The Database of Integer Sequences, Part 10 

     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 )

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

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


(start)



%I A137633
%S A137633 1,2,1,4,1,1,7,2,1,1,12,3,1,1,1,19,5,2,1,1,1,30,7,3,1,1,1,1,45,11,4,2,1,
%T A137633 1,1,1,67,15,6,3,1,1,1,1,1,97,22,8,4,2,1,1,1,1,1,139,30,11,5,3,1,1,1,1,
%U A137633 1,1
%N A137633 Triangle read by rows, A000012 * A026794.
%C A137633 Left border = A000070: (1, 2, 4, 7, 12, 19, 30, 45,...); next column = A000041, the partition numbers: (1, 1, 2, 3, 5, 7, 11,...). Row sums = A016905: (1, 3, 6, 11, 18, 29, 44,...).
%F A137633 A000012 * A026794 as infinite lower triangular matrices.
%e A137633 First few rows of the triangle are:
%e A137633 1;
%e A137633 2, 1;
%e A137633 4, 1, 1;
%e A137633 7, 2, 1, 1;
%e A137633 12, 3, 1, 1, 1;
%e A137633 19, 5, 2, 1, 1, 1;
%e A137633 30, 7, 3, 1, 1, 1, 1;
%e A137633 ...
%Y A137633 Cf. A000041, A000070, A016905, A026794.
%Y A137633 Adjacent sequences: A137630 A137631 A137632 this_sequence A137634 A137635 A137636
%Y A137633 Sequence in context: A055327 A105260 A099510 this_sequence A066633 A088443 A117352
%K A137633 nonn,tabl
%O A137633 1,2
%A A137633 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008

%I A066633
%S A066633 1,2,1,4,1,1,7,3,1,1,12,4,2,1,1,19,8,4,2,1,1,30,11,6,3,2,1,1,45,19,9,6,
%T A066633 3,2,1,1,67,26,15,8,5,3,2,1,1,97,41,21,13,8,5,3,2,1,1,139,56,31,18,12,
%U A066633 7,5,3,2,1,1,195,83,45,28,17,12,7,5,3,2,1,1,272,112,63,38,25,16,11,7,5
%N A066633 Triangle T(n,k), n>=1, 1<=k<=n, giving number of k's in all partitions of n.
%C A066633 1; 2,1; 4,1,1; 7,3,1,1; 12,4,2,1,1;....
%H A066633 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EldersTheorem.html">Elder's Theorem</a>
%F A066633 T(n, 1) + ... + T(n, n) = A006128(n).
%F A066633 G.f. for the number of k's in all partitions of n is x^k/(1-x^k)*Product_{m>=1} 1/(1-x^m). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 15 2002
%F A066633 T(n, k) = Sum_{j<n, j=n (mod k)} P(j), P(j) = number of partitions of j, P(0) = 1 - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
%e A066633 For n = 3, k = 1; 3 = 2+1 = 1+1+1. T(3,1) = (zero 1's) + (one 1's) + (three 1's), so T(3,1) = 4.
%Y A066633 Diagonals: A000070, A024786-A024794.
%Y A066633 Adjacent sequences: A066630 A066631 A066632 this_sequence A066634 A066635 A066636
%Y A066633 Sequence in context: A105260 A099510 A137633 this_sequence A088443 A117352 A137710
%K A066633 easy,nonn,tabl
%O A066633 1,2
%A A066633 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 09 2002
%E A066633 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 11 2002

%I A088443
%S A088443 1,2,1,4,1,1,7,8,8,1,2,1,2,5,14,14,17,17,17,17,14,2,1,4,1,1,2,4,4,5,11,
%T A088443 32,31,34,31,31,37,38,38,38,41,37,38,37,38,31,31,1,4,5,1,7,8,8,1,2,1,2,
%U A088443 5,4,1,8,8,8,8,11,11,20,23,25,71,71,68,70,68,76,74,68,68,68,70,82,83,82
%N A088443 A linear version of the Josephus problem: a(n) = the function w_3(n).
%C A088443 The survivor w(n,3) in a modified Josephus problem, with a step of 3.
%C A088443 See A090569 or the reference for the definition of w(n,q).
%D A088443 Chris Groer, The Mathematics of Survival: From Antiquity to the Playground, Amer. Math. Monthly, 110 (No. 9, 2003), 812-825.
%F A088443 A recurrence is given in the reference.
%Y A088443 Cf. A006257, A088442, A088452, A090569.
%Y A088443 Adjacent sequences: A088440 A088441 A088442 this_sequence A088444 A088445 A088446
%Y A088443 Sequence in context: A099510 A137633 A066633 this_sequence A117352 A137710 A068009
%K A088443 nonn
%O A088443 1,2
%A A088443 njas, Nov 09 2003
%E A088443 Terms computed by Chris Groer (cgroer(AT)math.uga.edu)
%E A088443 More terms from John W. Layman (layman(AT)math.vt.edu), Feb 05 2004

%I A117352
%S A117352 1,2,1,4,1,1,8,1,0,1,16,1,1,1,1,32,1,2,2,2,1,64,1,3,2,2,3,1,128,1,4,1,0,
%T A117352 1,4,1,256,1,5,1,3,3,1,5,1,512,1,6,4,6,6,6,4,6,1,1024,1,7,8,8,6,6,8,8,7,
%U A117352 1
%V A117352 1,2,1,4,1,1,8,1,0,1,16,1,-1,-1,1,32,1,-2,-2,-2,1,64,1,-3,-2,-2,-3,1,128,1,-4,-1,0,-1,
%W A117352 -4,1,256,1,-5,1,3,3,1,-5,1,512,1,-6,4,6,6,6,4,-6,1,1024,1,-7,8,8,6,6,8,8,-7,1
%N A117352 Riordan array (1/(1-2x), x(1-2x)/(1-x)).
%C A117352 Row sums are A038505. Diagonal sums are A117353. Inverse is A117354.
%F A117352 Number triangle T(n,k)=sum{j=0..n-k, C(j-k,j)C(n-k,j)}
%e A117352 Triangle begins
%e A117352 1,
%e A117352 2, 1,
%e A117352 4, 1, 1,
%e A117352 8, 1, 0, 1,
%e A117352 16, 1, -1, -1, 1,
%e A117352 32, 1, -2, -2, -2, 1,
%e A117352 64, 1, -3, -2, -2, -3, 1
%Y A117352 Adjacent sequences: A117349 A117350 A117351 this_sequence A117353 A117354 A117355
%Y A117352 Sequence in context: A137633 A066633 A088443 this_sequence A137710 A068009 A059119
%K A117352 easy,sign,tabl
%O A117352 0,2
%A A117352 Paul Barry (pbarry(AT)wit.ie), Mar 09 2006

%I A137710
%S A137710 1,2,1,4,1,1,8,2,1,1,16,4,2,1,1,32,8,3,2,1,1,64,16,6,2,2,1,1,64,16,6,2,
%T A137710 2,1,1,128,32,12,5,2,2,1,1,256,64,24,10,4,2,2,1,1,512,128,48,21,9,4,2,2,
%U A137710 1,1,1024,256,96,42,19,18,4,2,2,1,1,2048,512,192,84,40,18,18,4,2,2,1,1
%N A137710 Triangle read by rows: T(n,k) = T(n-1, k-1) - T(n-k, k-1); left border = (1, 2, 4, 8, 16, 32,...).
%C A137710 Row sums = A137711: (1, 3, 6, 12, 24, 47, 92, 183,...).
%F A137710 The triangle is generated by two rules: T(n,k) = T(n-1, k-1) - T(n-k, k-1); and left border = 1, 2, 4, 8, 16,...
%e A137710 First few rows of the triangle are:
%e A137710 1;
%e A137710 2, 1;
%e A137710 4, 1, 1;
%e A137710 8, 2, 1, 1;
%e A137710 16, 4, 2, 1, 1;
%e A137710 32, 8, 3, 2, 1, 1;
%e A137710 64, 16, 6, 2, 2, 1, 1;
%e A137710 128, 32, 12, 5, 2, 2, 1, 1;
%e A137710 256, 64, 24, 10, 4, 2, 2, 1, 1;
%e A137710 512, 128, 48, 21, 9, 4, 2, 2, 1, 1;
%e A137710 ...
%Y A137710 Cf. A137711.
%Y A137710 Adjacent sequences: A137707 A137708 A137709 this_sequence A137711 A137712 A137713
%Y A137710 Sequence in context: A066633 A088443 A117352 this_sequence A068009 A059119 A127772
%K A137710 nonn,tabl
%O A137710 1,2
%A A137710 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 08 2008

%I A068009
%S A068009 1,2,1,4,1,1,8,2,1,1,16,4,2,1,1,32,8,4,1,1,1,64,16,6,2,1,1,1,128,32,12,
%T A068009 4,2,1,1,1,256,64,24,8,4,2,1,1,1,512,128,44,16,8,3,1,1,1,1,1024,256,88,
%U A068009 32,14,6,3,1,1,1,1,2048,512,176,64,26,12,5,2,1,1,1,1,4096,1024,344,128
%N A068009 Square array T(m,n) with m (row) >= 1 and n (column) >= 0 read by antidiagonals: number of subsets of {1,2,3,...n} that sum to 0 mod m. (Including the empty set whose sum is 0).
%C A068009 When p is an odd prime, T(p,k+p) = 2*T(p,k) + (2^k * ((2^p) - 2)/p) for all k >= 0 [Sophie LeBlanc]
%C A068009 When m divides n (with n >= m), T(m,n) = (1/m) Sum_{d | m, and d is odd} phi(d) * 2^(n/d) [N. Kitchloo and L. Pachter; D. Rusin]
%C A068009 A068009[C(i+1,2),i] = 2, A068009[C(i,2)+1,i] = A000009[i-1]+1 [AK, cf. A068049]
%D A068009 N. Kitchloo and L. Pachter, An interesting result about subset sums.
%D A068009 Bill Pet, Sophie LeBlanc, Will Self et al., 2002 [See the sci.math thread given above]
%H A068009 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a068009s.txt">Scheme code for computing this table and its rows.</a>
%H A068009 Lior Pachter, <a href="http://www.math.berkeley.edu/~lpachter/combinatorics.html">Subset sums</a>
%H A068009 Bill Pet, Sophie LeBlanc, Will Self et al., <a href="http://groups.google.com/groups?hl=en&threadm=36abe133.0201181401.410577fe%40posting.google.com&rnum=1&prev=/groups%3Fhl%3Den%26selm%3D36abe133.0201181401.410577fe%2540posting.google.com">Subsets of {1,2,3,...,n}</a> (discussion in sci.math)
%H A068009 <a href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#subsetsums">Index entries for sequences related to subset sums mod m</a>
%Y A068009 Main diagonal: A000016, super-diagonal: A063776. The first term greater than one occurs on each row m in the position A002024[m] and these are given in A068049.
%Y A068009 Row 1: A000079, row 2: A011782, row 3: A068010, row 5: A068011, row 6: A068012, row 7: A068013, row 9: A068030, row 10: A068031, row 11: A068032, row 12: A068033, row 13: A068034, row 14: A068035, row 15: A068036, row 16: A068037, row 17: A068038, row 18: A068039, row 19: A068040, row 20: A068041, row 21: A068042, row 25: A068043, row 32: A068044, row 64: A068045.
%Y A068009 Adjacent sequences: A068006 A068007 A068008 this_sequence A068010 A068011 A068012
%Y A068009 Sequence in context: A088443 A117352 A137710 this_sequence A059119 A127772 A086256
%K A068009 nonn,nice,tabl
%O A068009 0,2
%A A068009 This entry and Scheme-code created by Antti Karttunen, Feb 11 2002

%I A059119
%S A059119 1,1,1,2,1,4,1,1,8,9,2,1,16,55,64,25,6,1,1,32,285,1090,2020,2146,1380,
%T A059119 490,115,20,2,1,64,1351,14000,82115,304752,759457,1308270,1613250,
%U A059119 1484230,1067771,635044,326990,147440,57675,19238,5325,1170,190,20,1,1
%N A059119 Triangle a(n,m)=number of m-element antichains on a labeled n-set; number of monotone n-variable Boolean functions with m mincuts (lower units), m=0..binomial(n,floor(n,2)).
%C A059119 Row sums give A000372.
%D A059119 V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
%D A059119 V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%H A059119 K. S. Brown, <a href="http://www.mathpages.com/home/kmath030.htm">Dedekind's Problem</a>
%F A059119 a(n, 0) = 1; a(n, 1) = 2^n; a(n, 2) = A016269(n); a(n, 3) = A047707(n); a(n, 4) = A051112(n); a(5, n) = A051113(n); a(6, n) = A051114(n); a(7, n) = A051115(n); a(8, n) = A051116(n); a(9, n) = A051117(n); a(10, n) = A051118(n).
%e A059119 [1, 1], [1, 2], [1, 4, 1], [1, 8, 9, 2], [1, 16, 55, 64, 25, 6, 1], [1, 32, 285, 1090, 2020, 2146, 1380, 490, 115, 20, 2], ...
%Y A059119 Cf. A000372, A016269, A047707, A051112-A051118.
%Y A059119 Adjacent sequences: A059116 A059117 A059118 this_sequence A059120 A059121 A059122
%Y A059119 Sequence in context: A117352 A137710 A068009 this_sequence A127772 A086256 A057550
%K A059119 nonn,tabf
%O A059119 0,4
%A A059119 Vladeta Jovovic, Goran Kilibarda (vladeta(AT)Eunet.yu), Jan 06 2001

%I A127772
%S A127772 1,1,1,1,2,1,4,1,2,0,6,7,8,2,2,1,8,8,12,10,4,4,12,21,12,0,6,8,16,18,22,
%T A127772 15,4,4,8,20,24,6,6,18,24,24,30,18,4,10,24,51,26,10,12,8,28,28,22,24,12,
%U A127772 8,30,60
%V A127772 1,1,1,1,2,-1,4,-1,2,0,6,-7,8,-2,2,-1,8,-8,12,-10,4,4,12,-21,12,0,6,-8,16,-18,22,-15,4,
%W A127772 4,8,-20,24,-6,6,-18,24,-24,30,-18,4,10,24,-51,26,-10,12,-8,28,-28,22,-24,12,8,30,-60
%N A127772 Row sums of inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/Euler_phi(n+1),0),0).
%C A127772 Row sums of A127771.
%Y A127772 Adjacent sequences: A127769 A127770 A127771 this_sequence A127773 A127774 A127775
%Y A127772 Sequence in context: A137710 A068009 A059119 this_sequence A086256 A057550 A059150
%K A127772 sign
%O A127772 0,5
%A A127772 Paul Barry (pbarry(AT)wit.ie), Jan 28 2007

%I A086256
%S A086256 0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,2,1,4,1,2,1,1,0,13,4,5,0,2,2,1,1,
%T A086256 13,1,1,4,7,1,11,4,14,9,4,4,28,0,12,11,12,4,2,5,28,4,26,1,63,0,1,5,12,1,
%U A086256 29,1,12,2,44,4,101,4,11,27,12,1,26,4,15,4,11,1,75,1,11,14,36,0,40,11
%N A086256 Number of base-2 Fermat pseudoprimes that divide 2^n-1.
%C A086256 A base-2 Fermat pseudoprime is a composite number x such that 2^x = 2 mod x.
%H A086256 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pseudoprime.html">Pseudoprime</a>
%F A086256 a(n) = Sum{d|n} A086249(d), the Mobius transform of A086249.
%t A086256 Table[d=Divisors[2^n-1]; cnt=0; Do[m=d[[i]]; If[ !PrimeQ[m]&&PowerMod[2, m, m]==2, cnt++ ], {i, Length[d]}]; cnt, {n, 100}]
%Y A086256 Cf. A001567 (base-2 pseudoprimes), A086249.
%Y A086256 Adjacent sequences: A086253 A086254 A086255 this_sequence A086257 A086258 A086259
%Y A086256 Sequence in context: A068009 A059119 A127772 this_sequence A057550 A059150 A133186
%K A086256 hard,nonn
%O A086256 1,20
%A A086256 T. D. Noe (noe(AT)sspectra.com), Jul 14 2003

%I A057550
%S A057550 1,2,1,4,1,2,1,1,9,1,2,1,1,4,1,2,1,1,2,1,1,1,23,1,2,1,1,4,1,2,1,1,2,1,
%T A057550 1,1,9,1,2,1,1,4,1,2,1,1,2,1,1,1,4,1,2,1,1,2,1,1,1,2,1,1,1,1,65,1,2,1,
%U A057550 1,4,1,2,1,1,2,1,1,1,9,1,2,1,1,4,1,2,1,1,2,1,1,1,4,1,2,1,1,2,1,1,1,2,1
%N A057550 First differences of A057549.
%Y A057550 Terms a(A014138[n]) seem to give A014137[n+1]. Cf. A057551.
%Y A057550 Adjacent sequences: A057547 A057548 A057549 this_sequence A057551 A057552 A057553
%Y A057550 Sequence in context: A059119 A127772 A086256 this_sequence A059150 A133186 A084236
%K A057550 nonn
%O A057550 0,2
%A A057550 Antti Karttunen Sep 07 2000

%I A059150
%S A059150 1,2,1,4,1,2,1,2,4,2,8,2,4,2,1,2,1,4,1,2,1,4,8,4,16,4,8,4,1,2,1,4,1,2,
%T A059150 1,2,4,2,8,2,4,2,1,2,1,4,1,2,1,8,16,8,32,8,16,8,1,2,1,4,1,2,1,2,4,2,8,
%U A059150 2,4,2,1,2,1,4,1,2,1,4,8,4,16,4,8,4,1,2,1,4,1,2,1,2,4,2,8,2,4,2,1,2,1
%N A059150 A hierarchical sequence (W'2{3}* - see A059126).
%H A059150 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059150 Adjacent sequences: A059147 A059148 A059149 this_sequence A059151 A059152 A059153
%Y A059150 Sequence in context: A127772 A086256 A057550 this_sequence A133186 A084236 A068057
%K A059150 easy,nonn
%O A059150 0,2
%A A059150 Jonas Wallgren (jonwa(AT)ida.liu.se), Feb 01 2001

%I A133186
%S A133186 1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,
%T A133186 4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,1,4,1,2,
%U A133186 1,4,1,2,1,4
%V A133186 1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,
%W A133186 1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4,1,2,1,-4
%N A133186 Period 4: repeat 1, 2, 1, -4.
%F A133186 a(n)=(1/4)*{-5*(n mod 4)+5*[(n+1) mod 4]+[(n+2) mod 4]-[(n+3) mod 4]}, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Oct 24 2007
%Y A133186 Cf. A109008.
%Y A133186 Adjacent sequences: A133183 A133184 A133185 this_sequence A133187 A133188 A133189
%Y A133186 Sequence in context: A086256 A057550 A059150 this_sequence A084236 A068057 A003484
%K A133186 sign
%O A133186 0,2
%A A133186 Paul Curtz (bpcrtz(AT)free.fr), Oct 07 2007

