n, a(n+1) <= a(n) + 1 (an easy construction). Using exhaustive search, may be able to extend the sequence to (most of) the terms up to and a bit beyond a(107) = 4, but going much further would likely require a more sophisticated method (see A052200).
%C A125269 If BB(n) = A060843(n), then a(BB(n)) = n, and that is the last occurrence of n in this sequence. a(n) will not become monotonic; if it did, we could compute BB(n), since a(n) is computable. a(n) diverges properly, but does so very slowly. a(n+1) <= a(n) + 1 (an easy construction). - Martin Fuller (martin_n_fuller(AT)btinternet.com), Feb 14 2007
%H A125269 Martin Fuller, <a href="http://www.research.att.com/~njas/sequences/b125269.txt">Table of n, a(n) for n = 1..626</a>
%H A125269 Martin Fuller, <a href="http://www.research.att.com/~njas/sequences/a125269.txt">Illustration file listing a Turing machine for each n</a>
%H A125269 H. Marxen, <a href="http://www.drb.insel.de/~heiner/BB/">Info on the Busy Beaver problem</a>
%Y A125269 Cf. A060843, A052200.
%Y A125269 Adjacent sequences: A125266 A125267 A125268 this_sequence A125270 A125271 A125272
%Y A125269 Sequence in context: A064876 A105517 A056813 this_sequence A077430 A105513 A004233
%K A125269 nice,nonn
%O A125269 1,2
%A A125269 Dustin Wehr (robert.wehr(AT)mail.mcgill.ca), Jan 16 2007, Jan 28 2007
%E A125269 More terms from Martin Fuller (martin_n_fuller(AT)btinternet.com), Feb 14 2007

%I A077430
%S A077430 1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A077430 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A077430 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A077430 Floor(Log10(2*n^2)) + 1.
%Y A077430 Cf. A077431, A077432, A004216, A077429, A077433.
%Y A077430 Adjacent sequences: A077427 A077428 A077429 this_sequence A077431 A077432 A077433
%Y A077430 Sequence in context: A105517 A056813 A125269 this_sequence A105513 A004233 A130259
%K A077430 nonn
%O A077430 1,3
%A A077430 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 05 2002

%I A105513
%S A105513 0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,
%T A105513 5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A105513 8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105513 a(n) = #{k: A008963(k) = 3 and 0<=k<=n};
%C A105513 a(A105503(n)) = a(A105503(n) - 1) + 1;
%C A105513 n = A105511(n) + A105512(n) + a(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105513 Cf. A000030, A000045.
%Y A105513 Adjacent sequences: A105510 A105511 A105512 this_sequence A105514 A105515 A105516
%Y A105513 Sequence in context: A056813 A125269 A077430 this_sequence A004233 A130259 A068549
%K A105513 nonn,base
%O A105513 0,10
%A A105513 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005

%I A004233
%S A004233 0,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,
%T A004233 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A004233 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A004233 ln(n) rounded up.
%Y A004233 Adjacent sequences: A004230 A004231 A004232 this_sequence A004234 A004235 A004236
%Y A004233 Sequence in context: A125269 A077430 A105513 this_sequence A130259 A068549 A132173
%K A004233 nonn
%O A004233 1,3
%A A004233 njas

%I A130259
%S A130259 0,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A130259 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A130259 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A130259 Maximal index k of an even Fibonacci number (A001906) such that A001906(k)=Fib(2k)<=n (the 'lower' even Fibonacci Inverse).
%C A130259 Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130260 for another version). a(n)+1 is the number of even Fibonacci numbers (A001906) <=n.
%F A130259 a(n)=floor(arsinh(sqr(5)*n/2)/(2*ln(phi))), where phi=(1+sqr(5))/2.
%F A130259 a(n)=A130260(n+1)-1.
%F A130259 G.f.: g(x)=1/(1-x)*sum(k>=1, x^Fib(2k)).
%F A130259 a(n)=floor(1/2*log_phi(sqr(5)*n+1)) for n>=0.
%e A130259 a(10)=3 because A001906(3)=8<=10, but A001906(4)=21>10.
%Y A130259 Cf. partial sums A130261. Other related sequences: A000045, A001519, A130233, A130237, A130239, A130255, A130260, A104160. Lucas inverse: A130241 - A130248.
%Y A130259 Adjacent sequences: A130256 A130257 A130258 this_sequence A130260 A130261 A130262
%Y A130259 Sequence in context: A077430 A105513 A004233 this_sequence A068549 A132173 A023968
%K A130259 nonn
%O A130259 0,4
%A A130259 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 25 2007, Jul 02 2007

%I A068549
%S A068549 2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%T A068549 5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A068549 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,11,11,11,11,11,11,11,11,11,11,11
%N A068549 Largest prime <= sqrt(2n-4) - 1.
%D A068549 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 19.
%Y A068549 Adjacent sequences: A068546 A068547 A068548 this_sequence A068550 A068551 A068552
%Y A068549 Sequence in context: A105513 A004233 A130259 this_sequence A132173 A023968 A000196
%K A068549 nonn
%O A068549 10,1
%A A068549 njas, Mar 22 2002

%I A132173
%S A132173 1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A132173 4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A132173 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6
%N A132173 Maternal generation number of A063882(n).
%H A132173 B. Balamohan, A. Kuznetsov and S. Tanny, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On the behavior of a variant of Hofstadter's Q-sequence</a>, J. Intger Sequences, Vol. 10 (2007), #07.7.1.
%H A132173 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst">Index entries for Hofstadter-type sequences</a>
%Y A132173 Adjacent sequences: A132170 A132171 A132172 this_sequence A132174 A132175 A132176
%Y A132173 Sequence in context: A004233 A130259 A068549 this_sequence A023968 A000196 A111850
%K A132173 nonn
%O A132173 1,5
%A A132173 njas, Nov 07 2007

%I A023968
%S A023968 0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A023968 4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A023968 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A023968 First digit after decimal point of 9-th root of n.
%t A023968 Array[ Function[ n, RealDigits[ N[ Power[ n, 1/9 ], 10 ], 10 ]// (#[ [ 1, #[ [ 2 ] ]+1 ] ])& ], 110 ]
%Y A023968 Adjacent sequences: A023965 A023966 A023967 this_sequence A023969 A023970 A023971
%Y A023968 Sequence in context: A130259 A068549 A132173 this_sequence A000196 A111850 A059396
%K A023968 nonn,base
%O A023968 1,6
%A A023968 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A000196
%S A000196 0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,
%T A000196 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,
%U A000196 8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10
%N A000196 Integer part of square root of n. Or, number of squares <= n. Or, n appears 2n+1 times.
%C A000196 Also the integer part of the geometric mean of the divisors of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 19 2001
%C A000196 a(n)=Card(k, 0<k<=n such that k is relatively prime to core(k)) where core(x) is the square-free part of x. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 02 2002
%C A000196 Number of numbers k (<=n) with an odd number of divisors - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2002
%D A000196 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 23.
%D A000196 K. Atanassov, On the 100-th, 101-st and the 102-th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 94-96.
%D A000196 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
%D A000196 N. J. A. Sloane and A. R. Wilks, On sequences of Recaman type, paper in preparation, 2006.
%D A000196 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
%H A000196 Franklin T. Adams-Watters, <a href="http://www.research.att.com/~njas/sequences/b000196.txt">Table of n, a(n) for n = 0..10000</a>
%H A000196 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of Smarandache's Problems</a>
%H A000196 H. Bottomley, <a href="http://www.research.att.com/~njas/sequences/a000196.gif">Illustration of A000196, A048760, A053186</a>
%H A000196 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.
%F A000196 a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 12 2004
%F A000196 a(n)=sum{0<k<=n, A010052(k)}. G.f.: g(x)=1/(1-x)*sum{j>=1, x^(j^2)}=(theta_3(0,x)-1)/(1-x)/2 where theta_3 is a Jacobi theta function. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 26 2007
%p A000196 Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
%o A000196 (MAGMA) [ Isqrt(n) : n in [0..100]];
%o A000196 (PARI) a(n)=floor(sqrt(n))
%o A000196 (PARI) a(n)=sqrtint(n)
%Y A000196 [A000267(n)/2]=A000196(n). Cf. A028391, A048766, A003056.
%Y A000196 Cf. A079051.
%Y A000196 Adjacent sequences: A000193 A000194 A000195 this_sequence A000197 A000198 A000199
%Y A000196 Sequence in context: A068549 A132173 A023968 this_sequence A111850 A059396 A108602
%K A000196 nonn,easy,nice
%O A000196 0,5
%A A000196 njas

%I A111850
%S A111850 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,
%T A111850 7,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%U A111850 11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,14,14,14,14,14
%N A111850 Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.
%C A111850 For n>1: if A023961(n)=0 then a(n)=a(n-1)+1 else a(n)=a(n-1).
%C A111850 a(n)/n --> 1/10.
%D A111850 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.
%e A111850 a(10) = 3, a(100) = 15, a(1000) = 118, a(10000) = 1050.
%Y A111850 Cf. A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111859, A111890.
%Y A111850 Adjacent sequences: A111847 A111848 A111849 this_sequence A111851 A111852 A111853
%Y A111850 Sequence in context: A132173 A023968 A000196 this_sequence A059396 A108602 A085290
%K A111850 nonn,base
%O A111850 1,4
%A A111850 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 20 2005

%I A059396
%S A059396 0,0,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%T A059396 5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,
%U A059396 7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9
%N A059396 Number of primes less than square root of n-th prime; i.e. number of trial divisions by smaller primes to show that n-th prime is indeed prime.
%C A059396 Perhaps close to 2*(n/loge(n))^(1/2)
%F A059396 a(n) = A000720(A000196(A000040(n)))
%e A059396 a(32) = 5 since the 32nd prime is 131 which is not divisible by 2, 3, 5, 7 or 11 (and does not need to be tested against 13, 17, 19 etc. since 13^2 = 169>131).
%Y A059396 Adjacent sequences: A059393 A059394 A059395 this_sequence A059397 A059398 A059399
%Y A059396 Sequence in context: A023968 A000196 A111850 this_sequence A108602 A085290 A108611
%K A059396 nonn
%O A059396 0,5
%A A059396 Henry Bottomley (se16(AT)btinternet.com), Jan 29 2001

%I A108602
%S A108602 0,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,4,4,5,5,5,5,5,5,5,5,
%T A108602 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,6,7,7,7,7,7,7,7,7,7,7,8,7,7,8,8,7,
%U A108602 8,8,8,8,8,8,8,8,8,8,8,8,9,8,9,8,9,8,9,9,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A108602 Number of distinct prime factors of highly composite numbers (definition 1, A002182).
%C A108602 n appears A086334(n) times. - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 02 2006
%F A108602 a(n) = A001221(A002182(n)).
%e A108602 A002182(8) = 48 = 2^4*3, which has 2 distinct prime factors, so a(8)=2.
%Y A108602 Cf. A002182, A002183.
%Y A108602 Adjacent sequences: A108599 A108600 A108601 this_sequence A108603 A108604 A108605
%Y A108602 Sequence in context: A000196 A111850 A059396 this_sequence A085290 A108611 A133875
%K A108602 nonn
%O A108602 1,4
%A A108602 Jud McCranie (j.mccranie(AT)comcast.net), Jun 12 2005
%E A108602 Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 11 2005

%I A085290
%S A085290 2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,8,8,8,
%T A085290 8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,
%U A085290 11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,16,16,16,16,16,16,16
%N A085290 Max[p1^b1] over all sorted multiplicative partitions of n! of length n.
%H A085290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Alladi-GrinsteadConstant.html">Alladi-Grinstead Constant</a>
%e A085290 6! = 2*2*2*2*5*9 = 2*2*3*3*4*5, the smallest terms of which are 2 and 2, so a(6)=Max[2,2]=2.
%o A085290 (PARI) works(n, m) = local(f, s, l, p, x); f = factor(n!); s = 0; l = matsize(f)[1]; for (i = 1, l, p = f[i, 1]; x = 1; while (p^x < m, x++); s += f[i, 2]\x; if (f[i, 2] < x, return(0))); s >= n; a(n) = local(f, m); f = factor(n); m = 2; while (works(n, m), m++); m - 1 (Wasserman)
%Y A085290 Cf. A085288, A085289, A085291.
%Y A085290 Cf. A103332.
%Y A085290 Adjacent sequences: A085287 A085288 A085289 this_sequence A085291 A085292 A085293
%Y A085290 Sequence in context: A111850 A059396 A108602 this_sequence A108611 A133875 A104355
%K A085290 nonn
%O A085290 4,1
%A A085290 Eric Weisstein (eric(AT)weisstein.com), Jun 23, 2003
%E A085290 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jan 31 2005

%I A108611
%S A108611 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,
%T A108611 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,
%U A108611 12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16
%N A108611 Excess of Beatty-function of 1/sin(1) over n.
%F A108611 a(n) = A108120[n] - n.
%Y A108611 Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120.
%Y A108611 Adjacent sequences: A108608 A108609 A108610 this_sequence A108612 A108613 A108614
%Y A108611 Sequence in context: A059396 A108602 A085290 this_sequence A133875 A104355 A092278
%K A108611 nonn
%O A108611 0,11
%A A108611 Zak Seidov (zakseidov(AT)yahoo.com), Jun 13 2005

%I A133875
%S A133875 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,
%T A133875 3,3,3,3,3,4,4,4,4,4,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,
%U A133875 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0,0,0,0,0,1,1,1,1,1
%N A133875 n modulo 5 repeated 5 times.
%C A133875 Periodic with length 5^2=25.
%F A133875 a(n)=(1+floor(n/5)) mod 5.
%F A133875 a(n)=A010874(A002266(n+5)).
%F A133875 a(n)=1+floor(n/5)-5*floor((n+5)/25).
%F A133875 a(n)=(((n+5) mod 25)-(n mod 5))/5.
%F A133875 a(n)=((n+5-(n mod 5))/5) mod 5.
%F A133875 a(n)=A010874((n+5-A010874(n))/5).
%F A133875 a(n)=binomial(n+5,n) mod 5 =binomial(n+5,5) mod 5.
%F A133875 G.f. g(x)=(1-x^5)(1+2x^5+3x^10+4x^15)/((1-x)(1-x^25)).
%F A133875 G.f. g(x)=(4x^25-5x^20+1)/((1-x)(1-x^5)(1-x^25)).
%Y A133875 Cf. A000040, A133620-A133625, A133630, A133633-A133636.
%Y A133875 Cf. A133885, A133880, A133890, A133900, A133910.
%Y A133875 Adjacent sequences: A133872 A133873 A133874 this_sequence A133876 A133877 A133878
%Y A133875 Sequence in context: A108602 A085290 A108611 this_sequence A104355 A092278 A105512
%K A133875 nonn
%O A133875 0,6
%A A133875 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 10 2007

%I A104355
%S A104355 0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,
%T A104355 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,
%U A104355 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A104355 Number of trailing zeros in decimal representation of A104350(n).
%C A104355 a(A104356(n)) = n and a(m) < n for m < A104356(n).
%Y A104355 Cf. A027868.
%Y A104355 Adjacent sequences: A104352 A104353 A104354 this_sequence A104356 A104357 A104358
%Y A104355 Sequence in context: A085290 A108611 A133875 this_sequence A092278 A105512 A002266
%K A104355 nonn,base
%O A104355 1,10
%A A104355 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 06 2005

%I A092278
%S A092278 0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,
%T A092278 6,7,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12,
%U A092278 12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16
%N A092278 Floor( (3*n+4)/16 ).
%D A092278 J. O'Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987, p. 82.
%Y A092278 Adjacent sequences: A092275 A092276 A092277 this_sequence A092279 A092280 A092281
%Y A092278 Sequence in context: A108611 A133875 A104355 this_sequence A105512 A002266 A075249
%K A092278 nonn
%O A092278 0,11
%A A092278 njas, Feb 18 2004

%I A105512
%S A105512 0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,
%T A105512 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,
%U A105512 12,12,13,13,13,13,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16
%N A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105512 a(n) = #{k: A008963(k) = 2 and 0<=k<=n};
%C A105512 a(A105502(n)) = a(A105502(n) - 1) + 1;
%C A105512 n = A105511(n) + a(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105512 Cf. A000030, A000045.
%Y A105512 Adjacent sequences: A105509 A105510 A105511 this_sequence A105513 A105514 A105515
%Y A105512 Sequence in context: A133875 A104355 A092278 this_sequence A002266 A075249 A008648
%K A105512 nonn,base
%O A105512 0,9
%A A105512 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005

%I A002266
%S A002266 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,
%T A002266 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,
%U A002266 12,12,12,12,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16
%N A002266 Integers repeated 5 times.
%C A002266 For n>3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000002 (see example). E.g. the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ....] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 08 2006
%C A002266 Complement of A010874, since A010874(n)+5*a(n)=n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
%F A002266 Floor(n/5), n>=0.
%F A002266 G.f.: x^5/((1-x)(1-x^5)).
%F A002266 a(n)= -1 + Sum_{k=0..n} {[8*(sin(2*Pi*k/5))^2-5]^2-5}/20, with n>=0. a(n)= -1 + Sum_{k=0..n} 1/50*{-9*[k mod 5]+[(n+1) mod 5]+[(n+2) mod 5]+[(n+3) mod 5]+11*[(n+4) mod 5]}, with n>=0. - Paolo P. Lava (ppl(AT)spl.at), May 15 2007
%F A002266 a(n)=(n-A010874(n))/5. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 29 2007
%F A002266 Also, floor(n^5-1/5n^4) will produce this sequence. Moreover, floor[(n^5-n^4)/(5n^4-4n^3)] will produce this sequence as well. - Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 08 2007
%Y A002266 Cf. A008648.
%Y A002266 a(n)=A010766(n,5).
%Y A002266 Cf. A004526, A002264, A002265, A010761, A010762, A110532, A110533.
%Y A002266 Partial sums: A130520. Other related sequences: A004526, A010872, A010873, A010874.
%Y A002266 Adjacent sequences: A002263 A002264 A002265 this_sequence A002267 A002268 A002269
%Y A002266 Sequence in context: A104355 A092278 A105512 this_sequence A075249 A008648 A105511
%K A002266 nonn,easy
%O A002266 0,11
%A A002266 njas

%I A075249
%S A075249 1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,
%T A075249 8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,14,13,
%U A075249 13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18
%N A075249 x-value of the solution (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075250 and A075251.
%C A075249 See A075248 for more details.
%F A075249 Is a(n) = A047252(n-3)-n+4 ? - Ralf Stephan, Feb 24 2004
%t A075249 For[xLst={}; yLst={}; zLst={}; n=3, n<=100, n++, cnt=0; xr=n/5; If[IntegerQ[xr], x=xr+1, x=Ceiling[xr]]; While[yr=1/(5/n-1/x); If[IntegerQ[yr], y=yr+1, y=Ceiling[yr]]; cnt==0&&y>x, While[zr=1/(5/n-1/x-1/y); cnt==0&&zr>y, If[IntegerQ[zr], z=zr; cnt++; AppendTo[xLst, x]; AppendTo[yLst, y]; AppendTo[zLst, z]]; y++ ]; x++ ]]; xLst
%Y A075249 Cf. A075248, A075250, A075251.
%Y A075249 Adjacent sequences: A075246 A075247 A075248 this_sequence A075250 A075251 A075252
%Y A075249 Sequence in context: A092278 A105512 A002266 this_sequence A008648 A105511 A027868
%K A075249 hard,nice,nonn
%O A075249 3,3
%A A075249 T. D. Noe (noe(AT)sspectra.com), Sep 10 2002