%I A084236
%S A084236 1,0,1,2,1,4,1,2,1,4,4,7,19,22,32,26,14,20,24,125,257,362,228,10,211,
%T A084236 1042,329,330,1703,6222,10374
%V A084236 1,0,-1,-2,-1,-4,-1,-2,-1,-4,-4,7,-19,22,-32,26,14,-20,24,-125,257,-362,228,-10,211,
%W A084236 -1042,329,330,-1703,6222,-10374
%N A084236 a(n) = M(2^n), where M(n) is Mertens's function.
%F A084236 Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius function (A008683).
%t A084236 s = 0; i = 1; Do[ While[i <= 2^n, s = s + MoebiusMu[i]; i++ ]; Print[s], {n, 0, 50}]
%Y A084236 Cf. A002321.
%Y A084236 Adjacent sequences: A084233 A084234 A084235 this_sequence A084237 A084238 A084239
%Y A084236 Sequence in context: A057550 A059150 A133186 this_sequence A068057 A003484 A006519
%K A084236 more,sign
%O A084236 0,4
%A A084236 Robert G. Wilson v (rgwv(AT)rgwv.com), May 15 2003

%I A068057
%S A068057 1,2,1,4,1,2,1,6,1,2,1,4,1,2,1,11,1,2,1,4,1,2,1,6,1,2,1,4,1,2,1,13,1,2,
%T A068057 1,4,1,2,1,6,1,2,1,4,1,2,1,11,1,2,1,4,1,2,1,6,1,2,1,4,1,2,1,26,1,2,1,4,
%U A068057 1,2,1,6,1,2,1,4,1,2,1,11,1,2,1,4,1,2,1,6,1,2,1,4,1,2,1,13,1,2,1,4,1,2
%N A068057 First differences of A068056.
%C A068057 Every subsequence a(1) - a((2^k)-1) (with k >= 1) is palindromic, and their middle terms a(2^(k-1)) give A068059.
%H A068057 A. Karttunen, <A HREF="http://www.research.att.com/~njas/sequences/a066425s.txt">Scheme functions for computing A066425 and related sequences.</A>
%Y A068057 Adjacent sequences: A068054 A068055 A068056 this_sequence A068058 A068059 A068060
%Y A068057 Sequence in context: A059150 A133186 A084236 this_sequence A003484 A006519 A055975
%K A068057 nonn
%O A068057 1,2
%A A068057 Antti Karttunen (firstname.surname ), Feb 26 2002

%I A003484 M0161
%S A003484 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,10,1,2,
%T A003484 1,4,1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,12,1,2,1,4,
%U A003484 1,2,1,8,1,2,1,4,1,2,1,9,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,10,1,2,1,4,1,2
%N A003484 Radon function, also called Hurwitz-Radon numbers.
%C A003484 Simon Plouffe (plouffe(AT)math.uqam.ca) observes that this sequence and A006519 (greatest power of 2 dividing n) are very similar, the difference being all zeros except for every 16-th term (see A101119 for nonzero differences). Dec 02, 2004.
%D A003484 J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
%D A003484 J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
%D A003484 J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
%D A003484 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
%D A003484 A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
%D A003484 M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
%D A003484 T. Y. Lam, The Algebraic Theory of Quadratic Forms. Benjamin, Reading, MA, 1973, p. 131.
%D A003484 J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
%D A003484 T. Ono, Variations on a Theme of Euler, Plenum, NY, 1994, p. 192.
%D A003484 J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.
%D A003484 A. R. Rajwade, Squares, Camb. Univ. Press, London Math. Soc. Lecture Notes Series 171, 1993; see p. 127.
%H A003484 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b003484.txt">Table of n, a(n) for n = 1..10000</a>
%H A003484 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>
%H A003484 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A003484 If n=2^{4b+c}*d, 0<=c<=3, d odd, then a(n) = 8b + 2^c.
%F A003484 If n=2^m*d, d odd, then a(n) = 2m+1 if m=0 mod 4, = 2m if m=1 or 2 mod 4, = 2m+2 if m=3 mod 4.
%F A003484 Multiplicative with a(p^e) = 2e + a_(e mod 4) if p = 2; 1 if p > 2; where a = (1, 0, 0, 2). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%p A003484 readlib(ifactors): for n from 1 to 150 do if n mod 2 = 1 then printf(`%d,`,1) fi: if n mod 2 = 0 then m := ifactors(n)[2][1][2]: if m mod 4 = 0 then printf(`%d,`,2*m+1) fi: if m mod 4 = 1 then printf(`%d,`,2*m) fi: if m mod 4 = 2 then printf(`%d,`,2*m) fi: if m mod 4 = 3 then printf(`%d,`,2*m+2) fi: fi: od: # from James A. Sellers Dec 07 2000
%o A003484 (PARI) a(n)=8*(valuation(n,2)\4)+2^(valuation(n,2)%4) /* Paul D. Hanna (pauldhanna(AT)juno.com), Dec 02 2004 */
%Y A003484 See A053381 for a closely related sequence. Cf. A003485.
%Y A003484 a(n) = A003485(A007814(n)).
%Y A003484 Cf. A006519, A101119.
%Y A003484 Adjacent sequences: A003481 A003482 A003483 this_sequence A003485 A003486 A003487
%Y A003484 Sequence in context: A133186 A084236 A068057 this_sequence A006519 A055975 A118827
%K A003484 nonn,easy,core,nice,mult
%O A003484 1,2
%A A003484 njas
%E A003484 More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

%I A006519 M0162
%S A006519 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A006519 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A006519 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%N A006519 Highest power of 2 dividing n.
%C A006519 Least positive k such that m^k+1 divides m^n+1(with fixed base m). - Vladimir Baltic (baltic(AT)galeb.etf.bg.ac.yu), Mar 25 2002
%C A006519 To construct the sequence: start with 1, concatenate 1,1, and double last term gives 1,2. Concatenate those 2 terms, 1,2,1,2 and double last term 1,2,1,2 ->1,2,1,4. Concatenate those 4 terms: 1,2,1,4,1,2,1,4 and double last term -> 1,2,1,4,1,2,1,8 etc. - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 17 2002
%C A006519 a(n)=GCD(seq(binomial(2*n,2*m+1)/2,m=0..n-1)) (odd numbered entries of even numbered rows of Pascal's triangle A007318 divided by 2), where GCD denotes the greatest common divisor of a set of numbers. Due to the symmetry of the rows it suffices to consider m=0..floor((n-1)/2). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 23 2004
%C A006519 Equals the continued fraction expansion of a constant x (cf. A100338) such that the continued fraction expansion of 2*x interleaves this sequence with 2's: contfrac(2*x) = [2; 1,2, 2,2, 1,2, 4,2, 1,2, 2,2, 1,2, 8,2,...].
%C A006519 Simon Plouffe (plouffe(AT)math.uqam.ca) observes that this sequence and A003484 (Radon function) are very similar, the difference being all zeros except for every 16-th term (see A101119 for nonzero differences). Dec 02, 2004.
%C A006519 Comment from Jim Caprioli, Feb 04 2005: This sequence arises when calculating the next odd number in a Collatz sequence: Next(x) = (3 * x + 1) / A006519, or simply (3 x + 1) / BitAnd(3x+1,-3x-1).
%C A006519 a(n) = n iff n = 2^k. This sequence can be obtained by taking a(2^n) = 2^n inplace of a(2^n) = n and using the same sequence building approach as in A001511. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 08 2005
%C A006519 Also smallest m such that m + n - 1 = m XOR (n - 1); A086799(n)=a(n)+n-1. - Reinhard Zumkeller, Feb 02 2007
%H A006519 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b006519.txt">Table of n, a(n) for n=1..10000</a>
%H A006519 Beeler, M., Gosper, R. W. and Schroeppel, R., <a href="http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item175">Item 175 in Beeler, M., Gosper, R. W., and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972</a>
%H A006519 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A006519 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A006519 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A006519 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EvenPart.html">Link to a section of The World of Mathematics.</a>
%F A006519 a(n) = n AND -n (where "AND" is bitwise) - Marc LeBrun (mlb(AT)well.com), Sep 25 2000
%F A006519 Also: a(n)=gcd[2^n, n]. - Labos E. (labos(AT)ana.sote.hu), Apr 22 2003
%F A006519 Multiplicative with a(p^e) = p^e if p = 2; 1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A006519 G.f.: sum(k>=0, 2^k*x^2^k/(1-x^2^(k+1))). - Ralf Stephan, May 06 2003
%F A006519 Dirichlet g.f.: zeta(s)*(2^s-1)/(2^s-2). - Ralf Stephan, Jun 17 2007
%e A006519 2^3 divides 24, but 2^4 does not divide 24, so a(24)=8.
%p A006519 with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d,`,1) else printf(`%d,`,2^ifactors(n)[2][1][2]) fi; od:
%t A006519 f[n_] := Block[{k = 0}, While[Mod[n, 2^k] == 0, k++ ]; 2^(k - 1)]; Table[ f[n], {n, 102}] (from Robert G. Wilson v Nov 17 2004)
%o A006519 (PARI) a(n)=2^valuation(n,2)
%Y A006519 Partial sums are in A006520, second partial sums in A022560.
%Y A006519 Cf. A007814, A100338, A003484, A101119.
%Y A006519 This is Guy Steele's sequence GS(5,2) (see A135416).
%Y A006519 Adjacent sequences: A006516 A006517 A006518 this_sequence A006520 A006521 A006522
%Y A006519 Sequence in context: A084236 A068057 A003484 this_sequence A055975 A118827 A118830
%K A006519 nonn,easy,nice,mult
%O A006519 1,2
%A A006519 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A006519 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000

%I A055975
%S A055975 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A055975 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A055975 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%V A055975 1,2,-1,4,1,-2,-1,8,1,2,-1,-4,1,-2,-1,16,1,2,-1,4,1,-2,-1,-8,1,2,-1,-4,1,-2,-1,32,1,2,
%W A055975 -1,4,1,-2,-1,8,1,2,-1,-4,1,-2,-1,-16,1,2,-1,4,1,-2,-1,-8,1,2,-1,-4,1,-2,-1,64,1,2,-1,
%X A055975 4,1,-2,-1,8,1,2,-1,-4,1,-2,-1,16,1,2,-1,4,1,-2,-1,-8,1,2,-1,-4,1,-2,-1,-32,1,2,-1,4
%N A055975 First differences of A003188 (decimal equivalent of the Gray Code).
%C A055975 Multiplicative with a(2^e) = 2^e, a(p^e) = (-1)^((p^e-1)/2) otherwise. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) May 17, 2005.
%H A055975 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A055975 a(2n) = 2a(n), a(2n+1) = (-1)^n. G.f. sum(k>=0, 2^k*t/(1+t^2), t=x^2^k). a(n) = 2^A007814(n) * (-1)^((n/2^A007814(n)-1)/2). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Oct 29 2003
%e A055975 Since A003188(n) is 0 1 3 2 6 7 5 4 12 13 15 14 10 ..., a(n) begins 1 2 -1 4 1 -2 -1 8 1 2 -1 4 ...
%o A055975 (PARI) a(n)=((-1)^((n/2^valuation(n,2)-1)/2)*2^valuation(n,2) (from Ralf Stephan)
%Y A055975 The unsigned sequence |a(n)| is A006519(n) = 2^A007814(n)
%Y A055975 Cf. A003188, A006519 and A007814.
%Y A055975 MASKTRANSi transform of A053644 (conjectural).
%Y A055975 Cf. A119972, A119974.
%Y A055975 Adjacent sequences: A055972 A055973 A055974 this_sequence A055976 A055977 A055978
%Y A055975 Sequence in context: A068057 A003484 A006519 this_sequence A118827 A118830 A087258
%K A055975 easy,nice,sign,mult
%O A055975 1,2
%A A055975 Alford Arnold (Alford1940), Jul 22 2000
%E A055975 More terms from Larry Reeves (larryr(AT)acm.org), Sep 05 2000

%I A118827
%S A118827 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A118827 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A118827 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%V A118827 1,-2,1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-16,1,-2,1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-32,1,
%W A118827 -2,1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-16,1,-2,1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-64,1,-2,
%X A118827 1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-16,1,-2,1,-4,1,-2,1,-8,1,-2,1,-4,1,-2,1,-32,1,-2,1
%N A118827 2-adic continued fraction of zero, where a(n) = if n=1(mod 2), +1, else -2*A006519(n/2).
%C A118827 Limit of convergents equals zero; only the 6-th convergent is indeterminate. Other 2-adic continued fractions of zero are: A118821, A118824, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.
%e A118827 For n>=1, convergents A118828(k)/A118829(k) are:
%e A118827 at k = 4*n: -1/(2*A080277(n));
%e A118827 at k = 4*n+1: -1/(2*A080277(n)-1);
%e A118827 at k = 4*n+2: -1/(2*A080277(n)-2);
%e A118827 at k = 4*n-1: 0.
%e A118827 Convergents begin:
%e A118827 1/1, -1/-2, 0/-1, -1/2, -1/1, 1/0, 0/1, 1/-8,
%e A118827 1/-7, -1/6, 0/-1, -1/10, -1/9, 1/-8, 0/1, 1/-24,
%e A118827 1/-23, -1/22, 0/-1, -1/26, -1/25, 1/-24, 0/1, 1/-32,
%e A118827 1/-31, -1/30, 0/-1, -1/34, -1/33, 1/-32, 0/1, 1/-64, ...
%o A118827 (PARI) a(n)=local(p=+1,q=-2);if(n%2==1,p,q*2^valuation(n/2,2))
%Y A118827 Cf. A006519, A080277; convergents: A118828/A118829; variants: A118821, A118824, A118830; A100338.
%Y A118827 Adjacent sequences: A118824 A118825 A118826 this_sequence A118828 A118829 A118830
%Y A118827 Sequence in context: A003484 A006519 A055975 this_sequence A118830 A087258 A076775
%K A118827 cofr,sign
%O A118827 1,2
%A A118827 Paul D. Hanna (pauldhanna(AT)juno.com), May 01 2006

%I A118830
%S A118830 1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,
%T A118830 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,64,1,2,1,4,
%U A118830 1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2
%V A118830 -1,2,-1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,16,-1,2,-1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,32,-1,
%W A118830 2,-1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,16,-1,2,-1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,64,-1,2,
%X A118830 -1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,16,-1,2,-1,4,-1,2,-1,8,-1,2,-1,4,-1,2,-1,32,-1,2,-1
%N A118830 2-adic continued fraction of zero, where a(n) = if n=1(mod 2), -1, else +2*A006519(n/2).
%C A118830 Limit of convergents equals zero; only the 6-th convergent is indeterminate. Other 2-adic continued fractions of zero are: A118821, A118824, A118827. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.
%e A118830 For n>=1, convergents A118831(k)/A118832(k) are:
%e A118830 at k = 4*n: 1/(2*A080277(n));
%e A118830 at k = 4*n+1: 1/(2*A080277(n)-1);
%e A118830 at k = 4*n+2: 1/(2*A080277(n)-2);
%e A118830 at k = 4*n-1: 0.
%e A118830 Convergents begin:
%e A118830 -1/1, -1/2, 0/-1, -1/-2, 1/1, 1/0, 0/1, 1/8,
%e A118830 -1/-7, -1/-6, 0/-1, -1/-10, 1/9, 1/8, 0/1, 1/24,
%e A118830 -1/-23, -1/-22, 0/-1, -1/-26, 1/25, 1/24, 0/1, 1/32,
%e A118830 -1/-31, -1/-30, 0/-1, -1/-34, 1/33, 1/32, 0/1, 1/64, ...
%o A118830 (PARI) a(n)=local(p=-1,q=+2);if(n%2==1,p,q*2^valuation(n/2,2))
%Y A118830 Cf. A006519, A080277; convergents: A118831/A118832; variants: A118821, A118824, A118827; A100338.
%Y A118830 Adjacent sequences: A118827 A118828 A118829 this_sequence A118831 A118832 A118833
%Y A118830 Sequence in context: A006519 A055975 A118827 this_sequence A087258 A076775 A079891
%K A118830 cofr,sign
%O A118830 1,2
%A A118830 Paul D. Hanna (pauldhanna(AT)juno.com), May 01 2006

%I A087258
%S A087258 1,2,1,4,1,2,1,8,1,2,1,4,1,2,5,16,1,2,1,20,1,2,1,24,1,2,1,4,1,10,1,32,1,
%T A087258 2,5,4,1,2,1,40,1,2,1,4,1,2,1,48,1,2,1,52,1,2,1,56,1,2,1,20,1,2,1,64,1,
%U A087258 2,1,68,1,10,1,72,1,2,5,4,1,2,1,80,1,2,1,84,1,2,1,88,1,2,1,4,1,2,1,96,1
%N A087258 a(n)=GCD[n, A025586(n)], greatest common divisor of n and largest value in 3x+1 iteration list started at n.
%t A087258 c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1)c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] Table[GCD[w, Max[fpl[w]]], {w, 1, 256}]
%Y A087258 Cf. A025586.
%Y A087258 Adjacent sequences: A087255 A087256 A087257 this_sequence A087259 A087260 A087261
%Y A087258 Sequence in context: A055975 A118827 A118830 this_sequence A076775 A079891 A108738
%K A087258 nonn
%O A087258 1,2
%A A087258 Labos E. (labos(AT)ana.sote.hu), Sep 09 2003

%I A076775
%S A076775 1,2,1,4,1,2,1,8,1,10,1,4,1,2,1,16,1,2,1,20,21,2,1,8,1,2,1,4,1,10,1,32,
%T A076775 11,2,1,4,1,2,1,40,1,42,1,4,1,2,1,16,1,10,1,4,1,2,1,8,1,2,1,20,1,2,21,
%U A076775 64,1,22,1,4,3,10,1,8,1,2,1,4,1,2,1,80,3,2,1,84,1,2,1,8,1,10,1
%N A076775 Greatest common divisor of n and the binary representation of n interpreted decimally.
%e A076775 12 is '1100' in binary representation: a(12)=gcd(12,1100)=gcd(3*2^2,11*5^2*2^2)=4.
%Y A076775 Cf. A007088.
%Y A076775 Adjacent sequences: A076772 A076773 A076774 this_sequence A076776 A076777 A076778
%Y A076775 Sequence in context: A118827 A118830 A087258 this_sequence A079891 A108738 A064405
%K A076775 nonn,base
%O A076775 1,2
%A A076775 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 14 2002