%I A008648
%S A008648 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,7,
%T A008648 7,7,7,7,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,15,15,
%U A008648 15,15,15,18,18,18,18,18,21,21,21,21,21,24,24,24,24,24
%N A008648 Molien series of 3 X 3 upper triangular matrices over GF( 5 ).
%D A008648 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008648 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=221">Encyclopedia of Combinatorial Structures 221</a>
%H A008648 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A008648 floor((n+4)/5), n>0.
%p A008648 1/(1-x)/(1-x^5)/(1-x^25)
%Y A008648 Cf. A002266.
%Y A008648 Adjacent sequences: A008645 A008646 A008647 this_sequence A008649 A008650 A008651
%Y A008648 Sequence in context: A105512 A002266 A075249 this_sequence A105511 A027868 A060384
%K A008648 nonn,easy
%O A008648 0,6
%A A008648 njas

%I A105511
%S A105511 0,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,7,7,7,7,8,9,9,9,9,10,10,10,
%T A105511 10,10,11,11,11,11,12,13,13,13,13,14,15,15,15,15,16,16,16,16,16,17,17,
%U A105511 17,17,17,18,18,18,18,19,20,20,20,20,21,22,22,22,22,23,23,23,23,23,24
%N A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105511 a(n) = #{k: A008963(k) = 1 and 0<=k<=n};
%C A105511 a(A105501(n)) = a(A105501(n) - 1) + 1;
%C A105511 n = a(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105511 Cf. A000030, A000045, A105501.
%Y A105511 Adjacent sequences: A105508 A105509 A105510 this_sequence A105512 A105513 A105514
%Y A105511 Sequence in context: A002266 A075249 A008648 this_sequence A027868 A060384 A105564
%K A105511 nonn,base
%O A105511 0,3
%A A105511 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005

%I A027868
%S A027868 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,6,6,6,6,6,7,7,7,
%T A027868 7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,12,12,12,12,12,13,13,13,13,
%U A027868 13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,18,18,18,18,18,19
%N A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.
%C A027868 Also the highest power of 10 dividing n! (different from A054899). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
%H A027868 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b027868.txt">Table of n, a(n) for n=0..1000</a>
%H A027868 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Link to a section of The World of Mathematics.</a>
%F A027868 Floor[n/5] + floor[n/25] + floor[n/125] + floor[n/625] + ....
%F A027868 Sum [ n/5^i ] from i=1 to infinity.
%F A027868 a(n)=(n-A053824(n))/4
%F A027868 G.f.: g(x)=sum{k>0, x^(5^k)/(1-x^(5^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
%F A027868 a(n)=sum{5<=k<=n, sum{j|k,j>=5, floor(log_5(j))-floor(log_5(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A027868 G.f.: g(x)=L[b(k)](x)/(1-x) where L[b(k)](x)=sum{k>=0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k)=1, if k>1 is a power of 5, else b(k)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A027868 G.f.: g(x)=sum{k>0, c(k)*x^k}/(1-x), where c(k)=sum{j>1,j|k, floor(log_5(j))-floor(log_5(j-1))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A027868 Recurrence: a(n)=floor(n/5)+a(floor(n/5)); a(5*n)=n+a(n); a(n*5^m)=n*(5^m-1)/4+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 a(k*5^m)=k*(5^m-1)/4, for 0<=k<5, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 Asymtotic behavior: a(n)=n/4+O(log(n)), a(n+1)-a(n)=O(log(n)), which follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 a(n)<=(n-1)/4; equality holds for powers of 5. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 a(n)>=n/4-1-floor(log_5(n)); equality holds for n=5^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 lim inf (n/4-a(n))=1/4, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 lim sup (n/4-log_5(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 lim sup (a(n+1)-a(n)-log_5(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
%F A027868 a(n) <= A027869(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 27 2008
%t A027868 Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
%Y A027868 See A000966 for the missing numbers. Cf. A011371 and A054861 for analogues involving powers of 2 and 3.
%Y A027868 Cf. A054899, A007953, A112765, A067080, A098844, A132027.
%Y A027868 Cf. A067080, A098844, A132029, A054999.
%Y A027868 Adjacent sequences: A027865 A027866 A027867 this_sequence A027869 A027870 A027871
%Y A027868 Sequence in context: A075249 A008648 A105511 this_sequence A060384 A105564 A025811
%K A027868 nonn,base,nice,easy
%O A027868 0,11
%A A027868 Warut Roonguthai (warut822(AT)yahoo.com)

%I A060384
%S A060384 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,
%T A060384 7,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,
%U A060384 13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18
%N A060384 Number of decimal digits in n-th Fibonacci number.
%H A060384 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/b060384.txt">Table of n, a(n) for n = 0..500</a>
%F A060384 a(n) = ceiling(n*ln(tau)/ln(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 29 2002
%F A060384 a(n)= floor(n*log10(gold)-log10(5)/2)+1 for n>=2, where gold is (1+sqrt(5))/2 - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
%p A060384 with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 19 2007
%o A060384 (PARI) print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
%Y A060384 Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.
%Y A060384 Adjacent sequences: A060381 A060382 A060383 this_sequence A060385 A060386 A060387
%Y A060384 Sequence in context: A008648 A105511 A027868 this_sequence A105564 A025811 A034258
%K A060384 base,nonn
%O A060384 0,8
%A A060384 Labos E. (labos(AT)ana.sote.hu), Apr 03 2001
%E A060384 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

%I A105564
%S A105564 0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,
%T A105564 8,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,13,
%U A105564 13,13,14,14,14,14,14,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18,18
%N A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.
%C A105564 a(n)/n --> 5 - 1/Log10((1+Sqrt(5))/2) = 0.215....
%C A105564 a(n) = Sum(A105563(k): 1<=k<=n); a(n) = n - A105566(n);
%D A105564 Juergen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhaeuser 2003).
%Y A105564 Adjacent sequences: A105561 A105562 A105563 this_sequence A105565 A105566 A105567
%Y A105564 Sequence in context: A105511 A027868 A060384 this_sequence A025811 A034258 A090663
%K A105564 nonn
%O A105564 1,8
%A A105564 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 14 2005

%I A025811
%S A025811 1,0,1,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,4,5,5,6,
%T A025811 6,6,6,7,7,8,8,8,9,9,10,10,10,11,11,12,12,13,13,14,14,15,
%U A025811 15,16,16,17,17,18,19,19,20,20,21,22,22,23,23,24,25,26
%N A025811 Expansion of 1/((1-x^2)(1-x^5)(1-x^11)).
%Y A025811 Adjacent sequences: A025808 A025809 A025810 this_sequence A025812 A025813 A025814
%Y A025811 Sequence in context: A027868 A060384 A105564 this_sequence A034258 A090663 A111890
%K A025811 nonn
%O A025811 0,11
%A A025811 njas

%I A034258
%S A034258 1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,8,8,8,9,9,10,
%T A034258 10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,15,15,15,15,15,16,
%U A034258 17,17,17,17,18,18,18,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22
%N A034258 Write n! as a product of n numbers, n = k(1)*k(2)*...*k(n) with k(1)<=k(2)<=..., in all possible ways; a(n) = max value of k(1).
%C A034258 36, 49, 52 and 55 are not in this sequence. - Don Reble (djr(AT)nk.ca), Nov 29 2001
%C A034258 a(n) >= a(n-1). - Larry Reeves (larryr(AT)acm.org), Jan 06 2005
%D A034258 R. K. Guy and J. L. Selfridge, Factoring factorial n, Amer. Math. Monthly, 105 (1998), 766-767.
%e A034258 3! = 6 = 1*2*3 is the only possible factorization, so a(3) = 1.
%e A034258 27! = 8^4 * 9^6 * 10^6 * 11^2 * 12 * 13^2 * 14^3 * 17 * 19 * 23, with 4 + 6 + 6 + 2 + 1 + 2 + 3 + 1 + 1 + 1 = 27 factors, which is the required number. Since the first factor is 8, a(27) >= 8. In fact no larger value can be obtained, and a(27) = 8.
%Y A034258 Cf. A034259, A034260.
%Y A034258 Adjacent sequences: A034255 A034256 A034257 this_sequence A034259 A034260 A034261
%Y A034258 Sequence in context: A060384 A105564 A025811 this_sequence A090663 A111890 A104277
%K A034258 nonn,nice
%O A034258 1,4
%A A034258 njas
%E A034258 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 12 2001
%E A034258 Verified by Don Reble (djr(AT)nk.ca), Apr 22 2007

%I A090663
%S A090663 2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,
%T A090663 9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,13,14,
%U A090663 14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,17,18,18
%N A090663 Second term in continued fraction for the n-th root of n.
%C A090663 The number of n's is: 5,4,4,5,4,5,5,5,5,5,5,6,5,6,5,6,6,5,6,6,6,6,...,
%t A090663 Table[ ContinuedFraction[n^(1/n), 2][[ -1]], {n, 2, 83}] (from Robert G. Wilson v Dec 22 2003)
%o A090663 (PARI) f(n) = for(x=2,n,a=contfrac(x^(1/x));print1(a[2]","))
%Y A090663 Adjacent sequences: A090660 A090661 A090662 this_sequence A090664 A090665 A090666
%Y A090663 Sequence in context: A105564 A025811 A034258 this_sequence A111890 A104277 A125893
%K A090663 nonn
%O A090663 2,1
%A A090663 Cino Hilliard (hillcino368(AT)gmail.com), Dec 15 2003