%I A079891
%S A079891 1,2,1,4,1,2,1,8,3,2,1,4,1,2,3,16,1,6,1,4,3,1,1,8,1,1,9,4,1,6,1,32,3,2,
%T A079891 7,12,1,2,3,8,1,6,1,2,9,2,1,16,1,2,1,2,1,18,1,8,3,1,1,12,1,1,9,64,1,3,1,
%U A079891 1,1,1,1,24,1,1,3,1,1,3,1,16,27,2,1,12,1,2,1,4,1,18,1,4,1,2,1,32,1,2,1
%N A079891 GCD(n, A079890(n)).
%Y A079891 Cf. A079893, A079894.
%Y A079891 Adjacent sequences: A079888 A079889 A079890 this_sequence A079892 A079893 A079894
%Y A079891 Sequence in context: A118830 A087258 A076775 this_sequence A108738 A064405 A059147
%K A079891 nonn
%O A079891 1,2
%A A079891 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 14 2003

%I A108738
%S A108738 1,2,1,4,1,2,1,8,3,2,1,4,1,2,5,16,1,6,1,4,7,2,1,8,5,2,9,4,1,10,1,32,11,
%T A108738 2,7,12,1,2,13,8,1,14,1,4,15,2,1,16,7,10,17,4,1,18,11,8,19,2,1,20,1,2,
%U A108738 21,64,13,22,1,4,23,14,1,24,1,2,25,4,11,26,1,16,27,2,1,28,17,2,29,8,1
%N A108738 a(n) = n/(smallest odd prime divisor of n).
%C A108738 a(n) = n if n has no odd prime divisor, i.e. for n = 2^k (k>=0).
%D A108738 Z. Nedev and S. Muthukrishnan, The Nagger-Mover Game, DIMACS Tech. Report 2005-22.
%e A108738 a(21) = 21/3 = 7.
%p A108738 with(numtheory): a:=proc(n) local nn: nn:=factorset(n): if n=1 then 1 elif nn={2} then n elif nn[1]=2 then n/nn[2] else n/nn[1] fi end: seq(a(n),n=1..100); (Deutsch)
%t A108738 f[n_] := If[IntegerQ@Log[2, n], n, pf = First /@ FactorInteger@n; If[ EvenQ@n, n/pf[[2]], n/pf[[1]] ]]; Array[f, 89] (* Robert G. Wilson v Sep 02 2006 *)
%Y A108738 a(n)=n/A078701(n). Cf. A108514.
%Y A108738 Adjacent sequences: A108735 A108736 A108737 this_sequence A108739 A108740 A108741
%Y A108738 Sequence in context: A087258 A076775 A079891 this_sequence A064405 A059147 A091891
%K A108738 nonn,easy
%O A108738 1,2
%A A108738 S. Muthukrishnan (muthu(AT)research.att.com), Jun 23 2005
%E A108738 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu) and Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 24 2005

%I A064405
%S A064405 1,2,1,4,1,2,1,8,5,2,3,4,5,2,1,16,13,10,11,4,13,6,7,8,17,10,11,4,13,2,
%T A064405 1,32,29,26,27,20,29,22,23,8,33,26,27,12,29,14,15,16,41,34,35,20,37,22,23,8,41,26,
%U A064405 27,4,29,2,1,64,61,58,59,52,61,54,55,40,65,58,59,44,61,46,47,16,73,66,67,52,69,54
%V A064405 -1,-2,-1,-4,1,-2,-1,-8,5,2,3,-4,5,-2,-1,-16,13,10,11,4,13,6,7,-8,17,10,11,-4,13,-2,
%W A064405 -1,-32,29,26,27,20,29,22,23,8,33,26,27,12,29,14,15,-16,41,34,35,20,37,22,23,-8,41,26,
%X A064405 27,-4,29,-2,-1,-64,61,58,59,52,61,54,55,40,65,58,59,44,61,46,47,16,73,66,67,52,69,54
%N A064405 Number of even entries (A048967) minus the number of odd entries (A001316) in row n of Pascal's triangle (A007318).
%F A064405 a(n) = sum(k=0, n, (-1)^C(n, k) ); a(2^n) = 2^n-3; a(2^n+1)=2^n-6; more generally there's a sequence z(k) such that for any k>=0, and for 2^n >k, a(2^n+k) = 2^n+z(k); for k=0, 1, 2, 3, 4, 5, 6, 7, 8... z(k) = -3, -6, -5, -12, -3, -10, -9, -24, 1, ... - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 18 2002
%F A064405 a(2n) = a(n) + n, a(2n+1) = 2a(n). - Ralf Stephan, Mar 05 2004
%F A064405 a(n)=-sum{k=0..n, mu(binomial(n, k) mod 2)}; - Paul Barry (pbarry(AT)wit.ie), Apr 29 2005
%t A064405 Table[ n + 1 - 2Sum[ Mod[ Binomial[ n, k ], 2 ], {k, 0, n} ], {n, 0, 100} ]
%o A064405 (PARI) a(n)=sum(i=0,n,(-1)^binomial(n,i))
%o A064405 (PARI) a(n)=if(n<1,-1,if(n%2==0,a(n/2)+n/2,2*a((n-1)/2)))
%Y A064405 Adjacent sequences: A064402 A064403 A064404 this_sequence A064406 A064407 A064408
%Y A064405 Sequence in context: A076775 A079891 A108738 this_sequence A059147 A091891 A070194
%K A064405 easy,sign
%O A064405 0,2
%A A064405 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 29 2001

%I A059147
%S A059147 1,2,1,4,1,2,1,8,16,8,32,8,16,8,1,2,1,4,1,2,1,64,128,64,256,64,128,64,
%T A059147 1,2,1,4,1,2,1,8,16,8,32,8,16,8,1,2,1,4,1,2,1,512,1024,512,2048,512,
%U A059147 1024,512,1,2,1,4,1,2,1,8,16,8,32,8,16,8,1,2,1,4,1,2,1,64,128,64,256
%N A059147 A hierarchical sequence (W'2{3} - see A059126).
%H A059147 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059147 Adjacent sequences: A059144 A059145 A059146 this_sequence A059148 A059149 A059150
%Y A059147 Sequence in context: A079891 A108738 A064405 this_sequence A091891 A070194 A105584
%K A059147 easy,nonn
%O A059147 0,2
%A A059147 Jonas Wallgren (jonwa(AT)ida.liu.se), Feb 01 2001

%I A091891
%S A091891 1,2,1,4,1,2,1,10,3,2,1,5,1,2,1,36,6,12,1,11,3,2,1,24,3,3,1,5,1,2,1,202,
%T A091891 67,55,9,93,4,5,1,112,8,13,1,10,3,2,1,304,22,18,1,20,3,3,1,34,3,3,1,5,1,
%U A091891 2,1,1828,1267,1456,71,1629,77,100,2,2342,99,123,9,132,4,3,1
%N A091891 Number of partitions of n into sums of exactly as many distinct powers of 2 as n has 1's in binary representation.
%C A091891 a(A000079(n))=A018819(n); a(A018900(n))=A091889(n); a(A014311(n))=A091890(n);
%C A091891 a(A091892(n)) = 1.
%Y A091891 Cf. A000120, A000041, A091893.
%Y A091891 Adjacent sequences: A091888 A091889 A091890 this_sequence A091892 A091893 A091894
%Y A091891 Sequence in context: A108738 A064405 A059147 this_sequence A070194 A105584 A072064
%K A091891 nonn
%O A091891 1,2
%A A091891 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 10 2004

%I A070194
%S A070194 1,2,1,4,1,2,2,4,1,4,1,4,3,2,1,4,1,4,3,4,1,4,2,4,2,4,1,6,1,2,3,4,3,4,1,
%T A070194 4,3,4,1,6,1,4,3,4,1,4,2,4,3,4,1,4,3,4,3,4,1,6,1,4,3,2,3,6,1,4,3,6,1,4,
%U A070194 1,4,3,4,3,6,1,4,2,4,1,6,3,4,3,4,1,6,3,4,3,4,3,4,1,4,3,4,1,6,1,4,5,4,1
%N A070194 List the phi(n) numbers from 1 to n-1 which are relatively prime to n; sequence gives size of maximal gap.
%C A070194 Maximal gap in reduced residue system mod n.
%C A070194 It is an unsolved problem to determine the rate of growth of this sequence.
%D A070194 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 200.
%H A070194 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b070194.txt">Table of n, a(n) for n=3..10000</a>
%e A070194 For n = 10 the reduced residues are 1, 3, 7, 9; the maximal gap is a(10) = 7-3 = 4.
%t A070194 f[n_] := Block[{a = Select[ Table[i, {i, n - 1}], GCD[ #, n] == 1 & ], b = {}, k = 1, l = EulerPhi[n]}, While[k < l, b = Append[b, Abs[a[[k]] - a[[k + 1]]]]; k++ ]; Max[b]]; Table[ f[n], {n, 3, 100}]
%Y A070194 Cf. A000010.
%Y A070194 Adjacent sequences: A070191 A070192 A070193 this_sequence A070195 A070196 A070197
%Y A070194 Sequence in context: A064405 A059147 A091891 this_sequence A105584 A072064 A105498
%K A070194 nonn,nice,easy
%O A070194 3,2
%A A070194 njas, May 13 2002
%E A070194 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com) and John W. Layman (layman(AT)math.vt.edu), May 13 2002

%I A105584
%S A105584 1,2,1,4,1,2,3,2,1,2,1,4,3,4,1,4,1,2,1,4,1,2,3,2,3,4,3,2,1,2,3,2,1,2,1,
%T A105584 4,1,2,3,2,1,2,1,4,3,4,1,4,3,4,3,2,3,4,1,4,1,2,1,4,3,4,1,4,1,2,1,4,1,2,
%U A105584 3,2,1,2,1,4,3,4,1,4,1,2,1,4,1,2,3,2,3,4,3,2,1,2,3,2,3,4,3,2,3,4,1,4,3
%N A105584 Fixed point of the morphism 1 -> 34, 2 -> 32, 3 -> 12, 4 -> 14, starting from a(0) = 1.
%C A105584 A triangle space fill substitution: characteristic polynomial:x^4-2*x^3-2*x^2-4*x.
%C A105584 This triangle set was obtained by shifting the Heighway's dragon matrix about: M(Heighways's)={{1, 1, 0, 0}, {0, 1, 1, 0}, {0, 0, 1, 1}, {1, 0, 0, 1}} M(triangle)={{0, 0, 1, 1}, {0, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 1}} This result is a permutation of the rows of the matrix. I have obtained three triangle sets and two Heighway's sets by experiments like these.
%D A105584 F. M. Dekking, "Recurrent Sets", Advances in Mathematics, vol. 44, no.1, 1982, page 96, section 4.11
%t A105584 Flatten[ Nest[ Flatten[ # /. {1 -> {3, 4}, 2 -> {3, 2}, 3 -> {1, 2}, 4 -> {1, 4}} &], {1}, 8]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 07 2005)
%Y A105584 Adjacent sequences: A105581 A105582 A105583 this_sequence A105585 A105586 A105587
%Y A105584 Sequence in context: A059147 A091891 A070194 this_sequence A072064 A105498 A083414
%K A105584 nonn
%O A105584 0,2
%A A105584 Roger Bagula (rlbagulatftn(AT)yahoo.com), May 03 2005
%E A105584 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), May 07 2005

%I A072064
%S A072064 1,1,2,1,4,1,2,3,2,3,2,2,2,2,4,3,4,1,6,3,4,1,10,1,4,1,2,2,2,2,4,1,4,1,
%T A072064 6,2,6,2,6,3,24,1,2,2,6,3,8,1,6,3,8,5,2,2,2,3,2,4,6,2,16,3,2,2,2,1,4,3,
%U A072064 6,1,10,1,4,2,6,6,16,3,8,2,4,1,6,2,10,3,4,4,18,2,6,1,2
%N A072064 Least k>0 such that prime(n)+k*n is prime.
%e A072064 n=3, prime(3)=5: 5+1*3=8 is not prime, but 5+2*3=11, therefore a(3)=2 and A072063(3)=11.
%Y A072064 Cf. A034693, A000040, A072063.
%Y A072064 Adjacent sequences: A072061 A072062 A072063 this_sequence A072065 A072066 A072067
%Y A072064 Sequence in context: A091891 A070194 A105584 this_sequence A105498 A083414 A106616
%K A072064 nonn
%O A072064 1,3
%A A072064 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 12 2002

%I A105498
%S A105498 1,2,1,4,1,2,3,4,1,2,1,4,3,4,3,4,1,2,1,4,1,2,3,4,3,4,3,4,3,4,3,4,1,2,1,
%T A105498 4,1,2,3,4,1,2,1,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,1,2,1,4,1,2,
%U A105498 3,4,1,2,1,4,3,4,3,4,1,2,1,4,1,2,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3
%N A105498 Trajectory of 1 under the morphism 1->{1,2}, 2->{1,4}, 3->{3,4}, 4->{3,4}.
%C A105498 Dekking's topological Markov sets: characteristic polynomial x*(x^3-3*x^2+x+2).
%D A105498 F. M. Dekking, "Recurrent Sets", Advances in Mathematics, vol. 44, no.1, 1982, page 100, section 4.16
%t A105498 Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1, 4}, 3 -> {3, 4}, 4 -> {3, 4}} &], {1}, 7]]
%Y A105498 Adjacent sequences: A105495 A105496 A105497 this_sequence A105499 A105500 A105501
%Y A105498 Sequence in context: A070194 A105584 A072064 this_sequence A083414 A106616 A030652
%K A105498 nonn
%O A105498 0,2
%A A105498 Roger Bagula (rlbagulatftn(AT)yahoo.com), May 02 2005

%I A083414
%S A083414 0,1,1,2,1,4,1,2,3,5,2,6,1,5,5,5,2,10,2,6,5,8,3,9,5,8,5,9,4,17,3,9,
%T A083414 7,9,6,15,4,9,8,13,4,21,3,11,10,11,4,17,5,15,9,14,5,20,8,14,9,14,6,
%U A083414 27,6,15,12,14,9,26,6,15,12,23,5,25,3,15,13,17,8,29,7,20,12,17,7,32
%N A083414 Write the numbers from 1 to n^2 consecutively in n rows of length n; let c(k) = number of primes in k-th column; a(n) = minimal c(k) for gcd(k,n) = 1.
%C A083414 Conjectured to be always positive for n>1.
%C A083414 Note that a(n) is large when phi(n), the number of integers relatively prime to n, is small and vice versa. - T. D. Noe (noe(AT)sspectra.com), Jun 10 2003
%C A083414 The conjecture is true for all n <= 40000.
%D A083414 See A083382 for references and links.
%H A083414 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b083414.txt">Table of n, a(n) for n=1..2000</a>
%e A083414 For n = 4 the array is
%e A083414 1 2 3 4
%e A083414 5 6 7 8
%e A083414 9 10 11 12
%e A083414 13 14 15 16
%e A083414 in which columns 1 and 3 contain 2 and 3 primes; therefore a(4) = 2.
%t A083414 Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r-1)*n], s++ ], {r, n}]; minP=Min[s, minP]], {c, n}]; minP, {n, 100}]
%Y A083414 Cf. A083415, and A083382 for primes in rows.
%Y A083414 A084927 generalizes this to three dimensions.
%Y A083414 Adjacent sequences: A083411 A083412 A083413 this_sequence A083415 A083416 A083417
%Y A083414 Sequence in context: A105584 A072064 A105498 this_sequence A106616 A030652 A077904
%K A083414 nonn
%O A083414 1,4
%A A083414 njas, Jun 10 2003
%E A083414 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu) and T. D. Noe (noe(AT)sspectra.com), Jun 10 2003

%I A106616
%S A106616 0,1,2,1,4,1,2,7,8,3,2,11,4,13,14,1,16,17,6,19,4,7,22,23,8,5,26,9,28,29,
%T A106616 2,31,32,11,34,7,12,37,38,13,8,41,14,43,44,3,46,47,16,49,10,17,52,53,18,
%U A106616 11,56,19,58,59,4,61,62,21,64,13,22,67,68,23,14,71,24,73,74,5,76,77,26
%N A106616 Numerator of n/(n+15).
%Y A106616 Adjacent sequences: A106613 A106614 A106615 this_sequence A106617 A106618 A106619
%Y A106616 Sequence in context: A072064 A105498 A083414 this_sequence A030652 A077904 A088964
%K A106616 nonn,frac
%O A106616 0,3
%A A106616 njas, May 15 2005

%I A030652
%S A030652 1,2,1,4,1,2,8,1,1,4,4,1,6,12,1,5,1,1,1,3,2,1,1,1,18,9,1,42,1,1,2,1,
%T A030652 1,10,3,2,4,6,2,11,1,1,8,65,9,4,1,11,2,3,1,4,3,1,2,1,2,1,5,1,1,1,2,
%U A030652 1,14,1,5,1,6,2,7,1,29,1,1,1,3,1,2,1,26,1,1,7,13,1,2,2,8,3,4,2,2,2
%N A030652 Continued fraction for GAMMA(2/3).
%H A030652 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A030652 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC">Index entries for continued fractions for constants</a>
%F A030652 Note that 3*GAMMA(1/3)*GAMMA(2/3)=2*Pi*sqrt(3).
%Y A030652 Cf. A030651.
%Y A030652 Adjacent sequences: A030649 A030650 A030651 this_sequence A030653 A030654 A030655
%Y A030652 Sequence in context: A105498 A083414 A106616 this_sequence A077904 A088964 A124331
%K A030652 nonn,cofr
%O A030652 1,2
%A A030652 Paolo Dominici (pl.dm(AT)libero.it)

%I A077904
%S A077904 1,0,1,2,1,4,1,2,11,8,5,18,33,44,7,58,147,160,45,250,569,660,159,978,2299,
%T A077904 2616,661,3938,9169,10492,2615,15722,36707,41936,10493,62922,146793,167780,
%U A077904 41935,251650,587211,671080,167781,1006642,2348801,2684364,671079,4026522
%V A077904 1,0,1,2,-1,4,1,-2,11,-8,5,18,-33,44,-7,-58,147,-160,45,250,-569,660,-159,-978,2299,
%W A077904 -2616,661,3938,-9169,10492,-2615,-15722,36707,-41936,10493,62922,-146793,167780,
%X A077904 -41935,-251650,587211,-671080,167781,1006642,-2348801,2684364,-671079,-4026522
%N A077904 Expansion of (1-x)^(-1)/(1+x-2*x^3).
%Y A077904 Adjacent sequences: A077901 A077902 A077903 this_sequence A077905 A077906 A077907
%Y A077904 Sequence in context: A083414 A106616 A030652 this_sequence A088964 A124331 A095248
%K A077904 sign
%O A077904 0,4
%A A077904 njas, Nov 17 2002

%I A088964
%S A088964 1,2,1,4,1,2,13,8,9,2,1,4,1,26,1,16,33,18,1,4
%N A088964 Number of non-congruent solutions to x^2 - 2y^2 == 0 mod n.
%Y A088964 Cf. A087561, A062803.
%Y A088964 Adjacent sequences: A088961 A088962 A088963 this_sequence A088965 A088966 A088967
%Y A088964 Sequence in context: A106616 A030652 A077904 this_sequence A124331 A095248 A122458
%K A088964 mult,nonn
%O A088964 1,2
%A A088964 Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 28 2003

%I A124331
%S A124331 1,2,1,4,1,3,1,1,1,1,1,6,1,7,1,8,1,1,1,4,1,11,1,1,25,1,9,1,1,1,1,16,1,1,
%T A124331 1,4,1,19,1,1,1,7,1,4,1,23,1,12,1,5,1,1,1,3,1,1,1,1,1,5,1,31,1,16,1,11,
%U A124331 1,4,1,1,1,1,1,1,25,1,1,1,1,4,81,1,1,1,1,43,1,1,1,1,1,4,1,47,1,24,1,1,1
%N A124331 a(n)= ((phi(n) mod d(n)) +1)th positive divisor of n, where phi(n) is number of positive integers which are <= n and are coprime to n, and d(n) is the number of positive divisors of n.
%t A124331 f[n_] := Block[{d = Divisors[n]}, d[[Mod[EulerPhi[n], Length[d]] + 1]]];Table[f[n], {n, 100}] (*Chandler*)
%Y A124331 Cf. A000005, A000010, A124219, A124330.
%Y A124331 Adjacent sequences: A124328 A124329 A124330 this_sequence A124332 A124333 A124334
%Y A124331 Sequence in context: A030652 A077904 A088964 this_sequence A095248 A122458 A127461
%K A124331 nonn
%O A124331 1,2
%A A124331 Leroy Quet (qq-quet(AT)mindspring.com) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 26 2006

%I A095248
%S A095248 1,2,1,4,1,3,1,3,2,2,1,3,1,3,2,2,1,3,1,3,2,2,1,3,2,2,2,2,1,3,1,3,2,2,2,
%T A095248 2,1,3,2,2,1,3,1,3,2,2,1,3,2,2,2,2,1,3,2,2,2,2,1,3,1,3,2,2,2,2,1,3,2,2,
%U A095248 1,3,1,3,2,2,2,2,1,3,2,2,1,3,2,2,2,2,1,3,2,2,2,2,2,2,1,3,2,2,1,3,1,3,2
%N A095248 a(n) = least k > 0 such that n-th partial sum is divisible by n if and only if n is not prime.
%e A095248 n = 1 is not prime, k = 1 results in first partial sum 1 divisible by 1 as required, so a(1) = 1.
%e A095248 n = 2 is prime, k = 1 results in second partial sum 2 divisible by 2 and is excluded, but k = 2 results in second partial sum 3 not divisible by 2 as required, so a(2) = 2.
%o A095248 (PARI) {m=105;s=0;for(n=1,m,k=1;if(isprime(n), while((s+k)%n==0,k++), while((s+k)%n>0,k++));print1(k,",");s=s+k)}
%Y A095248 Adjacent sequences: A095245 A095246 A095247 this_sequence A095249 A095250 A095251
%Y A095248 Sequence in context: A077904 A088964 A124331 this_sequence A122458 A127461 A101261
%K A095248 nonn
%O A095248 1,2
%A A095248 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 17 2004
%E A095248 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 19 2004

%I A122458
%S A122458 0,2,1,4,1,3,1,4,1,2,1,3,1,37,1,35,1,2,1,5,1,3,1,34,1,2,1,3,1,4,1,34,1,
%T A122458 2,1,32,1,3,1,5,1,2,1,3,1,28,1,5,1,2,1,26,1,3,1,19,1,2,1,3,1,5,1,9,1,2,
%U A122458 1,4,1,3,1,4,1,2,1,3,1,25,1,13,1,2,1,18,1,3,1,5,1,2,1,3,1,4,1,8,1,2,1,5
%N A122458 "Dropping time" of the reduced Collatz iteration starting with 2n+1.
%C A122458 We count only the 3x+1 steps of the usual Collatz iteration. We stop counting when the iteration produces a number less than the initial 2n+1. For a fixed dropping time k, let N(k)=A100982(k) and P(k)=2^(A020914(k)-1). There are exactly N(k) odd numbers less than P(k) with dropping time k. Moreover, the sequence is periodic: if d is one of the N(k) odd numbers, then k=a(d)=a(d+i*P(k)) for all i>0. This periodicity makes it easy to compute the average dropping time of the reduced Collatz iteration: sum_{k>0} k*N(k)/P(k) = 3.492651852186...
%H A122458 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b122458.txt">Table of n, a(n) for n=0..10000</a>
%e A122458 a(3)=4 because, starting with 7, the iteration produces 11,17,13,5 and the last term is less than 7.
%t A122458 nextOddK[n_]:=Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; dt[n_]:=Module[{m=n, cnt=0}, If[n>1, While[m=nextOddK[m]; cnt++; m>n]]; cnt]; Table[dt[n],{n,1,301,2}]
%Y A122458 Cf. A060445, A075677 (one step of the reduced Collatz iteration), A075680.
%Y A122458 Adjacent sequences: A122455 A122456 A122457 this_sequence A122459 A122460 A122461
%Y A122458 Sequence in context: A088964 A124331 A095248 this_sequence A127461 A101261 A067614
%K A122458 nonn
%O A122458 0,2
%A A122458 T. D. Noe (noe(AT)sspectra.com), Sep 08 2006

%I A127461
%S A127461 1,1,2,1,4,1,3,1,6,2,2,2,6,2,3,2,6,3,5,1,6,1,5,2,8,2,4,4,5,1,6,2,7,4,4,
%T A127461 1,10,2,3,4,8,2,4,3,7,3,6,1,11,1,4,4,7,1,9,3,7,1,4,3,11,1,6,4,7,2,8,3,7,
%U A127461 2,4,4,12,1,6,5,5,2,7,2,10,6,3,1,9,5,4,2,9,2,11,3,8,3,3,1,14,3,3,5,10,3
%N A127461 a(0)=1. a(n) = number of earlier terms a(k), 0<=k<=n-1, such that (k+a(k)) divides n.
%e A127461 (a(0)+0) divides 6; (a(1)+1) divides 6; and (a(5)+5) divides 6. These 3 cases are the only cases where (a(k)+k) divides 6, for 0<=k<=5. So a(6)=3.
%t A127461 f[l_List] := Block[{n = Length[l]},Append[l, Count[Table[Mod[n, k - 1 + l[[k]]], {k, n}], 0]]];Nest[f, {1}, 101] (*Chandler*)
%Y A127461 Cf. A127460, A127463.
%Y A127461 Adjacent sequences: A127458 A127459 A127460 this_sequence A127462 A127463 A127464
%Y A127461 Sequence in context: A124331 A095248 A122458 this_sequence A101261 A067614 A113901
%K A127461 nonn
%O A127461 0,3
%A A127461 Leroy Quet (qq-quet(AT)mindspring.com), Jan 15 2007
%E A127461 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jan 22 2007

%I A101261
%S A101261 1,2,1,4,1,3,1,8,1,5,1,9,1,6,1,16,1,7,1,14,1,10,1,21,1,11,1,19,1,12,1,
%T A101261 32,1,13,1,24,1,15,1,34,1,17,1,30,1,18,1,45,1,20,1,36,1,22,1,47,1,23,1,
%U A101261 40,1,25,1,64,1,26,1,44,1,27,1,59,1,28,1,51,1,29,1,74,1,31,1,56,1,33,1
%N A101261 a(2n-1) = 1; a(2n) = a(n)th smallest positive integer not among the earlier terms of the sequence.
%C A101261 The sequence {a(2k)} forms a permutation of the integers >= 2.
%e A101261 a(12) = the a(6)th (the 3rd) smallest positive integer not among the first 11 terms of the sequence. Not among the first 11 terms are 6, 7, 9, 10,... The 3rd of these is 9, which is a(12).
%t A101261 a[1] = 1; a[n_] := a[n] = If[ OddQ[n], 1, Complement[ Range[100], Union[ Table[ a[i], {i, n - 1}]]][[a[n/2]] ]]; Table[ a[n], {n, 90}] (from Robert G. Wilson v Jan 13 2005)
%Y A101261 Adjacent sequences: A101258 A101259 A101260 this_sequence A101262 A101263 A101264
%Y A101261 Sequence in context: A095248 A122458 A127461 this_sequence A067614 A113901 A062799
%K A101261 easy,nonn
%O A101261 1,2
%A A101261 Leroy Quet (qq-quet(AT)mindspring.com), Dec 17 2004
%E A101261 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 13 2005

%I A067614
%S A067614 2,1,4,1,3,1,8,2,1,2,1,12,2,1,1,3,1,1,5,2,1,1,9,2,1,20,6,2,2,1,3,2,1,1,
%T A067614 4,3,1,1,1,6,2,2,1,1,28,9,1,1,15,7,3,2,1,1,32,4,2,2,1,1,1,8,1,1,1,1,5,
%U A067614 2,1,1,1,1,6,3,2,1,1,1,40,4,2,1,1,1,1,21,5,2,2,1,1,1,14,6,2,2,1,1,1
%N A067614 floor(1/(sqrt(prime(n))-floor(sqrt(prime(n))))), where prime(n) is the n-th prime.
%C A067614 a(n) is the second partial quotient in the simple continued fraction for sqrt(prime(n)).
%e A067614 For n=8, prime(n)=19, floor(sqrt(19))=4, and 1/(sqrt(19)-4) = 2.786..., so a(8)=2.
%t A067614 a[n_] := Floor[1/(Sqrt[Prime[n]]-Floor[Sqrt[Prime[n]]])]
%Y A067614 Adjacent sequences: A067611 A067612 A067613 this_sequence A067615 A067616 A067617
%Y A067614 Sequence in context: A122458 A127461 A101261 this_sequence A113901 A062799 A063647
%K A067614 nonn
%O A067614 1,1
%A A067614 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2002
%E A067614 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Feb 14 2002

%I A113901
%S A113901 0,1,1,2,1,4,1,3,2,4,1,6,1,4,4,4,1,6,1,6,4,4,1,8,2,4,3,6,1,9,1,5,4,4,4,
%T A113901 8,1,4,4,8,1,9,1,6,6,4,1,10,2,6,4,6,1,8,4,8,4,4,1,12,1,4,6,6,4,9,1,6,4,
%U A113901 9,1,10,1,4,6,6,4,9,1,10,4,4,1,12,4,4,4,8,1,12,4,6,4,4,4,12,1,6,6,8,1,9
%N A113901 Product of omega(n) and bigomega(n).
%C A113901 if a(n) = 1 then n is prime.
%F A113901 omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.
%o A113901 (PARI) omegaxbigomega(n) = { local(x); for(x=1,n, print1(omega(x)*bigomega(x)",") ) }
%Y A113901 Adjacent sequences: A113898 A113899 A113900 this_sequence A113902 A113903 A113904
%Y A113901 Sequence in context: A127461 A101261 A067614 this_sequence A062799 A063647 A077808
%K A113901 easy,nonn
%O A113901 1,4
%A A113901 Cino Hilliard (hillcino368(AT)gmail.com), Jan 29 2006

%I A062799
%S A062799 0,1,1,2,1,4,1,3,2,4,1,7,1,4,4,4,1,7,1,7,4,4,1,10,2,4,3,7,1,12,1,5,4,4,
%T A062799 4,12,1,4,4,10,1,12,1,7,7,4,1,13,2,7,4,7,1,10,4,10,4,4,1,20,1,4,7,6,4,
%U A062799 12,1,7,4,12,1,17,1,4,7,7,4,12,1,13,4,4,1,20,4,4,4,10,1,20,4,7,4,4,4
%N A062799 Inverse Moebius transform of A001221, the number of distinct prime factors of n.
%F A062799 a(n)=Sum{A001221[d]}, where d runs over divisors of n.
%e A062799 n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2++2+3 = 12
%t A062799 f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
%Y A062799 Cf. A001221.
%Y A062799 Adjacent sequences: A062796 A062797 A062798 this_sequence A062800 A062801 A062802
%Y A062799 Sequence in context: A101261 A067614 A113901 this_sequence A063647 A077808 A021471
%K A062799 nonn
%O A062799 1,4
%A A062799 Labos E. (labos(AT)ana.sote.hu), Jul 19 2001

%I A063647
%S A063647 0,1,1,2,1,4,1,3,2,4,1,7,1,4,4,4,1,7,1,7,4,4,1,10,2,4,3,7,1,13,1,5,4,4,
%T A063647 4,12,1,4,4,10,1,13,1,7,7,4,1,13,2,7,4,7,1,10,4,10,4,4,1,22,1,4,7,6,4,
%U A063647 13,1,7,4,13,1,17,1,4,7,7,4,13,1,13,4,4,1,22,4,4,4,10,1,22,4,7,4,4,4
%N A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.
%C A063647 If 1/n=1/b-1/c then n=bc/(c-b) and 1/n=1/(2n-b)+1/(c+2n) (though it is also the case that 1/n=1/(2n)+1/(2n) equivalent to b=c=0).
%C A063647 Also number of divisors of n^2 less than n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 13 2001
%C A063647 Also number of decompositions of divisors of n into coprime pairs. - K.B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 24 2001
%C A063647 Number of elements in the set {(x,y): x|n, y|n, x<y, GCD(x,y)=1}. - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 03 2002
%C A063647 Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 04 2002
%C A063647 This is similar to A062799, having the same first 29 terms. But they are different sequence.
%C A063647 If A001221(n)<=2, then a(n)=A062799(n); if A001221(n)>2, then a(n)>A062799(n). A001221(n), or omega(n), is the number of distinct primes dividing n. - Matthew Vandermast (ghodges14(AT)comcast.net), Aug 25 2004
%D A063647 Amarnath Murthy, Decomposition of The divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001. pp. 303-306.
%D A063647 Crux Mathematicorum, CMS Vol. 23 No. 7 Nov. 1997 pp. 443-4 Soln. Prob. 2175.
%D A063647 Problem 1051(b), American Mathematical Monthly, Vol. 105, No. 4, 1998 p. 372.
%H A063647 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b063647.txt">Table of n, a(n) for n=1..10000</a>
%H A063647 Canadian Math. Soc., "Crux Mathematicorum", Vol. 23, No. 7, Nov.,1997, <a href="http://journals.cms.math.ca/cgi-bin/vault/public/view/CRUXv23n7/body/HTML/443?template=CRUX">pp 443-4 Soln. to Prob. 2175</a>
%H A063647 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/">Smarandache Notions Journal</a>
%F A063647 a(n) = (tau(n^2)-1)/2.
%F A063647 a(n) = A018892(n)-1. If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1+1)(2*a2+1)...(2*at+1)-1)/2.
%F A063647 If n is prime a(n)=1. Conjecture: (1/n)*sum(i=1, n, a(i))=C*ln(n)*ln(ln(n))+o(ln(n)) with C=0.7...
%F A063647 Bisection of A046079. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 09 2004
%e A063647 a(10) = 4 since 1/10 = 1/5-1/10 = 1/6-1/15 = 1/8-1/40 = 1/9-1/90.
%e A063647 a(12) = 7: the divisors of 12 are 1, 2, 3, 4, 6 and 12, and the decompositions are (1, 2), (1, 3), (1, 4), (1, 6), (1, 12), (2, 3), (3, 4).
%t A063647 Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
%o A063647 (PARI) for(n=1,100,print1(sum(i=1,n^2,if((n*i)%(i+n),0,1)),","))
%Y A063647 Cf. A018892, A063427, A063428. First twenty-nine terms identical to those of A062799 (offset).
%Y A063647 Cf. A063717, A063718, A048691.
%Y A063647 Adjacent sequences: A063644 A063645 A063646 this_sequence A063648 A063649 A063650
%Y A063647 Sequence in context: A067614 A113901 A062799 this_sequence A077808 A021471 A088372
%K A063647 nonn,easy,nice
%O A063647 1,4
%A A063647 Henry Bottomley (se16(AT)btinternet.com), Jul 23 2001

%I A077808
%S A077808 0,2,1,4,1,3,2,6,3,3,2,5,2,4,2,8,2,5,3,5,4,4,1,7,3,4,4,6,2,4,2,10,2,4,
%T A077808 4,7,3,5,3,7,3,6,2,6,4,3,1,9,3,5,3,6,2,6,2,8,2,4,2,6,2,4,5,12,2,4,2,6,
%U A077808 2,6,2,9,2,5,4,7,2,5,2,9,2,5,3,8,3,4,2,8,1,6,3,5,4,3,4,11,5,5,3,7,1,5
%N A077808 Number of prime factors of numbers containing in their decimal representation only the digits 0 and 1 (counted with multiplicity).
%F A077808 a(n) = A001222(A007088(n)).
%e A077808 a(36) = A001222(A007088(36)) = A001222(100100) = A001222(2*2*5*5*7*11*13) = 7.
%Y A077808 Cf. A077807.
%Y A077808 Adjacent sequences: A077805 A077806 A077807 this_sequence A077809 A077810 A077811
%Y A077808 Sequence in context: A113901 A062799 A063647 this_sequence A021471 A088372 A078072
%K A077808 nonn,base
%O A077808 1,2
%A A077808 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 16 2002

%I A021471
%S A021471 0,0,2,1,4,1,3,2,7,6,2,3,1,2,6,3,3,8,3,2,9,7,6,4,4,5,3,9,6,1,4,5,6,
%T A021471 1,0,2,7,8,3,7,2,5,9,1,0,0,6,4,2,3,9,8,2,8,6,9,3,7,9,0,1,4,9,8,9,2,
%U A021471 9,3,3,6,1,8,8,4,3,6,8,3,0,8,3,5,1,1,7,7,7,3,0,1,9,2,7,1,9,4,8,6,0
%N A021471 Decimal expansion of 1/467.
%Y A021471 Adjacent sequences: A021468 A021469 A021470 this_sequence A021472 A021473 A021474
%Y A021471 Sequence in context: A062799 A063647 A077808 this_sequence A088372 A078072 A049776
%K A021471 nonn,cons
%O A021471 0,3
%A A021471 njas

%I A088372
%S A088372 0,2,1,4,1,3,2,8,1,5,4,7,2,6,3,16,1,9,8,13,4,12,5,15,2,10,7,14,3,11,6,
%T A088372 32,1,17,16,25,8,24,9,29,4,20,13,28,5,21,12,31,2,18,15,26,7,23,10,30,3,
%U A088372 19,14,27,6,22,11,64,1,33,32,49,16,48,17,57,8,40,25,56,9,41,24,61,4,36
%N A088372 Table read by rows where T(0,0)=0; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. T(n,T088208(n,j))=2^n-j, where T088208 is the table described in A088208.
%C A088372 A Thue-Morse generator using the "Ordering of Iterates" algorithm.
%D A088372 Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W.H. Freeman, 1991, p. 282, 265.
%e A088372 0
%e A088372 2 1
%e A088372 4 1 3 2
%e A088372 8 1 5 4 7 2 6 3
%e A088372 ...
%Y A088372 Cf. A088208, A010060.
%Y A088372 Adjacent sequences: A088369 A088370 A088371 this_sequence A088373 A088374 A088375
%Y A088372 Sequence in context: A063647 A077808 A021471 this_sequence A078072 A049776 A079276
%K A088372 nonn,tabl
%O A088372 0,2
%A A088372 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28 2003
%E A088372 Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and njas, Oct 08 2003