%I A111890
%S A111890 1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,
%T A111890 7,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%U A111890 11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,13,13,14,14,14,14,14,14
%N A111890 Number of numbers m <= n such that 0 equals the second digit after decimal point of square root of n in decimal representation.
%C A111890 For n>1: if A111862(n)=4 then a(n)=a(n-1)+1 else a(n)=a(n-1).
%C A111890 a(n)/n --> 1/10.
%D A111890 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.
%e A111890 a(10) = 3, a(100) = 15, a(1000) = 104, a(10000) = 1006.
%Y A111890 Cf. A111891, A111892, A111893, A111894, A111895, A111896, A111897, A111898, A111899, A111850.
%Y A111890 Adjacent sequences: A111887 A111888 A111889 this_sequence A111891 A111892 A111893
%Y A111890 Sequence in context: A025811 A034258 A090663 this_sequence A104277 A125893 A005857
%K A111890 nonn,base
%O A111890 1,4
%A A111890 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 20 2005

%I A104277
%S A104277 1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,8,8,10,10,11,11,13,13,14,
%T A104277 14,14,16,16,18,18,20,20,22,23,23,25,25,28,30,30,33,35,35,38,39,43,43,
%U A104277 46,46,49,51,51,55,56,60,61
%N A104277 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice odd squares.
%F A104277 Gf: product_{k>0}((1+x^(2k)^2))/(1-x^(2k-1)^2)).
%e A104277 E.g. a(21)=7 because we can write 21 as 18+2+1=16+4+1=16+2+2+1=9+4+2+2+2+2=9+2+2+2+2+2+2=4+2+2+2+2+2+2+2+2+1=2+2+2+2+2+2+2+2+2+2+1.
%p A104277 series(product((1+x^((2*k)^2))/(1-x^((2*k-1)^2)),k=1..100),x=0,100);
%Y A104277 Adjacent sequences: A104274 A104275 A104276 this_sequence A104278 A104279 A104280
%Y A104277 Sequence in context: A034258 A090663 A111890 this_sequence A125893 A005857 A025809
%K A104277 easy,nonn
%O A104277 0,5
%A A104277 Noureddine Chair (n.chair(AT)rocketmail.com), Mar 01 2005

%I A125893
%S A125893 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,9,9,10,11,12,13,
%T A125893 14,15,17,18,20,21,23,25,27,30,32,35,38,41,44,48,52,56,61,66,71,77,83,
%U A125893 90,98,106,115,124,134,146,158,171,185,200,216,234,254,275,297,322,348
%N A125893 Floor((Pi^4)/90)^n).
%t A125893 Table[Floor[(Pi^4/90)^n], {n, 1, 100}]
%Y A125893 Cf. A125890-A125899.
%Y A125893 Adjacent sequences: A125890 A125891 A125892 this_sequence A125894 A125895 A125896
%Y A125893 Sequence in context: A090663 A111890 A104277 this_sequence A005857 A025809 A114575
%K A125893 nonn
%O A125893 1,9
%A A125893 Artur Jasinski (grafix(AT)csl.pl), Dec 13 2006

%I A005857 M0216
%S A005857 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,8
%N A005857 The coding-theoretic function A(n,12,7).
%D A005857 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005857 E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/codes/Andw/">A(n,d,w) tables</a>
%H A005857 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#Andw">Index entries for sequences related to A(n,d,w)</a>
%Y A005857 Adjacent sequences: A005854 A005855 A005856 this_sequence A005858 A005859 A005860
%Y A005857 Sequence in context: A111890 A104277 A125893 this_sequence A025809 A114575 A131849
%K A005857 nonn,hard
%O A005857 7,7
%A A005857 njas
%E A005857 The version in the Encyclopedia of Integer Sequences had 1 instead of 2 at n=13.

%I A025809
%S A025809 1,0,1,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,5,4,5,5,6,6,6,
%T A025809 7,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,16,16,17,
%U A025809 17,18,19,19,20,20,22,22,23,23,24,25,26,26,27,28,29,30
%N A025809 Expansion of 1/((1-x^2)(1-x^5)(1-x^9)).
%Y A025809 Adjacent sequences: A025806 A025807 A025808 this_sequence A025810 A025811 A025812
%Y A025809 Sequence in context: A104277 A125893 A005857 this_sequence A114575 A131849 A090735
%K A025809 nonn
%O A025809 0,10
%A A025809 njas

%I A114575
%S A114575 1,1,2,2,2,2,2,3,3,3,4,3,2,4,2,4,4,1,5,2,3,2,4,2,3,3,3,4,3,3,2,5,5,3,7,
%T A114575 4,3,3,4,5,2,5,4,3,6,5,3,4,4,1,4,5,5,6,4,5,6,3,4,2,4,5,7,9,3,6,7,8,5,3,
%U A114575 5,7,5,5,7,3,5,6,6,6
%N A114575 Number of distinct prime factors of floor(e^n).
%e A114575 floor(e^3) = floor(20.08553) = 20. 20 has two distinct prime factors (2 and 5), therefore a(3) = 2.
%t A114575 Table[Length[FactorInteger[Floor[E^n]]], {n, 1, 80}]
%Y A114575 Cf. A001113 [decimal expansion of e].
%Y A114575 Adjacent sequences: A114572 A114573 A114574 this_sequence A114576 A114577 A114578
%Y A114575 Sequence in context: A125893 A005857 A025809 this_sequence A131849 A090735 A090736
%K A114575 nonn
%O A114575 0,3
%A A114575 Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 17 2006

%I A131849
%S A131849 0,0,2,2,2,2,2,3,3,3,4,4,4,4,4,4
%N A131849 Cardinality of largest subset of A=(1,...,n) such that the difference between any two elements of the subset is never one less than a prime.
%C A131849 Suppose that A is a subset of (1,...,N) is such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^(1/4))) for some absolute c>0.
%H A131849 Imre Z. Ruzsa, Tom Sanders, <a href="http://arxiv.org/pdf/0710.0644">Difference sets and the primes</a>, October 2, 2007.
%e A131849 a(1) is undefined because we cannot have a difference between two elements of a set of 1 element.
%e A131849 a(2) = 0 because the only subset of size =>2 of (1,2) is (1,2), and 2-1 = 1 is 1 less thasn the prime 2.
%e A131849 a(3) = 0 because the only subset of size =>2 of (1,2,3) are (1,2), and 2-1 = 1 is 1 less thasn the prime 2; (1,3), and 3-1 = 2 is 1 less thasn the prime 3; and (2,3) and and 3-2 = 1 is 1 less thasn the prime 2.
%e A131849 a(4) = 2 because (1,4) is the unique subset of (1,2,3,4) with the desired property that 4-1 = 3 is not 1 less than a prime.
%e A131849 a(9) = 3 because (1,4,9) is the unique subset of (1,2,3,4,5,6,7,8,9) with the desired property that 4-1 = 3 is not 1 less than a prime, and 9-1 = 8 is not 1 less than a prime, and 9-4 = 5 is not 1 less than a prime.
%e A131849 For n=9, 10 and 11, the cardinality is limited to 3 (the subset {1,4,9}). For
%e A131849 12 <= n <= 17, the cardinality is limited to 4 (the subset {1,4,9,12}).
%Y A131849 Cf. A000040.
%Y A131849 Adjacent sequences: A131846 A131847 A131848 this_sequence A131850 A131851 A131852
%Y A131849 Sequence in context: A005857 A025809 A114575 this_sequence A090735 A090736 A094999
%K A131849 more,nonn
%O A131849 2,3
%A A131849 Jonathan Vos Post (jvospost2(AT)yahoo.com), Oct 04 2007
%E A131849 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 15 2008

%I A090735
%S A090735 0,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,7,7,7,7,7,8,8,
%T A090735 8,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,
%U A090735 13,13,13,13,14,14,14,14,14,14,14,14,15,16,16,16,16,16,16,16,16,17,17
%N A090735 Number of positive square-free numbers <=n that can be expressed as a sum of 2 squares >0.
%D A090735 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 100
%F A090735 a(n) is asymptotic to (6K/Pi^2)*n/sqrt(log(x)) where K is the Landau-Ramanujan constant
%o A090735 (PARI) a(n)=sum(i=1,n,issquarefree(i)*if(sum(u=1,i,sum(v=1,u,if(u^2+v^2-i,0,1))),1,0))
%Y A090735 Cf. A064533.
%Y A090735 Adjacent sequences: A090732 A090733 A090734 this_sequence A090736 A090737 A090738
%Y A090735 Sequence in context: A025809 A114575 A131849 this_sequence A090736 A094999 A120202
%K A090735 nonn
%O A090735 1,5
%A A090735 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 18 2004

%I A090736
%S A090736 0,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,9,9,
%T A090736 9,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,13,13,13,13,13,14,14,
%U A090736 14,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,18,18,19
%N A090736 Number of positive integers <=n that can be expressed as a sum of 2 coprime squares >0.
%D A090736 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 100
%F A090736 a(n) is asymptotic to (3/8/K)*n/sqrt(log(n)) where K is the Landau-Ramanujan constant
%o A090736 (PARI) a(n)=sum(i=1,n,if(sum(u=1,i,sum(v=1,u,if(abs(u^2+v^2-i)+abs(gcd(u,v)-1),0,1))),1,0))
%Y A090736 Cf. A064533.
%Y A090736 Adjacent sequences: A090733 A090734 A090735 this_sequence A090737 A090738 A090739
%Y A090736 Sequence in context: A114575 A131849 A090735 this_sequence A094999 A120202 A005861
%K A090736 nonn
%O A090736 1,5
%A A090736 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 18 2004

%I A094999
%S A094999 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5,5,6,6,7,8,8,9,10,11,12,13,14,
%T A094999 16,17,19,21,22,25,27,29,32,35,38,42,45,50,54,59,65,71,77,84,92,100,109,
%U A094999 119,130,142,155,169,185,201,220,240,262,285,311,340,371,404,441,481
%N A094999 [ 12^n / 11^n ].
%t A094999 Table[ Floor[(12/11)^n], {n, 0, 71}]
%Y A094999 Cf. A002379, A094969 - A094500.
%Y A094999 Adjacent sequences: A094996 A094997 A094998 this_sequence A095000 A095001 A095002
%Y A094999 Sequence in context: A131849 A090735 A090736 this_sequence A120202 A005861 A025788
%K A094999 easy,nonn
%O A094999 0,9
%A A094999 Robert G. Wilson v (rgwv(AT)rgwv.com), May 26 2004

%I A120202
%S A120202 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9,10,11,12,13,15,
%T A120202 16,18,20,22,25,28,31,34,38,42,47,52,58,64,71,79,88,98,109,121,134,149,
%U A120202 166,184,205,227,253,281,312,347,385,428,476,528,587,652,725,805,895
%N A120202 a(n)=ceiling( sum_{i=1..n-1} a(i)/8), a(1)=1.
%t A120202 f[s_] := Append[s, Ceiling[Plus @@ s/9]]; Nest[f, {1}, 70] (* Robert G. Wilson v *)
%Y A120202 Cf. A072493, A112088, A072493, A011782, A073941, A072493, A120160, A120170, A120178, A120186, A120194.
%Y A120202 Adjacent sequences: A120199 A120200 A120201 this_sequence A120203 A120204 A120205
%Y A120202 Sequence in context: A090735 A090736 A094999 this_sequence A005861 A025788 A071806
%K A120202 nonn
%O A120202 1,11
%A A120202 Graeme McRae (g_m(AT)mcraefamily.com), Jun 10 2006
%E A120202 Edited and extended by Robert G. Wilson v (rgwv(at)rgwv.com), Jul 07 2006

%I A005861 M0215
%S A005861 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,5,6,6,7
%N A005861 The coding-theoretic function A(n,14,9).
%D A005861 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005861 E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/codes/Andw/">A(n,d,w) tables</a>
%H A005861 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#Andw">Index entries for sequences related to A(n,d,w)</a>
%Y A005861 Adjacent sequences: A005858 A005859 A005860 this_sequence A005862 A005863 A005864
%Y A005861 Sequence in context: A090736 A094999 A120202 this_sequence A025788 A071806 A025781
%K A005861 nonn,hard
%O A005861 9,8
%A A005861 njas

%I A025788
%S A025788 1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,7,8,
%T A025788 8,9,9,9,10,10,11,11,12,13,13,14,14,15,15,16,17,17,18,18,
%U A025788 19,20,21,22,22,23,23,24,25,26,27,27,28,29,30,31,32,33
%N A025788 Expansion of 1/((1-x)(1-x^7)(1-x^12)).
%Y A025788 Adjacent sequences: A025785 A025786 A025787 this_sequence A025789 A025790 A025791
%Y A025788 Sequence in context: A094999 A120202 A005861 this_sequence A071806 A025781 A018119
%K A025788 nonn
%O A025788 0,8
%A A025788 njas

%I A071806
%S A071806 0,0,0,0,2,2,2,2,2,3,3,4,4,4,4,4,5,6,7,7,7,8,9,9,9,9,9,9,9,9,9,9,10,11,
%T A071806 12,12,12,15,18,18,19,19,20,20,20,20,20,20,20,21,21,22,22,23,23,23,23,
%U A071806 23,23,24,24,24,25,26,26,26,26,28,28,29,29,29,29,29,30,31,31,31,31,31
%N A071806 Number of pairs (x,y) such that prime(x) + prime(y) = y*tau(x) + x*tau(y), 1<=x<=y<=n.
%o A071806 (PARI) for(n=1,130,print1(sum(i=1,n,sum(j=1,i,if(prime(i)+prime(j)-j*numdiv(i)-i*numdiv(j),0,1))),","))
%Y A071806 Cf. A000005.
%Y A071806 Adjacent sequences: A071803 A071804 A071805 this_sequence A071807 A071808 A071809
%Y A071806 Sequence in context: A120202 A005861 A025788 this_sequence A025781 A018119 A120502
%K A071806 nonn
%O A071806 1,5
%A A071806 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 06 2002

%I A025781
%S A025781 1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9,10,10,
%T A025781 11,11,12,13,13,14,14,15,16,17,18,18,19,20,21,22,22,23,
%U A025781 24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,40,41,42
%N A025781 Expansion of 1/((1-x)(1-x^5)(1-x^12)).
%Y A025781 Adjacent sequences: A025778 A025779 A025780 this_sequence A025782 A025783 A025784
%Y A025781 Sequence in context: A005861 A025788 A071806 this_sequence A018119 A120502 A099480
%K A025781 nonn
%O A025781 0,6
%A A025781 njas

%I A018119
%S A018119 1,2,2,2,2,2,3,3,4,4,4,5,6,7,7,8,10,11,13,14,16,19,22,25,
%T A018119 28,32,37,43,49,56,64,74,85,98,112,128,148,169,195,223,
%U A018119 256,295,338,389,446,512,589,676,777,892,1024,1177,1352
%N A018119 Powers of fifth root of 2 rounded up.
%Y A018119 Adjacent sequences: A018116 A018117 A018118 this_sequence A018120 A018121 A018122
%Y A018119 Sequence in context: A025788 A071806 A025781 this_sequence A120502 A099480 A025783
%K A018119 nonn
%O A018119 0,2
%A A018119 njas

%I A120502
%S A120502 1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,5,6,6,7,8,8,8,8,8,8,8,9,10,10,11,12,12,
%T A120502 12,13,14,14,15,16,16,16,16,16,16,16,16,17,18,18,19,20,20,20,21,22,22,
%U A120502 23,24,24,24
%N A120502 Meta-fibonacci sequence a(n) with parameters s=3.
%D A120502 B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
%H A120502 C. Deugau and F. Ruskey, <a href="http://www.cs.uvic.ca/~ruskey/Publications/MetaFib/GenMetaFib.html">Complete k-ary Trees and Generalized Meta-Fibonacci Sequences</a>
%F A120502 If 1 <= n <= 4, a(n)=1. If n = 5, then a(n)=2. If n>5 then a(n)=a(n-3-a(n-1)) + a(n-4-a(n-2))
%F A120502 g.f.: A(z) = z * (1 - z^3) / (1 - z) * sum(prod(z^3 * (1 - z^(2 * [i])) / (1 - z^[i]), i=1..n), n=0..infinity), where [i] = (2^i - 1).
%p A120502 a := proc(n)
%p A120502 option remember;
%p A120502 if n <= 4 then return 1 end if;
%p A120502 if n <= 5 then return 2 end if;
%p A120502 return add(a(n - i - 2 - a(n - i)), i = 1 .. 2)
%p A120502 end proc
%Y A120502 Cf. A120513, A120524.
%Y A120502 Adjacent sequences: A120499 A120500 A120501 this_sequence A120503 A120504 A120505
%Y A120502 Sequence in context: A071806 A025781 A018119 this_sequence A099480 A025783 A025780
%K A120502 nonn
%O A120502 1,5
%A A120502 Frank Ruskey (http://www.cs.uvic.ca/~ruskey/) and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006

%I A099480
%S A099480 1,2,2,2,2,2,3,4,4,4,4,4,5,6,6,6,6,6,7,8,8,8,8,8,9,10,10,10,10,10,11,12,
%T A099480 12,12,12,12,13,14,14,14,14,14,15,16,16,16,16,16,17,18,18,18,18,18,19,
%U A099480 20,20,20,20,20,21,22,22,22,22,22,23,24,24,24,24,24,25,26,26,26,26,26
%N A099480 Count from 1, repeating 2n five times.
%C A099480 Could be called the Jones sequence of the knot 9_43, since the g.f. is the reciprocal of (a parameterisation of) the Jones polynomial for 9_43.
%F A099480 G.f.: 1/((1-x+x^2)(1-x-x^3+x^4))=1/(1-2x+2x^2-2x^3+2x^4-2x^5+x^6); a(n)=2a(n-1)-2a(n-2)+2a(n-3)-2a(n-4)+2a(n-5)-a(n-6); a(n)=-cos(pi*2n/3+pi/3)/6+sqrt(3)sin(pi*2n/3+pi/3)/18-sqrt(3)cos(pi*n/3+pi/6)/6 +sin(pi*n/3+pi/6)/2+(n+3)/3.
%Y A099480 Cf. A099479.
%Y A099480 Adjacent sequences: A099477 A099478 A099479 this_sequence A099481 A099482 A099483
%Y A099480 Sequence in context: A025781 A018119 A120502 this_sequence A025783 A025780 A109697
%K A099480 easy,nonn
%O A099480 0,2
%A A099480 Paul Barry (pbarry(AT)wit.ie), Oct 18 2004

%I A025783
%S A025783 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,5,6,6,6,6,7,8,9,9,9,
%T A025783 9,10,11,12,12,12,13,14,15,16,16,16,17,18,19,20,20,21,22,
%U A025783 23,24,25,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39
%N A025783 Expansion of 1/((1-x)(1-x^6)(1-x^11)).
%Y A025783 Adjacent sequences: A025780 A025781 A025782 this_sequence A025784 A025785 A025786
%Y A025783 Sequence in context: A018119 A120502 A099480 this_sequence A025780 A109697 A103373
%K A025783 nonn
%O A025783 0,7
%A A025783 njas

%I A025780
%S A025780 1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,6,6,6,6,7,8,9,9,9,10,11,
%T A025780 12,12,12,13,14,15,16,16,17,18,19,20,20,21,22,23,24,25,
%U A025780 26,27,28,29,30,31,32,33,34,35,37,38,39,40,41,43,44,45
%N A025780 Expansion of 1/((1-x)(1-x^5)(1-x^11)).
%Y A025780 Adjacent sequences: A025777 A025778 A025779 this_sequence A025781 A025782 A025783
%Y A025780 Sequence in context: A120502 A099480 A025783 this_sequence A109697 A103373 A038539
%K A025780 nonn
%O A025780 0,6
%A A025780 njas