%I A078072
%S A078072 2,1,4,1,3,5,2,2,2,4,3,2,1,3,2,1,1,1,8,2,2,28,1,1,4,10,1,9,62,4,1,1,
%T A078072 1,2,3,1,3,3,9,2,1,7,1,1,1,4,1,1,2,1,8,1,5,7,2,64,1,5,2,2,2,4,1,14,
%U A078072 1,3,2,2,2,23,1,9,1,20,2,2,12,1,1,3,1,2,1,3,7,4,3,12,1,4,2,3,1,4,1
%N A078072 Continued fraction for constant defined in A065485.
%Y A078072 Adjacent sequences: A078069 A078070 A078071 this_sequence A078073 A078074 A078075
%Y A078072 Sequence in context: A077808 A021471 A088372 this_sequence A049776 A079276 A126210
%K A078072 nonn,cofr
%O A078072 1,1
%A A078072 Benoit Cloitre, Dec 02 2002

%I A049776
%S A049776 1,1,2,1,4,1,3,6,8,1,7,10,16,1,5,11,15,18,22,28,32,1,13,19,31,34,46,52,
%T A049776 64,1,9,21,29,35,43,55,63,66,74,86,94,100,108,120,128,1,25,37,61,67,91,
%U A049776 103,127,130,154,166,190,196,220,232,256
%N A049776 Triangular array T read by rows: n-th row consists of fixed points, k, of n-th row of array t given by A049773; i.e. t(n,t(n,k))=t(n,k).
%e A049776 Rows: {1}; {1,2}; {1,4}; {1,3,6,8}; ...
%Y A049776 Adjacent sequences: A049773 A049774 A049775 this_sequence A049777 A049778 A049779
%Y A049776 Sequence in context: A021471 A088372 A078072 this_sequence A079276 A126210 A040005
%K A049776 nonn,tabl
%O A049776 1,3
%A A049776 Clark Kimberling (ck6(AT)evansville.edu)

%I A079276
%S A079276 1,2,1,4,1,3,15,18,20,12,18,27,7,5,43,2,4,10,38,3,60,20,53,62,52,83,11,
%T A079276 30,27,49,113,63,79,25,81,143,80,121,53,142,81,52,81,150,136,40,176,114,
%U A079276 167,138,84,46,239,213,137,4,122,136,255,141,273,30,22,25,179,9,43,12
%N A079276 Multiplicative inverse in the finite field F(prime(n)) of the product of the first n-1 primes modulo prime(n).
%H A079276 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial</a>
%F A079276 a(1) = 1; for n>1, a(n) = ( p(n-1)# (mod p(n)) )^(-1), where p(i) is the i-th prime number, p(i)# is the product of first i primes, (x^(-1) is the multiplicative inverse in the finite field F(p(n)).
%e A079276 a(6)=3 because 2*3*5*7*11=2310, 2310=9 (mod 13) and 9*(9^(-1))=9*3=1 (mod 13)
%t A079276 a[n_] := Module[{i}, Return[PowerMod[Product[Prime[i], {i, 1, n - 1}], -1, Prime[n]]]; ];
%Y A079276 Cf. A062347, A002110, A000040.
%Y A079276 Adjacent sequences: A079273 A079274 A079275 this_sequence A079277 A079278 A079279
%Y A079276 Sequence in context: A088372 A078072 A049776 this_sequence A126210 A040005 A053578
%K A079276 nonn
%O A079276 1,2
%A A079276 Valentin F. Schmid (v_schmid(AT)hotmail.com), Feb 07 2003

%I A126210
%S A126210 0,1,0,0,0,1,0,0,0,1,1,2,1,4,1,4,1,3,4,4,1,3,3,2,2,3,0,2,4,1,3,1,1,4,3,
%T A126210 3,3,4,5,3,4,4,5,4,4,3,5,7,4,10,5,3,7,7,4,8,3,6,7,4,10,6,5,3,5,10,8,9,7,
%U A126210 10,5,12,6,10,5,7,2,9,11,9,11,7,7,5,2,9,6,7,5,13,12,10,8,4,9,6,6,10,10
%N A126210 Number of 8's in decimal expansion of 8^n.
%p A126210 P:=proc(n) local i,k,x,y,w,cont; y:=8; k:=y; for i from 0 by 1 to n do x:=y^i; cont:=0; while x>0 do w:=x-trunc(x/10)*10; if w=k then cont:=cont+1; fi; x:=trunc(x/10); od; print(cont); od; end: P(100);
%Y A126210 Cf. A065710, A126205, A126206, A126207, A126208, A126209, A126211.
%Y A126210 Adjacent sequences: A126207 A126208 A126209 this_sequence A126211 A126212 A126213
%Y A126210 Sequence in context: A078072 A049776 A079276 this_sequence A040005 A053578 A029205
%K A126210 easy,nonn
%O A126210 0,12
%A A126210 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), Dec 20 2006

%I A040005
%S A040005 2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,
%T A040005 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,
%U A040005 4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4
%N A040005 Continued fraction for sqrt(8).
%H A040005 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A040005 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC">Index entries for continued fractions for constants</a>
%p A040005 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%Y A040005 Adjacent sequences: A040002 A040003 A040004 this_sequence A040006 A040007 A040008
%Y A040005 Sequence in context: A049776 A079276 A126210 this_sequence A053578 A029205 A072721
%K A040005 nonn,cofr,easy
%O A040005 0,1
%A A040005 njas

%I A053578
%S A053578 1,1,2,1,4,1,4,1,8,1,8,8,1,1,1,16,16,1,1,16,1,1,1,1,32,1,32,1,1,32,32,
%T A053578 1,1,1,1,1,1,64,1,1,1,1,1,64,1,64,1,64,1,1,1,1,1,1,1,1,1,1,1,1,128,1,1,
%U A053578 1,1,1,128,1,1,1,1,1,128,1,128,128,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A053578 Values of cototient function for A053577.
%C A053578 Except for 2^0=1, there are only finitely many values of k such that cototient(k) = 2^m for fixed m.
%e A053578 For p prime, cototient[p]=1. Smallest values for which cototient[x]=2^w are 6,12,24,96,192,..,49152 for w=2,3,4,5,6,...,15
%Y A053578 Cf. A051953, A053577.
%Y A053578 Adjacent sequences: A053575 A053576 A053577 this_sequence A053579 A053580 A053581
%Y A053578 Sequence in context: A079276 A126210 A040005 this_sequence A029205 A072721 A035092
%K A053578 nonn
%O A053578 1,3
%A A053578 Labos E. (labos(AT)ana.sote.hu), Jan 18 2000

%I A029205
%S A029205 1,0,1,0,1,1,1,1,2,1,4,1,4,2,4,4,5,4,7,4,10,5,10,7,11,10,
%T A029205 13,10,16,11,20,13,21,16,23,20,26,21,30,23,36,26,38,30,
%U A029205 41,36,45,38,51,41,59,45,62,51,66,59,72,62,80,66,90,72
%N A029205 Expansion of 1/((1-x^2)(1-x^5)(1-x^8)(1-x^10)).
%Y A029205 Adjacent sequences: A029202 A029203 A029204 this_sequence A029206 A029207 A029208
%Y A029205 Sequence in context: A126210 A040005 A053578 this_sequence A072721 A035092 A107457
%K A029205 nonn
%O A029205 0,9
%A A029205 njas

%I A072721
%S A072721 1,0,1,1,2,1,4,1,4,2,6,1,10,1,8,4,10,1,15,1,17,5,16,1,26,2,22,5,29,1,
%T A072721 37,1,36,7,38,4,57,1,48,9,65,1,73,1,77,13,76,1,108,2,99,11,117,1,130,5,
%U A072721 145,14,142,1,189,1,168,19,202,5,223,1,241,17,247,1,309,1,286,24,333,4
%N A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).
%F A072721 a(n) =A072721(n)-A072721(n-1). a(p)=1 for p prime.
%e A072721 a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
%Y A072721 Cf. A072720.
%Y A072721 Adjacent sequences: A072718 A072719 A072720 this_sequence A072722 A072723 A072724
%Y A072721 Sequence in context: A040005 A053578 A029205 this_sequence A035092 A107457 A112350
%K A072721 nonn
%O A072721 0,5
%A A072721 Henry Bottomley (se16(AT)btinternet.com), Jul 05 2002

%I A035092
%S A035092 1,1,2,1,4,1,4,3,2,1,6,3,4,1,8,1,12,4,30,1,2,3,24,1,18,1,2,4,12,2,16,
%T A035092 12,2,3,6,1,4,13,6,1,10,2,12,6,2,6,4,8,6,9,6,9,28,1,4,1,10,3,6,4,46,4,
%U A035092 4,3,4,1,4,3,22,6,10,2,4,1,2,7,22,3,6,4,6,3,10,1,4,3,2,4,6,1,10,4,2,1
%N A035092 Smallest k - dependent on n - such that (n^2)*k+1 is prime where k is the subscript of the progressions.
%C A035092 This is one possible generalization of "the least prime problem" for nk+1 arithmetical progression when n is replaced by n^2, a special difference.
%H A035092 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%e A035092 a(40)=1 because in 1600k+1 at k=1 1601 is the smallest prime; a(61)=46 because in 46*46*k+1 sequence the first prime appears at k=46, it is 171167.
%Y A035092 Analogous case is A034693. See also A005574 and A002496.
%Y A035092 Adjacent sequences: A035089 A035090 A035091 this_sequence A035093 A035094 A035095
%Y A035092 Sequence in context: A053578 A029205 A072721 this_sequence A107457 A112350 A063717
%K A035092 nonn
%O A035092 1,3
%A A035092 Labos E. (labos(AT)ana.sote.hu)

%I A107457
%S A107457 1,0,0,1,2,1,4,1,4,3,2,3,4,3,5,6,7,2,7,5,8,8,8,6,8,6,10,9,11,7,13,6,12,
%T A107457 12,13,9,15,11,13,14,16,10,17,11,17,14,17,15,21,12,19,18,18,13,23,14,22,
%U A107457 20,22,16,26,15,24,21,25,16,26,21,26,24
%N A107457 Triangle read by rows: row n gived number of nonisomorphic generalized Petersen graphs P(n,k) with girth 8 on n vertices for 1<=k<=floor[(n-1)/2].
%C A107457 The generalized Petersen graph P(n,k) is a graph with vertex set $V(P(n,k)) = \{u_0,u_1,\dots,u_{n-1},v_0,v_1,\dots,v_{n-1}\}$ and edge set $E(P(n,k)) = \{u_i u_{i+1}, u_i v_i, v_i v_{i+k} : i=0,\dots,n-1\},$ where the subscripts are to be read modulo $n$.
%D A107457 I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.
%D A107457 M. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combin. Theory 6 (1969), 152-164.
%H A107457 Marko Boben, Tomaz Pisanski, Arjana Zitnik, <a href="http://www.ijp.si/ftp/pub/preprints/ps/2004/pp939.ps">I-graphs and the corresponding configurations</a>, Preprint series (University of Ljubljana, IMFM), Vol. 42 (2004), 939 (ISSN 1318-4865).
%e A107457 Any generalized Petersen graph P(n,k) has girth at most 8; it has girth 8 if and only if it has girth more than 7.
%e A107457 The smallest generalized Petersen graph with girth 8 is P(18,5)
%Y A107457 Cf. A077105, A107452-A107460.
%Y A107457 Adjacent sequences: A107454 A107455 A107456 this_sequence A107458 A107459 A107460
%Y A107457 Sequence in context: A029205 A072721 A035092 this_sequence A112350 A063717 A024994
%K A107457 nonn
%O A107457 18,5
%A A107457 Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski (Tomaz.Pisanski(AT)fmf.uni-lj.si) and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), May 26 2005
%E A107457 Example corrected by Greg Demand, Jan 17 2008

%I A112350
%S A112350 2,1,4,1,4,3,4,1,1,3,2,21,24,21,24,23,24,21,21,23,42,421,424,421,424,
%T A112350 423,424,421,421,423,12,121,124,121,124,123,124,121,121,123,42,421,424,
%U A112350 421,424,423,424,421,421,423,32,321,324,321,324,323,324,321,321,323,42
%N A112350 Pronunciation tones of the characters in the Chinese word for n.
%C A112350 Mandarin Chinese has four tones. The first tone is a flat, high tone. The second tone is a rising tone. The third tone first falls a little and then rises. The fourth tone is a falling tone.
%D A112350 Any Chinese dictionary.
%e A112350 a(21) = 421 because the characters in the Chinese word for 21, "er shi yi", are pronounced with the fourth, second, and first tone, respectively.
%Y A112350 Cf. A112348, A112350, A030166.
%Y A112350 Adjacent sequences: A112347 A112348 A112349 this_sequence A112351 A112352 A112353
%Y A112350 Sequence in context: A072721 A035092 A107457 this_sequence A063717 A024994 A051953
%K A112350 nonn,word
%O A112350 0,1
%A A112350 Wei Ji Ma (weijima(AT)gmail.com), Sep 05 2005

%I A063717
%S A063717 1,1,2,1,4,1,4,3,5,1,9,1,7,9,8,1,12,1,16,9,11,1,18,5,13,9,16,1,25,1,16,
%T A063717 11,17,25,27,1,19,13,32,1,36,1,22,27,23,1,36,7,25,17,26,1,36,25,49,19,
%U A063717 29,1,50,1,31,49,32,25,44,1,34,23,50,1,64,1,37,45,38,49,52,1,64,27,41
%N A063717 Greatest divisor of n^2 that is less than n.
%C A063717 Smaller of two distinct numbers with minimum sum whose geometric mean is n. E.g. a(12) = 9 as 12^2 = 144 = 1*144= 2*72 = 3*48 = 4*36=6*24=8*18=9*16 etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Feb 15 2003
%e A063717 a(45)=27 because set of divisors of 45^2 is {1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025} and the greatest element of the set less than 45 is 27.
%p A063717 with(numtheory): for n from 2 to 200 do a := divisors(n^2): b := a[(tau(n^2)-1)/2]: printf(`%d,`,b); od:
%t A063717 f[n_] := Module[{dn2 = Divisors[n^2]}, Last[Take[dn2, {1, Flatten[Position[dn2, n]][[ 1]] - 1}]]]; Table[f[i], {i, 2, 85}]
%Y A063717 A063649(n)=n+a(n), A063718(n)=n^2/A063717(n), A063428(n)=n-a(n).
%Y A063717 Cf. A063718.
%Y A063717 Adjacent sequences: A063714 A063715 A063716 this_sequence A063718 A063719 A063720
%Y A063717 Sequence in context: A035092 A107457 A112350 this_sequence A024994 A051953 A079277
%K A063717 nonn
%O A063717 2,3
%A A063717 Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 12 2001

%I A024994
%S A024994 1,1,1,2,1,4,1,4,3,5,1,10,1,7,6,10,1,16,1,17,8,14,1,31,4,20,11,31,1,48,1,42,
%T A024994 15,40,9,79,1,56,21,87,1,111,1,105,41,106,1,185,6,157,41,187,1,254,16,259,
%U A024994 57,258,1,425,1,342,92,432,22,557,1,554,107,627,1,875,1,762,175,922,18,1173
%N A024994 Number of periodic partitions of n: each part occurs more than once and the same number of times.
%F A024994 a(n) = Sum(q(k)), where k divides n, k < n, where q(n) = A000009(n), distinct partitions - Alford Arnold (Alford1940(AT)aol.com)
%e A024994 E.g. 6 = 1+1+1+1+1+1 = 2+2+2 = 3+3 = 2+1+2+1, so a(6)=4.
%Y A024994 Cf. A000009, A047966.
%Y A024994 Adjacent sequences: A024991 A024992 A024993 this_sequence A024995 A024996 A024997
%Y A024994 Sequence in context: A107457 A112350 A063717 this_sequence A051953 A079277 A066452
%K A024994 nonn
%O A024994 1,4
%A A024994 Clark Kimberling (ck6(AT)evansville.edu)

%I A051953
%S A051953 0,1,1,2,1,4,1,4,3,6,1,8,1,8,7,8,1,12,1,12,9,12,1,16,5,14,9,16,1,22,1,
%T A051953 16,13,18,11,24,1,20,15,24,1,30,1,24,21,24,1,32,7,30,19,28,1,36,15,32,
%U A051953 21,30,1,44,1,32,27,32,17,46,1,36,25,46,1,48,1,38,35,40,17,54,1,48,27
%N A051953 Cototient(n) := n - phi(n).
%C A051953 Unlike totients, cototient[x+1] = cototient[x] never holds - except 2-Phi[2] = 3-Phi[3] = 1 - because cototient[x] congruent x modulo 2. - Labos E. (labos(AT)ana.sote.hu), Aug 08 2001
%D A051953 J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.
%D A051953 R. E. Jamison, The Helly bound for singular sums, Discrete Math., 249 (2002), 117-133.
%H A051953 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b051953.txt">Table of n, a(n) for n = 1..10000</a>
%e A051953 n=12, Phi[12]=4=Card[{1,5,7,11}], a[12]=12-Phi[12]=8, numbers not exceeding 12 and not coprime to 12:{2,3,4,6,8,9,10,12}
%p A051953 with(numtheory); A051953 := n->n-phi(n);
%Y A051953 Cf. A000010, A005278, A001274, A098006.
%Y A051953 Adjacent sequences: A051950 A051951 A051952 this_sequence A051954 A051955 A051956
%Y A051953 Sequence in context: A112350 A063717 A024994 this_sequence A079277 A066452 A007104
%K A051953 nonn,easy,nice
%O A051953 1,4
%A A051953 Labos E. (labos(AT)ana.sote.hu), Dec 21 1999

%I A079277
%S A079277 0,1,1,2,1,4,1,4,3,8,1,9,1,8,9,8,1,16,1,16,9,16,1,18,5,16,9,16,1,27,1,
%T A079277 16,27,32,25,32,1,32,27,32,1,36,1,32,27,32,1,36,7,40,27,32,1,48,25,49,
%U A079277 27,32,1,54,1,32,49,32,25,64,1,64,27,64,1,64,1,64,45,64,49,72,1,64,27
%N A079277 a(1) = 0; for n>1, a(n) is the largest integer < n such that any prime factor of a(n) is also a prime factor of n.
%C A079277 The function a(n) complements Eulers phi-function: 1) a(n)+phi(n)=n if n is a power of a prime. 2) It seems also that a(n)+phi(n)>=n for "almost all numbers". 3) a(2n)=n+1 if and only if n is a Mersenne prime. 4) Lim a(n^k)/n^k =1 if n has at least two prime factors and k goes to infinity.
%e A079277 a(10)=8 since 8 is the largest integer< 10 that can be written using only the primes 2 and 5. a(78)=72 since 72 is the largest number less than 78 that can be written using only the primes 2, 3 and 13. (78=2*3*13).
%Y A079277 Adjacent sequences: A079274 A079275 A079276 this_sequence A079278 A079279 A079280
%Y A079277 Sequence in context: A063717 A024994 A051953 this_sequence A066452 A007104 A102627
%K A079277 nonn
%O A079277 1,4
%A A079277 Istvan Beck (istbe(AT)online.no), Feb 07 2003