%I A109697
%S A109697 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,6,7,7,7,8,10,11,12,12,13,15,17,18,19,
%T A109697 20,23,26,28,29,31,34,38,41,43,45,50,55,60,63,66,71,79,85,90,94,101,110,
%U A109697 120,127,133,141,153,165,176,184,195,210,227,241,254,267,286,307,327
%N A109697 Number of partitions of n into parts each equal to 1 mod 5.
%F A109697 G.f.=1/product(1-x^(1+5j), j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%e A109697 a(11)=3 since 11 = 11 = 6+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1
%p A109697 g:=1/product(1-x^(1+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%Y A109697 Adjacent sequences: A109694 A109695 A109696 this_sequence A109698 A109699 A109700
%Y A109697 Sequence in context: A099480 A025783 A025780 this_sequence A103373 A038539 A109368
%K A109697 nonn
%O A109697 0,7
%A A109697 Erich Friedman (efriedma(AT)stetson.edu), Aug 07 2005
%E A109697 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006

%I A103373
%S A103373 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,7,8,8,8,9,12,15,16,16,17,21,27,31,32,
%T A103373 33,38,48,58,63,65,71,86,106,121,128,136,157,192,227,249,264,293,349,
%U A103373 419,476,513,557,642,768,895,989,1070,1199,1410,1663,1884,2059,2269
%N A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).
%C A103373 k=5 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1) and k=4 case is A103372.
%C A103373 The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
%C A103373 For this k=5 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^6 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.134724138401519492605446054506472840279667226382801485925149551668236893999842671279689011614820249
%C A103373 The sequence of prime values in this k=5 case is A103383; The sequence of semiprime values in this k=5 case is A103393.
%D A103373 Selmer, E.S., "On the irreducibility of certain trinomials", Math. Scand., 4 (1956) 287-302
%D A103373 Shallit, J., "A generalization of automatic sequences", Theoretical Computer Science, 61(1988)1-16.
%D A103373 Zanten, A. J. van, "The golden ratio in the arts of painting, building, and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.
%H A103373 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103373 J.-P. Allouche and T. Johnson, <a href="http://www.lri.fr/~allouche/johnson2.pdf">Narayana's Cows and Delayed Morphisms</a>
%e A103373 a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.
%t A103373 k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]
%Y A103373 Cf. A000045, A000931, A079398, A103372-A103381, A103383, A103393.
%Y A103373 Adjacent sequences: A103370 A103371 A103372 this_sequence A103374 A103375 A103376
%Y A103373 Sequence in context: A025783 A025780 A109697 this_sequence A038539 A109368 A046774
%K A103373 nonn,easy
%O A103373 1,7
%A A103373 Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 03 2005
%E A103373 Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2005

%I A038539
%S A038539 1,0,1,1,0,1,1,1,2,2,2,2,2,3,4,4,4,5,5,5,8,8,7,12,12,9,13,15,15,19,
%T A038539 21,21,23,25,28,34,35,37,45,45,45,56,59,61,77,80,76,92,100,101,119,
%U A038539 130,133,147,159,170,188,200,216,243,251,260,298,317,329,379,400
%N A038539 Complex semisimple Lie algebras of dimension n.
%C A038539 Direct consequence of classification of complex finite-dimensional simple Lie algebras
%D A038539 N. Jacobson, Lie Algebras, Dover Publications.
%F A038539 G.f.: (1+x)/((1 - x^14)(1 - x^52)(1 - x^78)(1-x^133)(1 - x^248) prod( 1-x^(n^2 + 2n), n = 1..inf) prod(1 - x^(2n^2 + n), n=2..inf) prod(1-x^(2n^2+n), n=3..inf) prod( 1-x^(2n^2 - n), n=4..inf)).
%Y A038539 Adjacent sequences: A038536 A038537 A038538 this_sequence A038540 A038541 A038542
%Y A038539 Sequence in context: A025780 A109697 A103373 this_sequence A109368 A046774 A029105
%K A038539 nonn,easy,nice
%O A038539 1,9
%A A038539 Paolo Dominici (pl.dm(AT)libero.it)

%I A109368
%S A109368 1,1,1,1,1,2,2,2,2,2,3,4,4,5,5,6,7,8,9,10,11,12,14,16,18,20,22,24,27,30,
%T A109368 34,37,40,44,49,54,60,65,71,78,85,94,103,112,122,132,144,158,172,186,
%U A109368 201,218,237,258,279,302,326,352,381,412,445,480,516,556,599,646
%N A109368 Expansion of q^(-1/2)eta(q^2)*eta(q^3)*eta(q^7)*eta(q^42)/(eta(q)*eta(q^6)*eta(q^14)*eta(q^21)) in powers of q.
%F A109368 Euler transform of period 42 sequence [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, ...].
%F A109368 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^2+v^3-v*w^2-v*u^2+v^2*w^2+v^2*u^2-u^2*w^2-v*u^2*w^2.
%F A109368 G.f.: Product_{k>0} (1+x^k)(1+x^(21k))/((1+x^(3k))(1+x^(7k))) = Product_{k>0} P42(x^k) where P42 is the 42nd cyclotomic polynomial.
%o A109368 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)*eta(x^7+A)*eta(x^42+A)/ (eta(x+A)*eta(x^6+A)*eta(x^14+A)*eta(x^21+A)), n))}
%Y A109368 Adjacent sequences: A109365 A109366 A109367 this_sequence A109369 A109370 A109371
%Y A109368 Sequence in context: A109697 A103373 A038539 this_sequence A046774 A029105 A079954
%K A109368 nonn
%O A109368 0,6
%A A109368 Michael Somos, Jun 26 2005

%I A046774
%S A046774 1,1,1,1,1,1,2,2,2,2,2,3,4,4,5,5,6,7,8,12,13,14,16,17,28,33,35,37,40,
%T A046774 61,77,83,87,94,132,168,186,194,213,277,350,392,414,460,569,703,793,
%U A046774 843,953,1139,1375,1550,1663,1894,2226,2628,2952,3187,3655,4249,4932
%N A046774 Number of partitions of n with equal number of parts congruent to each of 0, 2, 3 and 4 (mod 5).
%Y A046774 Adjacent sequences: A046771 A046772 A046773 this_sequence A046775 A046776 A046777
%Y A046774 Sequence in context: A103373 A038539 A109368 this_sequence A029105 A079954 A079629
%K A046774 nonn
%O A046774 0,7
%A A046774 David W. Wilson (davidwwilson(AT)comcast.net)

%I A029105
%S A029105 1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,6,7,8,8,8,9,10,12,13,14,
%T A029105 15,16,18,19,20,21,22,24,26,28,30,32,34,36,38,40,42,44,
%U A029105 46,49,52,55,58,61,64,67,70,73,76,79,83,87,91,95,99,104
%N A029105 Expansion of 1/((1-x)(1-x^5)(1-x^11)(1-x^12)).
%Y A029105 Adjacent sequences: A029102 A029103 A029104 this_sequence A029106 A029107 A029108
%Y A029105 Sequence in context: A038539 A109368 A046774 this_sequence A079954 A079629 A029116
%K A029105 nonn
%O A029105 0,6
%A A029105 njas

%I A079954
%S A079954 0,1,2,2,2,2,2,3,4,5,6,7,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%T A079954 10,10,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,
%U A079954 32,33,34,35,36,37,38,39,40,41,42,42,42,42,42,42,42,42,42,42,42,42,42,42
%N A079954 Partial sums of A030301.
%F A079954 a(n) = (n-1-(2/3)*(4^e_4-1)-(-1)^e_2*(n-1-2*(4^e_4-1)))/2 where e_4=floor(log[4](n)) and e_2=floor(log[2](n))=floor(log[4](n^2)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
%Y A079954 Adjacent sequences: A079951 A079952 A079953 this_sequence A079955 A079956 A079957
%Y A079954 Sequence in context: A109368 A046774 A029105 this_sequence A079629 A029116 A064770
%K A079954 nonn
%O A079954 1,3
%A A079954 njas, Feb 22 2003

%I A079629
%S A079629 2,2,2,2,2,3,5,3,7,6,6,10,13,7,8,9,9,7,12,18,14,24,19,10,21,21,20,20,19,
%T A079629 22,19,24,24,27,25,30,27,23,34,29,21,35,38,30,32,30,33,36,33,30
%N A079629 Number of twin prime pairs between p^2 and q^2 where (p,q) is the n-th twin prime pair.
%C A079629 Conjecturally a(n) is always positive. It seems that a(n) might tend to infinity.
%C A079629 a(n) = A071538(A006512(n)^2) - A071538(A001359(n)^2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2008
%e A079629 a(3)=2 because the third twin prime pair is (11,13) and there are 2 twin prime pairs between 121 and 169, namely (137,139) and (149,151).
%Y A079629 Cf. A057767.
%Y A079629 Cf. A137859, A137860.
%Y A079629 Adjacent sequences: A079626 A079627 A079628 this_sequence A079630 A079631 A079632
%Y A079629 Sequence in context: A046774 A029105 A079954 this_sequence A029116 A064770 A060467
%K A079629 easy,nonn
%O A079629 1,1
%A A079629 Paul Boddington (psb(AT)maths.warwick.ac.uk), Jan 30 2003

%I A029116
%S A029116 1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,6,8,8,8,8,9,11,14,14,
%T A029116 14,14,15,17,20,20,20,21,23,26,30,30,30,31,33,36,40,40,
%U A029116 41,43,46,50,55,55,56,58,61,65,70,71,73,76,80,85,91,92
%N A029116 Expansion of 1/((1-x)(1-x^6)(1-x^11)(1-x^12)).
%Y A029116 Adjacent sequences: A029113 A029114 A029115 this_sequence A029117 A029118 A029119
%Y A029116 Sequence in context: A029105 A079954 A079629 this_sequence A064770 A060467 A125918
%K A029116 nonn
%O A029116 0,7
%A A029116 njas

%I A064770
%S A064770 0,1,1,1,2,2,2,2,2,3,10,11,11,11,12,12,12,12,12,13,10,11,11,11,12,12,
%T A064770 12,12,12,13,10,11,11,11,12,12,12,12,12,13,20,21,21,21,22,22,22,22,
%U A064770 22,23,20,21,21,21,22,22,22,22,22,23,20,21,21,21,22,22,22,22,22,23
%N A064770 Replace each digit of n by the floor of its square root.
%C A064770 The graph of this sequence is fractal-like.
%C A064770 a(A007088(n))=A007088(n); a(A136399(n))<>A136399(n); a(a(n))=A136400(n); a(A136400(n))=A136400(n); A136428(n)=a(n+1)-a(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2007
%H A064770 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/b064770.txt">Table of n, a(n) for n = 0..9999</a>
%H A064770 O. Gerard, <a href="http://www.seqfan.net/spadaro/A064770-1.gif">Fractal behavior of this sequence (1)</a>
%H A064770 O. Gerard, <a href="http://www.seqfan.net/spadaro/A064770-2.gif">Fractal behavior of this sequence (2)</a>
%e A064770 26 -> [1.414...][2.449...] -> 12, so a(26) = 12.
%t A064770 Table[ FromDigits[ Floor[ Sqrt[ IntegerDigits[ n]]]], {n, 0, 100} ]
%Y A064770 Adjacent sequences: A064767 A064768 A064769 this_sequence A064771 A064772 A064773
%Y A064770 Sequence in context: A079954 A079629 A029116 this_sequence A060467 A125918 A083533
%K A064770 base,nonn,nice
%O A064770 0,5
%A A064770 Santi Spadaro (spados(AT)katamail.com), Oct 19 2001

%I A060467
%S A060467 0,1,1,1,2,2,2,2,2,3,11,2,1626,2,3,3,3,16,2,3,3,3,3,3
%V A060467 0,1,1,1,2,2,2,2,2,3,-11,2,1626,2,3,3,3,16,2,3,3,3,3,3
%N A060467 Consider solutions to n = x^3 + y^3 + z^3 (for n not 4 or 5 mod 9) with 0 <= |x| <= |y| <= |z|; take solution with smallest |z| and smallest |y|; sequence give value of z.
%C A060467 Indexed by A060464.
%D A060467 R. K. Guy, Unsolved Problems in Number Theory, Section D5.
%H A060467 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm">About n=x^3+y^3+z^3</a>
%e A060467 For n=16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term 1626.
%Y A060467 Cf. A060465-A060466.
%Y A060467 Adjacent sequences: A060464 A060465 A060466 this_sequence A060468 A060469 A060470
%Y A060467 Sequence in context: A079629 A029116 A064770 this_sequence A125918 A083533 A076500
%K A060467 sign,nice,hard
%O A060467 0,5
%A A060467 njas, Apr 10 2001

%I A125918
%S A125918 1,1,0,1,1,2,2,2,2,2,4,1,1,1
%N A125918 Sprague-Grundy values for octal game .154.
%C A125918 The sequence is eventually periodic with period 11. The last exception is at n=3.
%D A125918 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 104.
%Y A125918 Adjacent sequences: A125915 A125916 A125917 this_sequence A125919 A125920 A125921
%Y A125918 Sequence in context: A029116 A064770 A060467 this_sequence A083533 A076500 A060594
%K A125918 nonn
%O A125918 1,6
%A A125918 Richard Sabey (richardsabey(AT)hotmail.co.uk), Jan 24 2007

%I A083533
%S A083533 1,2,2,2,2,2,4,2,2,2,2,4,2,2,4,4,2,2,2,2,4,2,2,2,2,4,2,4,2,6,2,2,2,4,4,
%T A083533 4,4,2,2,2,2,2,2,4,4,6,2,2,2,4,2,2,4,4,2,6,4,2,2,2,2,4,4,2,2,4,6,2,4,2,
%U A083533 2,4,4,2,2,4,4,2,2,2,2,4,6,2,10,2,4,4,2,2,4,2,2,4,4,2,6,4,2,2,4,6,4,2,4
%N A083533 First difference sequence of A002202. Difference between consecutive possible values for phi[n].
%F A083533 a(n)=A002202[n+1]-A02202[n]
%e A083533 12 and 16 are the 7th and 8th possible totient values
%e A083533 12=phi[13],16=phi[17],
%e A083533 while {13,14,15} are impossible ones;
%e A083533 thus 16-12=4=a(7)=A002202[8]-A002202[7].
%t A083533 t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
%Y A083533 Cf. A000010, A002202, A005277, A083531-A083536, A005277.
%Y A083533 Adjacent sequences: A083530 A083531 A083532 this_sequence A083534 A083535 A083536
%Y A083533 Sequence in context: A064770 A060467 A125918 this_sequence A076500 A060594 A104361
%K A083533 nonn
%O A083533 1,2
%A A083533 Labos E. (labos(AT)ana.sote.hu), May 20 2003

%I A076500
%S A076500 1,2,2,2,2,2,4,2,2,2,2,6,2,2,2,4,2,2,4,2,4,6,4,2,2,2,2,2,1,5,4,4,2,6,4,
%T A076500 2,6,2,10,8,2,2,2,1,1,2,2,4,4,2,4,2,4,2,6,8,4,12,4,2,2,10,6,8,1,13,2,6,
%U A076500 4,2,4,2,2,2,2,2,2,2,2,4,4,6,2,2,4,2,4,6,2,12,4,6,6,6,8,2,5,3,24,8,4,4
%N A076500 Distance between natural sculptures.
%C A076500 The 'sculpture' of a positive integer n is the infinite vector (c[1], c[2], ...), where c[k] is the number of prime factors p of n (counted with multiplicity) such that n^(1/(k+1)) < p <= n^(1/k). A number is in sequence A076450 if its sculpture is not equal to the sculpture of any smaller number. This sequence contains the first differences of A076450.
%H A076500 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/sculptures/sculptures.htm">Sculptures</a>
%e A076500 The first 8 terms of A076450 are 1,2,4,6,8,10,12,16, so a(1)=1, a(2)=...=a(6)=2, and a(7)=4.
%t A076500 sculpt[1]={}; sculpt[n_] := Module[{fn, v, i}, fn=FactorInteger[n]; v=Table[0, {Floor[Log[fn[[1, 1]], n]]}]; For[i=1, i<=Length[fn], i++, v[[Floor[Log[fn[[i, 1]], n]]]]+=fn[[i, 2]]]; v]; For[n=1; nlist=slist={}, n<500, n++, sn=sculpt[n]; If[ !MemberQ[slist, sn], AppendTo[slist, sn]; AppendTo[nlist, n]]]; Drop[nlist, 1]-Drop[nlist, -1]
%Y A076500 Cf. A076450.
%Y A076500 Adjacent sequences: A076497 A076498 A076499 this_sequence A076501 A076502 A076503
%Y A076500 Sequence in context: A060467 A125918 A083533 this_sequence A060594 A104361 A086876
%K A076500 nonn
%O A076500 1,2
%A A076500 Jon Perry (perry(AT)globalnet.co.uk), Nov 08 2002
%E A076500 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Nov 18 2002

%I A060594
%S A060594 1,1,2,2,2,2,2,4,2,2,2,4,2,2,4,4,2,2,2,4,4,2,2,8,2,2,2,4,2,4,2,4,4,2,4,
%T A060594 4,2,2,4,8,2,4,2,4,4,2,2,8,2,2,4,4,2,2,4,8,4,2,2,8,2,2,4,4,4,4,2,4,4,4,
%U A060594 2,8,2,2,4,4,4,4,2,8,2,2,2,8,4,2,4,8,2,4,4,4,4,2,4,8,2,2,4,4,2,4,2
%N A060594 Number of non-congruent solutions of x^2 == 1 mod n (square roots of unity mod n).
%C A060594 Sum(k=1,n,a(k)) appears to be asymptotic to C*n*Log(n) with C=0.6... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2002
%C A060594 a(q) = number of real characters modulo q. - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2003
%C A060594 Also number of real Dirichlet characters modulo n and sum(k=1,n,a(k)) is asymptotic to (6/pi^2)*n*ln(n). - S. R. Finch (Steven.Finch(AT)inria.fr), Feb 16 2006
%D A060594 G. Tenenbaum, "Introduction a la theorie analytique et probabiliste des nombres", Cours specialise, 1995, Collection SMF, p. 260
%H A060594 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b060594.txt">Table of n, a(n) for n=1..1000</a>
%H A060594 S. R. Finch and Pascal Sebah, <a href="http://arXiv.org/abs/math.NT/0604465">Squares and Cubes Modulo n</a> (arXiv: math.NT/0604465).
%H A060594 K. Matthews, <a href="http://www.numbertheory.org/php/squareroot.html">Solving the congruence x^2=a(mod m)</a>
%F A060594 If q is the number of distinct odd primes dividing n (sequence A005087) then: if 8 divides n a(n) = 2^(q+2) = 2^(A005087(n) + 2); if n == 4 (mod 8) a(n) = 2^(q+1) = 2^(A005087(n) + 1); otherwise a(n) = 2^q = 2^(A005087(n)) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
%F A060594 a(n)=2^omega(n)/2 if n==+/-2 (mod 8), a(n)=2^omega(n) if n==+/-1, +/-3, 4 (mod 8), a(n)=2*2^omega(n) if n==0 (mod 8), where omega(n)=A001221(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2003
%F A060594 For n>=2 A046073(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for A046073(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
%e A060594 The four numbers 1^2, 3^2, 5^2, and 7^2 are congruent to 1 mod 8, so a(8)=4.
%o A060594 (PARI) a(n)=sum(i=1,n,if((i^2-1)%n,0,1))
%Y A060594 Cf. A005087.
%Y A060594 Cf. A046073, A000010.
%Y A060594 Adjacent sequences: A060591 A060592 A060593 this_sequence A060595 A060596 A060597
%Y A060594 Sequence in context: A125918 A083533 A076500 this_sequence A104361 A086876 A066691
%K A060594 nonn,mult
%O A060594 1,3
%A A060594 Jud McCranie (j.mccranie(AT)comcast.net), Apr 11 2001