%I A066452
%S A066452 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
%T A066452 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23,25,20,12,40,17,34,
%U A066452 21,18,14,37,17,24,25,41,20,39,14,31,34,33,18,42,16,32,37,41,18,44,25
%N A066452 Anti-phi(n).
%C A066452 anti-phi(n) = the number of integers < n that are not divisible by any anti-divisor of n.
%C A066452 The old definition given for this sequence was: anti-phi(n) = number of integers <= n that are coprime to the anti-divisors of n. However this does not match the entries.
%C A066452 See A066272 for definition of anti-divisor.
%H A066452 Diana Mecum, <a href="http://www.research.att.com/~njas/sequences/b066452.txt">Table of n, a(n) for n = 2..129</a>
%H A066452 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm">Anti-phi function</a> [Broken link]
%H A066452 Jon Perry, <a href="http://www.research.att.com/~njas/sequences/a066272a.html">The Anti-divisor</a> [Cached copy]
%H A066452 Jon Perry, <a href="http://www.research.att.com/~njas/sequences/a066272.html">The Anti-divisor: Even More Anti-Divisors</a> [Cached copy]
%e A066452 10 has anti-divisors 3,4,7. The numbers not divisible by any of 3,4,7 and less than 10 are are 1,2,5. Therefore anti-phi(10)=3.
%Y A066452 Cf. A058838, A066241.
%Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453 A066454 A066455
%Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence A007104 A102627 A088296
%K A066452 nonn,easy
%O A066452 2,3
%A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
%E A066452 Better definition and more terms from Diana Mecum (diana.mecum(AT)gmail.com), Jul 01 2007

%I A007104
%S A007104 1,1,1,2,1,4,1,4,4,4,2,10,1,5,7,12,1,10,4,10,12,4,6,24,4,12,7,10,5,
%T A007104 26,6,20,13,12,7,30,11,4,15,32
%N A007104 Number of elements (a b, c d) in GL(2,Z) with det = -1, trace n and 0 <= a <= {b, c} <= d.
%Y A007104 Adjacent sequences: A007101 A007102 A007103 this_sequence A007105 A007106 A007107
%Y A007104 Sequence in context: A051953 A079277 A066452 this_sequence A102627 A088296 A093890
%K A007104 nonn
%O A007104 1,4
%A A007104 John Lewis (frc.mass.edu)

%I A102627
%S A102627 1,1,1,2,1,4,1,4,4,5,1,15,1,7,14,17,1,28,1,40,28,11,1,99,31,13,49,99,1,
%T A102627 186,1,152,76,17,208,425,1,19,109,699,1,584,1,433,823,23,1,1625,437,
%U A102627 1140,193,746,1,2003,1748,2749,244,29,1,7404,1,31,4158,3258,3766,6307,1
%N A102627 Number of partitions of n into distinct parts in which the number of parts divides n.
%Y A102627 Cf. A067538.
%Y A102627 Adjacent sequences: A102624 A102625 A102626 this_sequence A102628 A102629 A102630
%Y A102627 Sequence in context: A079277 A066452 A007104 this_sequence A088296 A093890 A006306
%K A102627 easy,nonn
%O A102627 1,4
%A A102627 Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 01 2005

%I A088296
%S A088296 1,2,1,4,1,5,1,5,7,4,5,7,2,1,7,4,11,8,7,2,11,13,5,7,14,1,5,7,16,13,1,5,
%T A088296 14,4,13,8,1,7,19,2,13,10,20,19,11,8,17,1,16,11,23,20,10,17,1,7,11,13,8,
%U A088296 22,5,7,16,11,26,2,25,19,5,7,23,13,25,8,19,1,26,20,17,10,19,14,5,7,23
%N A088296 Values of y - x, where x^2 + xy + y^2=p (x<y) is a prime of the form 6n + 1 (=A002476).
%Y A088296 Cf. A002476, A088241, A088242, A088243, A088298, A088299, A088977.
%Y A088296 Adjacent sequences: A088293 A088294 A088295 this_sequence A088297 A088298 A088299
%Y A088296 Sequence in context: A066452 A007104 A102627 this_sequence A093890 A006306 A083711
%K A088296 nonn
%O A088296 1,2
%A A088296 Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 03 2003
%E A088296 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 04 2003

%I A093890
%S A093890 0,2,1,4,1,5,1,6,2,7,1,9,1,5,4,11,1,12,1,13,5,5,1,17,2,5,4,16,1,20,1,18,
%T A093890 4,6,6,24,1,5,5,24,1,24,1,18,11,5,1,30,1,15,3,18,1,30,6,30,5,7,1,39,1,3,
%U A093890 18,31,6,34,1,16,3,34,1,44,1,4,13,16,4,39,1,42,5,5,1,48,5,5,2,41,1,51,2
%N A093890 Number of primes arising as the sum of one or more divisors of n.
%C A093890 a(2^n) = pi(2^(n+1)-1).
%e A093890 a(4) = 4, the divisors of 4 are 1, 2 and 4.
%e A093890 Primes arising are 2, 3= 1+2, 5 = 1+4 and 7 = 1 + 2 + 4.
%t A093890 Do[l = Subsets[Divisors[n]]; l = Union[Map[Plus @@ #&, l]]; Print[Length[Select[l, PrimeQ]]], {n, 100}] - Ryan Propper (rpropper(AT)stanford.edu), Jun 04 2006
%Y A093890 Cf. A093891, A093892.
%Y A093890 Adjacent sequences: A093887 A093888 A093889 this_sequence A093891 A093892 A093893
%Y A093890 Sequence in context: A007104 A102627 A088296 this_sequence A006306 A083711 A018783
%K A093890 nonn
%O A093890 1,2
%A A093890 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 23 2004
%E A093890 Corrected and extended by Ryan Propper (rpropper(AT)stanford.edu), Jun 04 2006

%I A006306 M0163
%S A006306 1,1,1,2,1,4,1,5,2,5,4,7,4,11,3,13,6,14,9,18,7,24,8,29,14,32,17,38,18,
%T A006306 50,20,58,25,63,33,77,35,94,36,108,48,122,60,141,63,170,70,195,87,215,
%U A006306 101,250,110,294,124,333,146,371,173,424,190,492,206,554,245,617,283
%V A006306 1,-1,1,2,-1,-4,1,5,-2,-5,4,7,-4,-11,3,13,-6,-14,9,18,-7,-24,8,29,-14,-32,17,38,-18,
%W A006306 -50,20,58,-25,-63,33,77,-35,-94,36,108,-48,-122,60,141,-63,-170,70,195,-87,-215,
%X A006306 101,250,-110,-294,124,333,-146,-371,173,424,-190,-492,206,554,-245,-617,283
%N A006306 Taylor series from Ramanujan's Lost Notebook.
%D A006306 G. E. Andrews, Mordell integrals and Ramanujan's "Lost" Notebook, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).
%F A006306 G.f.: sum for n >= 0 of (-1)^n q^n^2 (1-q)(1-q^3)...(1-q^(2n-1))/((1+q^2)^2 (1+q^4)^2 ... (1+q^(2n))^2)
%t A006306 Series[Sum[(-q)^n^2 Product[(1-q^(2k-1))/(1+q^(2k))^2, {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
%Y A006306 Cf. A006304, A006305.
%Y A006306 Adjacent sequences: A006303 A006304 A006305 this_sequence A006307 A006308 A006309
%Y A006306 Sequence in context: A102627 A088296 A093890 this_sequence A083711 A018783 A114326
%K A006306 sign,easy,nice
%O A006306 0,4
%A A006306 njas
%E A006306 Corrected and extended by Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 13 1999

%I A083711
%S A083711 1,1,1,2,1,4,1,5,3,7,1,14,1,13,8,20,1,33,1,40,14,44,1,85,6,79,25,117,1,181,
%T A083711 1,196,45,233,17,389,1,387,80,545,1,750,1,839,165,1004,1,1516,12,1612,234,
%U A083711 2040,1,2766,48,3142,388,3720,1,5295,1,5606,663,7038,83,9194,1,10379,1005
%N A083711 A083710(n) - A000041(n-1).
%D A083711 L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.
%F A083711 a(n) = Sum_{ d|n, d<n} A000041(d-1).
%p A083711 with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od:
%Y A083711 Cf. A083710, A000041.
%Y A083711 Adjacent sequences: A083708 A083709 A083710 this_sequence A083712 A083713 A083714
%Y A083711 Sequence in context: A088296 A093890 A006306 this_sequence A018783 A114326 A092986
%K A083711 nonn,easy
%O A083711 1,4
%A A083711 njas, Jun 16 2003
%E A083711 More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Jun 21 2003

%I A018783
%S A018783 0,0,1,1,2,1,4,1,5,3,8,1,14,1,16,9,22,1,38,1,45,17,57,1,94,7,102,30,138,
%T A018783 1,218,1,231,58,298,21,451,1,491,103,644,1,919,1,1005,203,1256,1,1784,15,
%U A018783 1993,299,2439,1,3365,62,3735,492,4566,1,6252,1,6843,819,8349,107,11096
%N A018783 Number of partitions of n into parts having a common factor.
%F A018783 a(n) = -Sum_{d|n, d<n} moebuis(n/d)*A000041(d) = A000041(n) - A000837(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 17 2003
%Y A018783 Cf. A083710.
%Y A018783 Adjacent sequences: A018780 A018781 A018782 this_sequence A018784 A018785 A018786
%Y A018783 Sequence in context: A093890 A006306 A083711 this_sequence A114326 A092986 A060680
%K A018783 nonn
%O A018783 0,5
%A A018783 David W. Wilson (davidwwilson(AT)comcast.net)

%I A114326
%S A114326 0,1,1,2,1,4,1,5,4,8,3,15,7,19,19,29,22,49,38,72,73,104,100,164,161,229,
%T A114326 252,343,363,510,539,716,806,1022,1153,1477,1639,2057,2353,2899,3281,
%U A114326 4054,4590,5586,6416,7691,8798,10575,12077,14376
%N A114326 Number of partitions p of n such that min(p) and max(p) have a common factor.
%e A114326 a(6)=4 since 6 has the following 4 partitions {6}, {4, 2}, {3, 3}, {2, 2, 2} with GCD of min and max greater than 1.
%Y A114326 Cf. A000041.
%Y A114326 Adjacent sequences: A114323 A114324 A114325 this_sequence A114327 A114328 A114329
%Y A114326 Sequence in context: A006306 A083711 A018783 this_sequence A092986 A060680 A057237
%K A114326 nonn
%O A114326 1,4
%A A114326 Giovanni Resta (g.resta(AT)iit.cnr.it), Feb 06 2006

%I A092986
%S A092986 1,1,2,1,4,1,6,1,2,1,10,1,12,1,2,1,16,1,18,1,2,1,22,1,4,1,2,1,28,1,30
%N A092986 Duplicate of A057237.
%Y A092986 Adjacent sequences: A092983 A092984 A092985 this_sequence A092987 A092988 A092989
%Y A092986 Sequence in context: A083711 A018783 A114326 this_sequence A060680 A057237 A049559
%K A092986 dead
%O A092986 1,3

%I A060680
%S A060680 1,2,1,4,1,6,1,2,1,10,1,12,1,2,1,16,1,18,1,2,1,22,1,4,1,2,1,28,1,30,1,
%T A060680 2,1,2,1,36,1,2,1,40,1,42,1,2,1,46,1,6,1,2,1,52,1,4,1,2,1,58,1,60,1,2,
%U A060680 1,4,1,66,1,2,1,70,1,72,1,2,1,4,1,78,1,2,1,82,1,4,1,2,1,88,1,6,1,2,1,4
%N A060680 Smallest difference between consecutive divisors of n (ordered by size).
%C A060680 a(n) = 1 if n is even, and a(n) is even if n is odd. A060684 contains just the odd-numbered terms.
%D A060680 A. Balog, P. Erdos and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors. In: Analytical Number Theory, pp. 77-90, Birkhauser, Basel, 1990.
%e A060680 For n=35, divisors={1,5,7,35}; differences={4,2,28}; a(35) = smallest difference = 2.
%t A060680 a[n_ ] := Min@@(Drop[d=Divisors[n], 1]-Drop[d, -1])
%Y A060680 Cf. A060681-A060684.
%Y A060680 Adjacent sequences: A060677 A060678 A060679 this_sequence A060681 A060682 A060683
%Y A060680 Sequence in context: A018783 A114326 A092986 this_sequence A057237 A049559 A063994
%K A060680 nonn
%O A060680 2,2
%A A060680 Labos E. (labos(AT)ana.sote.hu), Apr 19 2001
%E A060680 Corrected by David W. Wilson (davidwwilson(AT)comcast.net), May 04 2001
%E A060680 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Jan 22 2002

%I A057237
%S A057237 1,1,2,1,4,1,6,1,2,1,10,1,12,1,2,1,16,1,18,1,2,1,22,1,4,1,2,1,28,1,30,
%T A057237 1,2,1,4,1,36,1,2,1,40,1,42,1,2,1,46,1,6,1,2,1,52,1,4,1,2,1,58,1,60,1,
%U A057237 2,1,4,1,66,1,2,1,70,1,72,1,2,1,6,1,78,1,2,1,82,1,4,1,2,1,88,1,6,1,2,1
%N A057237 Maximum k <= n such that 1, 2, ..., k are all relatively prime to n.
%C A057237 In reduced residue system for n [=RRS(n)] the [initial] segment of consecutive integers, i.e. of which no number is missing is {1,2,....,a[n]}. The first missing term from RRS(n) is 1+a(n), the least prime divisor.. E.g. n=121 : RRS[121] = {1,2,3,4,5,6,7,8,9,10,lag,12,..}, i.e. no 11 is in RRS; a[n] is the length of longest lag-free number segment consisting of consecutive integers, since A020639[n] divides n. - Labos E. (labos(AT)ana.sote.hu), May 14 2003
%F A057237 For n >= 2, a(n) = (smallest prime dividing n) - 1
%F A057237 For n >= 2, a(n) = (n-1) mod (smallest prime dividing n); cf. A083218. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 22 2003
%e A057237 a(25) = 4 because 1, 2, 3, and 4 are relatively prime to 25.
%Y A057237 Cf. A020639, A083218.
%Y A057237 Cf. A066169.
%Y A057237 Adjacent sequences: A057234 A057235 A057236 this_sequence A057238 A057239 A057240
%Y A057237 Sequence in context: A114326 A092986 A060680 this_sequence A049559 A063994 A076512
%K A057237 nonn
%O A057237 1,3
%A A057237 Leroy Quet (qq-quet(AT)mindspring.com), Sep 20 2000

%I A049559
%S A049559 1,1,2,1,4,1,6,1,2,1,10,1,12,1,2,1,16,1,18,1,4,1,22,1,4,1,2,3,28,1,30,
%T A049559 1,4,1,2,1,36,1,2,1,40,1,42,1,4,1,46,1,6,1,2,3,52,1,2,1,4,1,58,1,60,1,
%U A049559 2,1,16,5,66,1,4,3,70,1,72,1,2,3,4,1,78,1,2,1,82,1,4,1,2,1,88,1,18,1,4
%N A049559 a(n)=GCD[n-1,Phi(n)].
%D A049559 R. K. Guy, Unsolved Problems in Number Theory, B37.
%e A049559 For n=p prime 1+a(p)=GCD[p-1,p-1]=p-1+1=p. Question: do nonprimes exist with this property? For several values like n=10, a(10)=GCD[9,Phi(10)]=GCD[9,8]=1. Other values result in values between n-1 or 1, like n=33, a(33)=GCD[32,Phi(33)]=GCD[32,20]=4.
%Y A049559 Cf. A000010, A058515, A009195.
%Y A049559 Adjacent sequences: A049556 A049557 A049558 this_sequence A049560 A049561 A049562
%Y A049559 Sequence in context: A092986 A060680 A057237 this_sequence A063994 A076512 A128707
%K A049559 nonn
%O A049559 0,3
%A A049559 Labos E. (labos(AT)ana.sote.hu), Dec 28 2000

%I A063994
%S A063994 1,1,2,1,4,1,6,1,2,1,10,1,12,1,4,1,16,1,18,1,4,1,22,1,4,1,2,3,28,1,30,
%T A063994 1,4,1,4,1,36,1,4,1,40,1,42,1,8,1,46,1,6,1,4,3,52,1,4,1,4,1,58,1,60,1,
%U A063994 4,1,16,5,66,1,4,3,70,1,72,1,4,3,4,1,78,1,2,1,82,1,16,1,4,1,88,1,36,1
%N A063994 Product_{primes p dividing n } GCD(p-1, n-1).
%C A063994 a(n) = number of bases b mod n for which b^{n-1} = 1 mod n.
%D A063994 Baillie and Wagstaff, Mathematics of Computation, 35 (1980), 1391-1417.
%D A063994 P. Erdos and C. Pomerance, Mathematics of Computation, 46 (1986), 259-279.
%D A063994 Keith Gibson, posting to Number Theory List, Sep 07, 2001.
%D A063994 Carl Pomerance, posting to Number Theory List, Sep 07, 2001.
%H A063994 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b063994.txt">Table of n, a(n) for n=1..1000</a>
%H A063994 W. R. Alford, A. Granville and C. Pomerance, <a href="http://www.dms.umontreal.ca/~andrew/Postscript/carmichael.ps">There are infinitely many Carmichael numbers</a>, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
%t A063994 f[n_ ] := If[n == 1, 1, Apply[ Times, GCD[n - 1, Transpose[ FactorInteger[n]] [[1]] - 1]]]; Table[f[n], {n, 1, 100} ]
%Y A063994 Cf. A002997.
%Y A063994 Adjacent sequences: A063991 A063992 A063993 this_sequence A063995 A063996 A063997
%Y A063994 Sequence in context: A060680 A057237 A049559 this_sequence A076512 A128707 A083258
%K A063994 nonn,easy,nice
%O A063994 1,3
%A A063994 njas, Sep 18 2001
%E A063994 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 21 2001