%I A104361
%S A104361 1,2,2,2,2,2,4,2,2,2,4,4,4,4,2,4,8,8,2,8,8,8,2,4,16,16,4,4,32,16,4,8,4,
%T A104361 8,2,8,16,8,32,8,16,16,32,2,8,16,4,4,4
%N A104361 Number of divisors of A104350(n) - 1.
%C A104361 a(n) = A000005(A104357(n)).
%H A104361 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a104350.txt">Products of largest prime factors of numbers <= n</a>
%Y A104361 Cf. A104362, A104369, A064145.
%Y A104361 Adjacent sequences: A104358 A104359 A104360 this_sequence A104362 A104363 A104364
%Y A104361 Sequence in context: A083533 A076500 A060594 this_sequence A086876 A066691 A064133
%K A104361 nonn
%O A104361 2,2
%A A104361 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 06 2005

%I A086876
%S A086876 1,2,2,2,2,2,4,2,2,4,4,4,2,2,4,4,4,4,4,4,4,2,2,4,4,4,4,4,4,4,4,4,4,4,4,
%T A086876 4,6,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,6,2,4,4,4,4,4,4,6,4,6,6,6,6,6,2,2,
%U A086876 4,4,4,4,4,4,4,4,4,4,4,4,4,6,2,4,4,4,4,4,4,6,4,6,6,6
%N A086876 Run lengths in A071542.
%C A086876 All a(n) are even for n>1.
%C A086876 Records occur at positions { 1,2,7,37,122,... } which correspond to run start positions { 2,4,16,126,512,... } in A071542.
%o A086876 (PARI) e1(n)=sum(k=0,floor(log2(n)),bittest(n,k))
%o A086876 f(n)=local(c):c=0:while(n,n=n-e1(n):c=c+1):c
%o A086876 p=1:r=1:for(n=1,150,c=0:while(f(r) == p,r=r+1:c=c+1):p=f(r):print1(c","))
%Y A086876 Adjacent sequences: A086873 A086874 A086875 this_sequence A086877 A086878 A086879
%Y A086876 Sequence in context: A076500 A060594 A104361 this_sequence A066691 A064133 A105674
%K A086876 nonn,easy
%O A086876 1,2
%A A086876 Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 21 2003

%I A066691
%S A066691 2,2,2,2,2,4,2,4,2,2,4,4,4,2,2,4,2,4,4,2,4,2,4,4,2,2,4,4,4,4,4,4,2,4,2,
%T A066691 6,4,4,2,2,4,4,4,2,4,4,4,4,4,2,4,4,4,4,4,2,6,2,4,4,2,4,4,6,4,4,2,4,4,4,
%U A066691 4,2,4,4,4,2,6,2,4,6,2,2,8,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,2,2,6,4,2,4,2
%N A066691 Value of tau(2n-1) when tau(2n-1)=tau(2n+1).
%o A066691 (PARI) for (n=2, 10000, if (numdiv(2*n-1) == numdiv(2*n+1), write1("tau=tau.txt", numdiv(2*n-1), ", ")))
%Y A066691 Adjacent sequences: A066688 A066689 A066690 this_sequence A066692 A066693 A066694
%Y A066691 Sequence in context: A060594 A104361 A086876 this_sequence A064133 A105674 A130496
%K A066691 nonn
%O A066691 0,1
%A A066691 Jon Perry (perry(AT)globalnet.co.uk), Jan 11 2002

%I A064133
%S A064133 2,2,2,2,2,4,4,4,4,2,4,2,2,8,4,2,8,4,8,4,4,2,8,4,4,2,8,4,16,8,16,2,8,8,
%T A064133 4,32,8,8,4,16,8,8,4,2,4,2,16,2,16,4,8,16,8,16,16,8,16,8,4,2,4,4,2,8,8,
%U A064133 4,32
%N A064133 Number of divisors of 6^n + 1 that are relatively prime to 6^m + 1 for all 0 < m < n.
%H A064133 Sam Wagstaff, Cunningham Project, <a href="http://www.cerias.purdue.edu/homes/ssw/cun/pmain501">Factorizations of 6^n-1, n odd, n<330</a>
%t A064133 a = {1}; Do[ d = Divisors[ 6^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 6^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 66} ]
%Y A064133 Adjacent sequences: A064130 A064131 A064132 this_sequence A064134 A064135 A064136
%Y A064133 Sequence in context: A104361 A086876 A066691 this_sequence A105674 A130496 A001299
%K A064133 nonn
%O A064133 0,1
%A A064133 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2001

%I A105674
%S A105674 2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,8,6,8,8,8,8,8,10,10,10,10,10
%N A105674 Highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
%D A105674 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.
%H A105674 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A105674 P. Gaborit, <a href="http://www.unilim.fr/pages_perso/philippe.gaborit/SD/">Tables of Self-Dual Codes</a>
%H A105674 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (<a href="http://www.research.att.com/~njas/doc/self.txt">Abstract</a>, <a href="http://www.research.att.com/~njas/doc/self.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/self.ps">ps</a>).
%e A105674 At length 8 the only strictly Type I self-dual code is {00,11}^4, which has d=2, so a(4) = 2.
%Y A105674 Cf. A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
%Y A105674 Cf. also A105685 for the number of such codes.
%Y A105674 Adjacent sequences: A105671 A105672 A105673 this_sequence A105675 A105676 A105677
%Y A105674 Sequence in context: A086876 A066691 A064133 this_sequence A130496 A001299 A001300
%K A105674 nonn,nice
%O A105674 1,1
%A A105674 njas, May 06 2005
%E A105674 The sequence continues: a(28) = either 10 or 12, then a(58) = 10, a(60) through a(68) = 12, ...

%I A130496
%S A130496 0,0,0,0,0,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,8,8,8,8,8,10,10,10,10,10,12,12,
%T A130496 12,12,12,14,14,14,14,14,16,16,16,16,16,18,18,18,18,18,20,20,20,20,20,
%U A130496 22,22,22,22,22,24,24,24,24,24,26,26,26,26,26,28,28,28,28,28,30,30,30
%N A130496 Repetition of even numbers, with initial zeros, five times.
%F A130496 a(n)= -2 + 2*Sum_{k=0..n} {[8*(sin(2*Pi*k/5))^2-5]^2-5}/20, with n>=0. a(n)= -2 + 2*Sum_{k=0..n} 1/50*{-9*[k mod 5]+[(n+1) mod 5]+[(n+2) mod 5]+[(n+3) mod 5]+11*[(n+4) mod 5]}, with n>=0.
%p A130496 P:=proc(n) local i,j,k; for i from 0 by 1 to n do j:=-2+2*sum('(8*(sin(2*Pi*k/5))^2-5)^2-5','k'=0..i)/20; print(j); od; end: P(100);
%Y A130496 Cf. A122461.
%Y A130496 Adjacent sequences: A130493 A130494 A130495 this_sequence A130497 A130498 A130499
%Y A130496 Sequence in context: A066691 A064133 A105674 this_sequence A001299 A001300 A001306
%K A130496 easy,nonn
%O A130496 0,6
%A A130496 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), May 31 2007

%I A001299
%S A001299 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,9,9,9,9,9,13,13,
%T A001299 13,13,13,18,18,18,18,18,24,24,24,24,24,31,31,31,31,31,
%U A001299 39,39,39,39,39,49,49,49,49,49,60,60,60,60,60,73,73,73
%N A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.
%D A001299 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001299 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001299 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=175">Encyclopedia of Combinatorial Structures 175</a>
%H A001299 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#change">Index entries for sequences related to making change.</a>
%F A001299 G.f.: 1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)).
%p A001299 1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25));
%t A001299 CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 65} ], x ]
%Y A001299 Adjacent sequences: A001296 A001297 A001298 this_sequence A001300 A001301 A001302
%Y A001299 Sequence in context: A064133 A105674 A130496 this_sequence A001300 A001306 A108105
%K A001299 nonn
%O A001299 1,6
%A A001299 njas

%I A001300
%S A001300 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,9,9,9,9,9,13,13,
%T A001300 13,13,13,18,18,18,18,18,24,24,24,24,24,31,31,31,31,31,
%U A001300 39,39,39,39,39,50,50,50,50,50,62,62,62,62,62,77,77,77
%N A001300 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 cents.
%D A001300 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001300 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001300 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=176">Encyclopedia of Combinatorial Structures 176</a>
%H A001300 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#change">Index entries for sequences related to making change.</a>
%p A001300 1/(1-x)/(1-x^5)/(1-x^10)/(1-x^25)/(1-x^50)
%t A001300 CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)(1 - x^50)), {x, 0, 65} ], x ]
%Y A001300 Adjacent sequences: A001297 A001298 A001299 this_sequence A001301 A001302 A001303
%Y A001300 Sequence in context: A105674 A130496 A001299 this_sequence A001306 A108105 A063468
%K A001300 nonn
%O A001300 1,6
%A A001300 njas

%I A001306
%S A001306 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,10,10,10,10,10,
%T A001306 14,14,14,14,14