%I A076512
%S A076512 1,1,2,1,4,1,6,1,2,2,10,1,12,3,8,1,16,1,18,2,4,5,22,1,4,6,2,3,28,4,30,
%T A076512 1,20,8,24,1,36,9,8,2,40,2,42,5,8,11,46,1,6,2,32,6,52,1,8,3,12,14,58,4,
%U A076512 60,15,4,1,48,10,66,8,44,12,70,1,72,18,8,9,60,4,78,2,2,20,82,2,64,21
%N A076512 Denominator of cototient(n)/totient(n).
%C A076512 a(n)=1 iff n=A007694(k) for some k.
%C A076512 Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Aug 26 2005
%F A076512 a(n) = A000010(n)/A009195(n).
%Y A076512 numerator = A076511, A051953.
%Y A076512 Adjacent sequences: A076509 A076510 A076511 this_sequence A076513 A076514 A076515
%Y A076512 Sequence in context: A057237 A049559 A063994 this_sequence A128707 A083258 A083259
%K A076512 nonn,frac
%O A076512 1,3
%A A076512 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 15 2002

%I A128707
%S A128707 1,1,2,1,4,1,6,1,2,3,10,1,12,5,4,1,16,1,18,3,5,9,22,1,4,11,2,5,28,1,30,
%T A128707 1,10,15,13,1,36,17,11,3,40,5,42,9,4,21,46,1,6,3,16,11,52,1,9,5,17,27,
%U A128707 58,1,60,29,5,1,24,7,66,15,22,3,70,1,72,35,4,17,20,11,78,3,2,39,82,5,33
%N A128707 Least number having the maximal distance between consecutive integers coprime to n.
%C A128707 Let j(n) be the Jacobsthal function (A048669): maximal distance between consecutive integers coprime to n. Then a(n) is the least k>0 such that k+1,k+2,...k+j(n)-1 are not coprime to n. If n is prime and e>0, then j(n^e)=2 and a(n^e)=n-1. If n>2 is prime, then a(2n)=n-2. If m is the squarefree kernal of n (A007947), then j(n)=j(m) and a(n)=a(m). For composite n, a(n)<n/2. Note that a(n)=1 iff n is in sequence A055932. When n is the product of the first r primes (A002110), then a(n)+1 begins (or is inside) a prime gap of size at least A048670(r).
%H A128707 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b128707.txt">Table of n, a(n) for n=1..10000</a>
%e A128707 The numbers coprime to 10 are 1,3,7,9,11,13,17,19,... Observe that the differences are periodic: 2,4,2,2,2,4,2,... The largest distance between the coprime numbers is 4, which first occurs between 3 and 7. Hence j(10)=4 and a(10)=3.
%t A128707 JacobsthalPos[n_] := Module[{g,d,mx,pos}, g=Select[Range[n+1], GCD[n,# ]==1&]; d=Rest[g]-Most[g]; mx=Max@@d; pos=Position[d,mx,1,1][[1,1]]; g[[pos]]]; Table[JacobsthalPos[n], {n,100}]
%Y A128707 Cf. A128708 (number of times the maximal value occurs).
%Y A128707 Adjacent sequences: A128704 A128705 A128706 this_sequence A128708 A128709 A128710
%Y A128707 Sequence in context: A049559 A063994 A076512 this_sequence A083258 A083259 A124625
%K A128707 nonn
%O A128707 1,3
%A A128707 T. D. Noe (noe(AT)sspectra.com), Mar 24 2007

%I A083258
%S A083258 1,2,1,4,1,6,1,8,1,2,1,12,1,2,3,16,1,6,1,4,3,2,1,24,1,2,1,4,1,30,1,32,3,
%T A083258 2,1,36,1,2,3,8,1,6,1,4,3,2,1,48,1,2,3,4,1,6,1,8,3,2,1,60,1,2,3,64,1,6,
%U A083258 1,4,3,10,1,72,1,2,3,4,1,6,1,16,1,2,1,12,1,2,3,8,1,30,1,4,3,2,1,96,1,2
%N A083258 a(n)=GCD[A046523(n),n].
%Y A083258 Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083255-A083260.
%Y A083258 Adjacent sequences: A083255 A083256 A083257 this_sequence A083259 A083260 A083261
%Y A083258 Sequence in context: A063994 A076512 A128707 this_sequence A083259 A124625 A009531
%K A083258 nonn
%O A083258 1,2
%A A083258 Labos E. (labos(AT)ana.sote.hu), May 09 2003

%I A083259
%S A083259 1,2,1,4,1,6,1,8,1,2,1,12,1,2,3,16,1,18,1,4,3,2,1,24,1,2,1,4,1,30,1,32,
%T A083259 3,2,1,36,1,2,3,8,1,6,1,4,3,2,1,48,1,2,3,4,1,54,1,8,3,2,1,60,1,2,3,64,1,
%U A083259 6,1,4,3,10,1,72,1,2,3,4,1,6,1,16,1,2,1,12,1,2,3,8,1,90,1,4,3,2,1,96,1
%N A083259 a(n)=GCD[A071364(n),n].
%Y A083259 Cf. A007395, A000040, A002808, A018252, A046523, A071364, A083255-A083260.
%Y A083259 Adjacent sequences: A083256 A083257 A083258 this_sequence A083260 A083261 A083262
%Y A083259 Sequence in context: A076512 A128707 A083258 this_sequence A124625 A009531 A131132
%K A083259 nonn
%O A083259 1,2
%A A083259 Labos E. (labos(AT)ana.sote.hu), May 09 2003

%I A124625
%S A124625 1,0,1,2,1,4,1,6,1,8,1,10,1,12,1,14,1,16,1,18,1
%N A124625 Even numbers sandwiched between 1's.
%C A124625 Created to simplify the definition of A129952.
%Y A124625 Adjacent sequences: A124622 A124623 A124624 this_sequence A124626 A124627 A124628
%Y A124625 Sequence in context: A128707 A083258 A083259 this_sequence A009531 A131132 A137374
%K A124625 nonn
%O A124625 0,4
%A A124625 njas, Jun 13 2007

%I A009531
%S A009531 0,1,2,1,4,1,6,1,8,1,10,1,12,1,14,1,16,1,18,1,20,1,22,1,24,1,26,1,28,
%T A009531 1,30,1,32,1,34,1,36,1,38,1,40,1,42,1,44,1,46,1,48,1,50,1,52,1,54,1,
%U A009531 56,1,58,1,60,1,62,1,64,1,66,1,68,1,70,1,72,1,74,1,76,1,78,1,80
%V A009531 0,1,2,-1,-4,1,6,-1,-8,1,10,-1,-12,1,14,-1,-16,1,18,-1,-20,1,22,-1,-24,1,26,-1,-28,
%W A009531 1,30,-1,-32,1,34,-1,-36,1,38,-1,-40,1,42,-1,-44,1,46,-1,-48,1,50,-1,-52,1,54,-1,
%X A009531 -56,1,58,-1,-60,1,62,-1,-64,1,66,-1,-68,1,70,-1,-72,1,74,-1,-76,1,78,-1,-80
%N A009531 Expansion of sin(x)*(1+x).
%F A009531 There's an obvious formula for the n-th term!
%F A009531 G.f.: x(1+x)^2/(1+x^2)^2.
%F A009531 abs(a(n))=sum{k=0..floor((n-1)/2), (C(n-k-1, k) mod 2)(-1)^k*2^A000120(n-2k-1)} - Paul Barry (pbarry(AT)wit.ie), Jan 06 2005
%F A009531 a(n) =(n^(n+1) mod (n+1)) * (-1)^[(n-1)/2] =a(n-1)-a(n-2)+(-1)^n*a(n-1) =-2a(n-2)-a(n-4). - Henry Bottomley (se16(AT)btinternet.com), May 07 2005
%F A009531 a(n+2) is the Hankel transform of A086622(n+1). - Paul Barry (pbarry(AT)wit.ie), Nov 06 2007
%Y A009531 Cf. A009001.
%Y A009531 Cf. A029578.
%Y A009531 Adjacent sequences: A009528 A009529 A009530 this_sequence A009532 A009533 A009534
%Y A009531 Sequence in context: A083258 A083259 A124625 this_sequence A131132 A137374 A131516
%K A009531 sign,easy
%O A009531 0,3
%A A009531 R. H. Hardin (rhh(AT)cadence.com)

%I A131132
%S A131132 1,0,1,2,1,4,1,6,1,8,1,10,1,12,1,14,1,16,1,18,1,20,1,22,1,24,1,26,1,28,
%T A131132 1,30,1,32,1,34,1,36,1,38,1,40,1,42,1,44,1,46,1,48,1,50,1,52,1,54,1,56,
%U A131132 1,58,1,60,1,62,1,64,1,66,1,68,1,70,1,72,1,74,1,76,1,78,1,80,1,82,1,84
%N A131132 Interleaving of A000012 and A005843.
%C A131132 a(n) = abs(A009531(n-1)).
%F A131132 a(n) = 1 for even n, a(n) = n-1 for odd n.
%F A131132 a(2*n) = 1, a(2*n+1) = 2*n.
%F A131132 G.f.: (1-x^2+2*x^3)/((1-x)^2*(1+x)^2).
%F A131132 a(n)=C(n+1,((n+2) mod 2))-1-(-1)^(n+1) - Paolo P. Lava (ppl(AT)spl.at), Aug 29 2007
%o A131132 (PARI) {for(n=0, 85, print1(if(n%2>0, n-1, 1), ","))}
%o A131132 (MAGMA) &cat[ [ 1, 2*k ]: k in [0..42] ];
%Y A131132 Cf. A000012 (all 1's), A005843 (even numbers), A009531.
%Y A131132 Adjacent sequences: A131129 A131130 A131131 this_sequence A131133 A131134 A131135
%Y A131132 Sequence in context: A083259 A124625 A009531 this_sequence A137374 A131516 A088140
%K A131132 nonn,easy
%O A131132 0,4
%A A131132 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 16 2007

%I A137374
%S A137374 2,1,4,1,6,1,8,8,1,20,10,1,16,36,12,1,56,56,14,1,32,128,80,16,1,144,240,
%T A137374 108,18,1,64,400,400,140,20,1
%N A137374 Triangular sequence of coefficients of Jacobsthal-Lucas polynomials as defined in the Lucas.m MathWorld package for Mathematica.
%C A137374 Row sums are A014551
%D A137374 Weisstein, Eric W. "Jacobsthal-Lucas Polynomial." http://mathworld.wolfram.com/Jacobsthal-LucasPolynomial.html
%F A137374 "The Jacobsthal polynomials are the w-polynomials obtained by setting p(x)=1 and q(x)=2x in the Lucas polynomial sequence. "; Jacobsthalj[n, x]
%e A137374 {2},
%e A137374 {1},
%e A137374 {4, 1},
%e A137374 {6, 1},
%e A137374 {8, 8, 1},
%e A137374 {20, 10, 1},
%e A137374 {16, 36, 12, 1},
%e A137374 {56, 56, 14, 1},
%e A137374 {32, 128, 80, 16, 1},
%e A137374 {144, 240, 108, 18, 1},
%e A137374 {64, 400, 400, 140, 20, 1}
%t A137374 << Lucas`; Table[ExpandAll[Jacobsthalj[n, x]], {n, 0, 10}]; a = Table[Reverse[CoefficientList[Jacobsthalj[n, x], x]], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[Jacobsthalj[n, x], x]], {n, 0, 10}]
%Y A137374 Cf. A014551.
%Y A137374 Adjacent sequences: A137371 A137372 A137373 this_sequence A137375 A137376 A137377
%Y A137374 Sequence in context: A124625 A009531 A131132 this_sequence A131516 A088140 A130758
%K A137374 nonn,uned
%O A137374 1,1
%A A137374 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 09 2008

%I A131516
%S A131516 0,1,2,1,4,1,6,1,8,9,10,1,12,1,14,15,16,1,18,1,20,21,22,1,24,25,26,27,
%T A131516 28,1,30,1,32,33,34,35,36,1,38,39,40,1,42,1,44,45,46,1,48,49,50,51,52,1,
%U A131516 54,55,56,57,58,1,60,1,62,63,64,65,66,1,68
%N A131516 a(n)=1 if n is an odd prime number, otherwise, a(n)=n.
%Y A131516 Cf. A000027; A001477; A006005.
%Y A131516 Adjacent sequences: A131513 A131514 A131515 this_sequence A131517 A131518 A131519
%Y A131516 Sequence in context: A009531 A131132 A137374 this_sequence A088140 A130758 A130892
%K A131516 nonn
%O A131516 2,3
%A A131516 Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 14 2007

%I A088140
%S A088140 2,1,4,1,6,1,8,9,10,1,12,1,14,15,16,1,18,1,20,21,22,1,24,25,26,27,28,1,
%T A088140 30,1,32,33,34,35,36,1,38,39,40,1,42,1,44,45,46,1,48,49,50,51,52,1,54,
%U A088140 55,56,57,58,1,60,1,62,63,64,65,66,1,68,69,70,1,72,1,74,75,76,77,78,1
%N A088140 a(n) = 1 if n is an odd prime otherwise a(n) = n.
%C A088140 From the factorial identity: n!=Product[Prime[i],{i,1,PrimePi[n]}]*Product[Composite[i],{i,1,n-PrimePi[n]}]
%F A088140 a(n) = Product[Composite[i], {i, 1, n-PrimePi[n]]/Product[Composite[i], {i, 1, n-1-PrimePi[n-1]]
%t A088140 (* Composite Product*) p[n_]=n!/Product[Prime[i], {i, 2, PrimePi[n]}] digits=200 a0=Table[p[n]/p[n-1], {n, 2, digits}] (* Composites by sorting out ones and two*) Delete[Delete[Union[a0], 1], 1]
%Y A088140 Cf. A002808.
%Y A088140 Adjacent sequences: A088137 A088138 A088139 this_sequence A088141 A088142 A088143
%Y A088140 Sequence in context: A131132 A137374 A131516 this_sequence A130758 A130892 A074643
%K A088140 nonn
%O A088140 2,1
%A A088140 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 04 2003

%I A130758
%S A130758 0,1,2,1,4,1,6,1,8,9,10,2,12,2,14,15,16,2,18,2,20,21,22,2,24,25,26,27,
%T A130758 28,2,30,2,32,33,34,35,36,2,38,39,40,2,42,2,44,45,46,2,48,49,50,51,52,2,
%U A130758 54,55,56,57,58,2,60,2,62,63,64,65,66,2,68
%N A130758 a(n)=n if n is not an odd prime number. Otherwise, a(n)=k, where k is the smallest integer such that n<10^k.
%Y A130758 Cf. A000027; A001477; A006005; A131516.
%Y A130758 Adjacent sequences: A130755 A130756 A130757 this_sequence A130759 A130760 A130761
%Y A130758 Sequence in context: A137374 A131516 A088140 this_sequence A130892 A074643 A060794
%K A130758 nonn
%O A130758 2,3
%A A130758 Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 17 2007

%I A130892
%S A130892 0,1,2,1,4,1,6,1,8,9,10,4,12,4,14,15,16,4,18,4,20,21,22,6,24,25,26,27,
%T A130892 28,6,30,8,32,33,34,35,36,8,38,39,40,10,42,10,44,45,46,10,48,49,50,51,
%U A130892 52,12,54,55,56,57,58,12,60,14,62,63,64,65,66,14,68
%N A130892 a(n)=n if n is not an odd prime number. Otherwise, a(n)=k*ceiling(n/10), where k is the smallest integer such that n<10^k.
%Y A130892 Cf. A131516; A130758; A000027; A001477; A006005.
%Y A130892 Adjacent sequences: A130889 A130890 A130891 this_sequence A130893 A130894 A130895
%Y A130892 Sequence in context: A131516 A088140 A130758 this_sequence A074643 A060794 A074919
%K A130892 nonn
%O A130892 2,3
%A A130892 Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 21 2007

%I A074643
%S A074643 1,1,2,1,4,1,6,2,2,1,10,1,4,3,2,4,16,3,18,2,3,1,22,1,4,3,6,1,28,2,30,8,
%T A074643 1,2,12,3,36,3,3,4,40,3,14,1,12,11,46,1,14,5,4,2,52,9,20,6,3,7,58,1,60,
%U A074643 5,9,16,24,5,66,2,11,3,70
%N A074643 Denominator of A074639(n)/A000010(n).
%Y A074643 Cf. A074642.
%Y A074643 Adjacent sequences: A074640 A074641 A074642 this_sequence A074644 A074645 A074646
%Y A074643 Sequence in context: A088140 A130758 A130892 this_sequence A060794 A074919 A138009
%K A074643 nonn,frac
%O A074643 1,3
%A A074643 Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 12, 2002

%I A060794
%S A060794 1,2,1,4,1,6,2,2,3,10,1,12,5,2,2,16,3,18,1,4,9,22,2,4,11,6,3,28,1,30,4,
%T A060794 8,15,2,2,36,17,10,3,40,1,42,7,4,21,46,2,6,5,14,9,52,3,6,1,16,27,58,4,
%U A060794 60,29,2,4,8,5,66,13,20,3,70,1,72,35,10,15,4,7,78,2,6,39,82,5,12,41,26
%N A060794 Difference between upper and lower central divisors of n.
%e A060794 Difference between upper and lower central divisors may be small or relatively large. So neither A060775 nor A033677 are always good central divisors as to their magnitude. n=182,D={1,2,7,13,14,26,91,182}; central divisors={13,14}, difference=1. n=254, D={1,2,127,254}, central divisors={2,127}, a(254)=125. n=p, D={1,p}. Here the central divisors are also marginal ones: a(p)=p-1.
%t A060794 a(n)=Part[Divisors[n], 1+cd[n]]-Part[Divisors[n], cd[n]], where cd[x_] := cd[x_] := Floor[DivisorSigma[0, x]/2]
%Y A060794 a(n)=A033677(n)-A060775(n).
%Y A060794 Cf. A060775-A060777, A033677, A000196.
%Y A060794 Adjacent sequences: A060791 A060792 A060793 this_sequence A060795 A060796 A060797
%Y A060794 Sequence in context: A130758 A130892 A074643 this_sequence A074919 A138009 A131755
%K A060794 nonn
%O A060794 2,2
%A A060794 Labos E. (labos(AT)ana.sote.hu), Apr 27 2001

%I A074919
%S A074919 1,1,2,1,4,1,6,2,4,2,10,1,12,3,5,4,16,2,18,3,7,5,22,3,16,6,12,5,28,2,
%T A074919 30,8,13,8,17,4,36,9,15,6,40,3,42,9,13,11,46,5,36,8,21,11,52,6,29,10,
%U A074919 23,14,58,4,60,15,20,16,36,6,66,15,29,8,70,8,72,18,21,17,47,7,78,13,36
%N A074919 Number of integers in {1, 2, ..., phi(n)} that are coprime to n.
%C A074919 Compare the definition of a(n) to phi(n) = number of integers in {1, 2, ..., n} that are coprime to n.
%e A074919 There are four numbers in {1, 2, ..., phi(8) = 4} that are coprime to 8, i.e. 1, 3. Hence a(8) = 2.
%t A074919 h[n_] := Module[{l}, l = {}; For[i = 1, i <= EulerPhi[n], i++, If[GCD[i, n] == 1, l = Append[l, i]]]; l]; Table[Length[h[i]], {i, 1, 100}]
%Y A074919 Adjacent sequences: A074916 A074917 A074918 this_sequence A074920 A074921 A074922
%Y A074919 Sequence in context: A130892 A074643 A060794 this_sequence A138009 A131755 A118275
%K A074919 nonn
%O A074919 1,3
%A A074919 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Oct 04 2002

%I A138009
%S A138009 1,1,2,1,4,1,6,2,4,3,10,1,12,5,6,2,16,2,18,3,10,11,22,1,15,13,14,5,28,2,
%T A138009 30,7,18,19,20,1,36,22,23,4,40,5,42,11,12,28,46,1,33,14,31,15,52,7,34,8,
%U A138009 36,37,58,1,60,39,19,10,42,10,66,22,45,11,70,2,72,48,25,26,51,13,78,4
%N A138009 a(n) = number of positive integers k, k <= n, where d(k) >= d(n); d(n) = number of positive divisors of n.
%e A138009 9 has 3 positive divisors. Among the first 9 positive integers, there are four that have more than or equal the number of divisors than 9 has: 4, with 3 divisors; 6, with 4 divisors; 8, with 4 divisors; and 9, with 3 divisors. So a(9) = 4.
%t A138009 Table[Length[Select[Range[n], Length[Divisors[ # ]]>=Length[Divisors[n]]&]], {n,1,100}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 29 2008
%Y A138009 Cf. A079788, A067004.
%Y A138009 Adjacent sequences: A138006 A138007 A138008 this_sequence A138010 A138011 A138012
%Y A138009 Sequence in context: A074643 A060794 A074919 this_sequence A131755 A118275 A128099
%K A138009 nonn
%O A138009 1,3
%A A138009 Leroy Quet (qq-quet(AT)mindspring.com), Feb 27 2008
%E A138009 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 29 2008

%I A131755
%S A131755 1,2,1,4,1,6,2,4,3,10,2,12,4,4,3,16,3,18,3,6,7,22,3,12,8,8,5,28,4,30,6,
%T A131755 10,11,11,4,36,12,12,5,40,5,42,8,8,15,46,5,24,9,16,10,52,7,18,7,18,19,
%U A131755 58,5,60,20,12,10,21,9,66,13,22,9,70,6,72,24,14,15,25,11,78,8
%N A131755 a(n) = floor of the average of distances between consecutive positive divisors of n. Also, a(n) = floor((n-1)/(d(n)-1)), where d(n) = A000005(n).
%C A131755 (n-1)/(d(n)-1) is an integer if and only if n is in sequence A096738.
%e A131755 The positive divisors of 12 are 1,2,3,4,6,12. The differences between the pairs of consecutive divisors are 2-1=1, 3-2=1, 4-3=1, 6-4=2, 12-6=6. The average of these differences is (1+1+1+2+6)/5 = 11/5. So a(12) = floor(11/5) = 2.
%p A131755 A131755 := proc(n) local dvs ; dvs := sort(convert(numtheory[divisors](n),list)) ; floor(add(op(i,dvs)-op(i-1,dvs) ,i=2..nops(dvs))/(nops(dvs)-1)) ; end: seq(A131755(n),n=2..80) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2007
%Y A131755 Cf. A000005, A096738.
%Y A131755 Adjacent sequences: A131752 A131753 A131754 this_sequence A131756 A131757 A131758
%Y A131755 Sequence in context: A060794 A074919 A138009 this_sequence A118275 A128099 A108952
%K A131755 nonn
%O A131755 2,2
%A A131755 Leroy Quet (qq-quet(AT)mindspring.com), Sep 17 2007
%E A131755 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2007

%I A118275
%S A118275 1,1,2,1,4,1,6,3,6,4,2,6,5,6
%N A118275 a(0) = 1. a(n) is the number of times the binary representation of a(n-1) appears in the concatenated string of the terms a(0) through a(n-1) written in binary. (The concatenated string is written from left to right, and each binary integer is written so the most significant 1 is on the left.).
%C A118275 Sequence A118274 is the string of terms of this sequence written in binary and concatenated.
%e A118275 The string of concatenated binary representations of a(0) through a(7) is
%e A118275 11101100111011. Now a(7)= 3, which is 11 in binary. '11' occurs 6 times in the string (with, in this case, some binary digits in the string being used more than once). (The six '11's occur at {with position 1 on the left} positions 1, 2, 5, 9, 10, and 13.) So a(8) = 6. (And '1,1,0' is appended to the end of sequence A118274.)
%Y A118275 Cf. A118274.
%Y A118275 Adjacent sequences: A118272 A118273 A118274 this_sequence A118276 A118277 A118278
%Y A118275 Sequence in context: A074919 A138009 A131755 this_sequence A128099 A108952 A088522
%K A118275 easy,more,nonn
%O A118275 0,3
%A A118275 Leroy Quet (qq-quet(AT)mindspring.com), Apr 21 2006

%I A128099
%S A128099 1,1,1,2,1,4,1,6,4,1,8,12,1,10,24,8,1,12,40,32,1,14,60,80,16,1,16,84,
%T A128099 160,80,1,18,112,280,240,32,1,20,144,448,560,192,1,22,180,672,1120,672,
%U A128099 64,1,24,220,960,2016,1792,448,1,26,264,1320,3360,4032,1792,128,1,28
%N A128099 Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0<=k<=floor(n/2)).
%C A128099 Row sums are the Jacobstahl numbers (A001045). Sum(k(T(n,k),k=0..floor(n/2))=A095977(n-1).
%F A128099 T(n,k)=2^k*binom(n-k,k). G.f.=1/(1-z-2tz^2).
%e A128099 Triangle starts:
%e A128099 1;
%e A128099 1;
%e A128099 1,2;
%e A128099 1,4;
%e A128099 1,6,4;
%e A128099 1,8,12;
%e A128099 1,10,24,8;
%e A128099 1,12,40,32;
%p A128099 T:=proc(n,k) if k<=n/2 then 2^k*binomial(n-k,k) else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%Y A128099 Cf. A001045, A095977.
%Y A128099 Adjacent sequences: A128096 A128097 A128098 this_sequence A128100 A128101 A128102
%Y A128099 Sequence in context: A138009 A131755 A118275 this_sequence A108952 A088522 A115124
%K A128099 nonn,tabf
%O A128099 0,4
%A A128099 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2007

%I A108952
%S A108952 0,2,1,4,1,6,9,3,7,7,0,6,2,3,2,6,4,9,2,4,7,8,9,3,4,8,1,8,8,9,3,1,6,1,7,
%T A108952 8,3,4,1,3,8,0,9,0,1,5,6,5,9,0,4,5,4,4,3,5,0,0,1,8,1,4,5,7,1,9,1,6,6,7,
%U A108952 7,3,6,1,8,1,1,3,9,2,7,9,7,9,8,7,0,3,5,3,3,8,7,1,3,8,9,8,4,0,0,6,2,5,9
%N A108952 Decimal expansion of 1/delta, where delta = Feigenbaum constant.
%C A108952 In a dynamical system exhibiting period-doubling cascades,1/delta is the limiting ratio by which the nonlinear parameter's range of stable values shrinks after each successive onset of period-doubling.
%e A108952 1/delta=0.21416937706232649247893481889316178341380
%Y A108952 Cf. A006890.
%Y A108952 Adjacent sequences: A108949 A108950 A108951 this_sequence A108953 A108954 A108955
%Y A108952 Sequence in context: A131755 A118275 A128099 this_sequence A088522 A115124 A115122
%K A108952 nonn,cons
%O A108952 1,2
%A A108952 Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 2005

%I A088522
%S A088522 2,1,4,1,6,12,2,10,19,0,11,23,36,7,22,38,55,12,31,51,72,15,38,62,87,12,
%T A088522 39,67,96,13,44,76,109,4,39,75,112,150,22,62,103,145,188,39,84,130,177,
%U A088522 2,51,101,152,204,16,70,125,181,238,25,84,144,205,267,23,87,152,218,285
%N A088522 a(1) = 2; for n > 1, a(n) = (a(n-1) + n) mod prime(n).
%t A088522 a[1] = 2; a[n_] := a[n] = Mod[a[n - 1] + n, Prime[n]]; Table[ a[n], {n, 67}]
%Y A088522 Cf. A088521, A088523, A006257.
%Y A088522 Adjacent sequences: A088519 A088520 A088521 this_sequence A088523 A088524 A088525
%Y A088522 Sequence in context: A118275 A128099 A108952 this_sequence A115124 A115122 A097360
%K A088522 nonn,easy
%O A088522 1,1
%A A088522 njas, Nov 14 2003
%E A088522 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 15 2003

%I A115124
%S A115124 0,0,1,1,2,1,4,1,7,3,14,1,40,1,86,15,257,1,797,1,2523,87,8360,1,29218,
%T A115124 13,101341,765
%N A115124 Number of imprimitive (periodic) 2n-bead black-white reversible complementable necklaces with n black beads.
%C A115124 a(p)=1 for prime p.
%F A115124 a(n)=A006840(n) - A045633(n).
%Y A115124 Adjacent sequences: A115121 A115122 A115123 this_sequence A115125 A115126 A115127
%Y A115124 Sequence in context: A128099 A108952 A088522 this_sequence A115122 A097360 A072345
%K A115124 easy,nonn
%O A115124 0,5
%A A115124 Valery A. Liskovets (liskov(AT)im.bas-net.by), Jan 17 2006

%I A115122
%S A115122 0,0,1,1,2,1,4,1,7,3,16,1,49,1,129,17,415,1,1408,1,4659,130,16081,1,
%T A115122 56858,15,200171,1367
%N A115122 Number of imprimitive (periodic) 2n-bead black-white complementable necklaces with n black beads.
%C A115122 a(p)=1 for prime p.
%F A115122 a(n)=A045629(n) - A045632(n).
%Y A115122 Adjacent sequences: A115119 A115120 A115121 this_sequence A115123 A115124 A115125
%Y A115122 Sequence in context: A108952 A088522 A115124 this_sequence A097360 A072345 A115120
%K A115122 easy,nonn
%O A115122 0,5
%A A115122 Valery A. Liskovets (liskov(AT)im.bas-net.by), Jan 17 2006

%I A097360
%S A097360 1,1,1,2,1,4,1,7,4,12,1,21,1,34,17,55,1,88,1,137,60,210,1,320,30,478,
%T A097360 191,708,1,1039,1,1507,556,2167,150,3094,1,4378,1510,6153,1,8591,1,
%U A097360 11914,3872,16424,1,22519,304,30701,9465,41646,1,56224,2464,75547,22210
%N A097360 Number of partitions of n into parts not less than the smallest prime factor of n.
%C A097360 a(n) = 1 iff n=1 or n is prime: a(A000040(n)) = 1.
%Y A097360 Cf. A020639, A000041, A097359.
%Y A097360 Adjacent sequences: A097357 A097358 A097359 this_sequence A097361 A097362 A097363
%Y A097360 Sequence in context: A088522 A115124 A115122 this_sequence A072345 A115120 A107061
%K A097360 nonn
%O A097360 1,4
%A A097360 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 09 2004

%I A072345
%S A072345 1,2,1,4,1,8,1,16,1,32,1,64,1,128,1,256,1,512,1,1024,1,2048,1,4096,1,8192,
%T A072345 1,16384,1,32768,1,65536,1,131072,1,262144,1,524288,1,1048576,1,2097152,
%U A072345 1,4194304,1,8388608,1,16777216,1,33554432,1,67108864,1,134217728,1,268435456
%N A072345 Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives numerator of C_n.
%C A072345 Answer to question of how to extend the sequence 1, 2 r, Pi r^2, 4 Pi r^3 / 3, Pi^2 r^4 / 2, ...
%C A072345 Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1). - see A072478/A072479.
%D A072345 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.
%H A072345 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>
%H A072345 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ball.html">Ball</a>
%H A072345 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Four-DimensionalGeometry.html">Four-Dimensional Geometry</a>
%F A072345 1 if n even, 2^((n+1)/2) if n odd.
%e A072345 Sequence of C_n's begins 1, 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, 1/24, 32/945, 1/120, 64/10395, ...
%p A072345 seq(seq(k^n, k=1..2), n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 29 2007
%t A072345 f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 55} ]
%Y A072345 Cf. A072346.
%Y A072345 Adjacent sequences: A072342 A072343 A072344 this_sequence A072346 A072347 A072348
%Y A072345 Sequence in context: A115124 A115122 A097360 this_sequence A115120 A107061 A112481
%K A072345 nonn,frac
%O A072345 0,2
%A A072345 njas, Jul 31 2002

%I A115120
%S A115120 0,0,1,1,2,1,4,1,8,3,17,1,56,1,134,18,440,1,1434,1,4758,135,16160,1,
%T A115120 57254,16
%N A115120 Number of imprimitive (periodic) 2n-bead black-white reversible necklaces with n black beads.
%C A115120 a(p)=1 for prime p.
%F A115120 a(n)=A005648(n) - A0045628(n).
%Y A115120 Adjacent sequences: A115117 A115118 A115119 this_sequence A115121 A115122 A115123
%Y A115120 Sequence in context: A115122 A097360 A072345 this_sequence A107061 A112481 A134851
%K A115120 easy,nonn
%O A115120 0,5
%A A115120 Valery A. Liskovets (liskov(AT)im.bas-net.by), Jan 17 2006

%I A107061
%S A107061 1,1,2,1,4,1,8,5,4,3,6,5,8,5,4,5,6,5,4,3,10,3,4,1,12,7,8,3,2,1,6,9,2,3,
%T A107061 12,1,4,3,8,27,10,1,6,9,2,3,6,5,8,17,4,5,8,13,8,3,16,9,6,11,28,3,32,9,2,
%U A107061 7,12,5,14,9,10,1,10,9,4,15,6,5,8,29,2,15,24,1,6,15,16,15,10,11,2,3,2
%N A107061 a(n) = largest number m >0 such that n*prime(n)-a(n) is a prime.
%C A107061 Cf. A107060: a(n)>=0 such that n*prime(n)-a(n) is the largest prime
%Y A107061 Cf. A107060.
%Y A107061 Adjacent sequences: A107058 A107059 A107060 this_sequence A107062 A107063 A107064
%Y A107061 Sequence in context: A097360 A072345 A115120 this_sequence A112481 A134851 A038001
%K A107061 nonn
%O A107061 1,3
%A A107061 Zak Seidov (zakseidov(AT)yahoo.com), May 10 2005

%I A112481
%S A112481 1,1,2,1,4,1,8,8,2,2,12,28,16,1,12,4
%N A112481 Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.
%C A112481 T(g, w, h) = |{m : A062537(m) = g, A001221(m) = w, A109301(m) = h}|.
%C A112481 This is the column that is labeled "r" in the tabulation of A112480.
%C A112481 a(n) is a permutation of the elements in A112096.
%C A112481 g = h > 0 implies w = 1, and T(j, 1, j) = 2^(j-1) = A000079(j-1).
%e A112481 Table T(g, w, h), omitting empty cells, starts out as follows:
%e A112481 --------+-------------------------------------------------------
%e A112481 g\(w,h) | (0,0) (1,1) (1,2) ` ` ` (1,3) ` ` ` (1,4) ` ` ` (1,5)
%e A112481 ` ` ` ` | ` ` ` ` ` ` ` ` ` (2,2) ` ` ` (2,3) ` ` ` (2,4) ` ` `
%e A112481 ========+=======================================================
%e A112481 0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 2 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 3 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
%e A112481 3 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` ` ` 8 ` ` ` ` ` ` `
%e A112481 4 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `12 ` ` ` ` `28 ` ` ` ` `16 `
%e A112481 5 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` `12 ` ` ` ` ` 4 ` ` ` `
%e A112481 --------+-------------------------------------------------------
%e A112481 Row sums = A111797. Horizontal section sums = A061396.
%Y A112481 Cf. A000079, A001221, A007097, A014221, A050924.
%Y A112481 Cf. A061396, A062504, A062537, A062860, A106177.
%Y A112481 Cf. A109300, A109301, A111791 to A111800.
%Y A112481 Cf. A112095, A112096, A112480.
%Y A112481 Adjacent sequences: A112478 A112479 A112480 this_sequence A112482 A112483 A112484
%Y A112481 Sequence in context: A072345 A115120 A107061 this_sequence A134851 A038001 A104620
%K A112481 nonn,tabf
%O A112481 1,3
%A A112481 Jon Awbrey (jawbrey(AT)att.net), Sep 27 2005

%I A134851
%S A134851 0,1,1,1,1,1,2,1,4,1,8,9,1,39,11,1,3,10,7,1
%N A134851 Number of primes between A001605(n) and A001605(n+1).
%t A134851 a = {}; k = {}; Do[If[PrimeQ[Fibonacci[n]], AppendTo[k, n]], {n, 1, 1000}]; Do[AppendTo[a, PrimePi[k[[n + 1]]] - PrimePi[k[[n]]]], {n, 1, 20}]; a( * Artur Jasinski *)
%Y A134851 Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851.
%Y A134851 Adjacent sequences: A134848 A134849 A134850 this_sequence A134852 A134853 A134854
%Y A134851 Sequence in context: A115120 A107061 A112481 this_sequence A038001 A104620 A024539
%K A134851 nonn
%O A134851 1,7
%A A134851 Artur Jasinski (grafix(AT)csl.pl), Nov 13 2007

%I A038001
%S A038001 1,1,1,1,2,1,4,1,9,2,20,1,48,4,115,1,286,9,719,2,1842,20,4766,1,
%T A038001 12486,48,32973,4,87811,115,235381,1,634847,286,1721159,9,4688676,
%U A038001 719,12826228,2,35221832,1842,97055181,20,268282855,4766,74372498
%N A038001 Inverse WEIGH transform of A038000.
%H A038001 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%Y A038001 Cf. A000081.
%Y A038001 Adjacent sequences: A037998 A037999 A038000 this_sequence A038002 A038003 A038004
%Y A038001 Sequence in context: A107061 A112481 A134851 this_sequence A104620 A024539 A128271
%K A038001 nonn
%O A038001 1,5
%A A038001 Christian G. Bower (bowerc(AT)usa.net)

%I A104620
%S A104620 1,2,1,4,1,9,6,1,8,2,3,1,4,2,19,10,1,7,2,5,31,8,1,6,2,10,18,3,14,1,7,2,
%T A104620 11,12,3,10,4,1,29,2,8,13,3,12,62,13,1,5,2,12,6,3,9,23,73,12,1,9,2,13,
%U A104620 11,3,16,7,80,4,22,1,8,2,6,15,3,18,19,10,4,37,11,1,9,2,13,70,3,7,26,16
%N A104620 Consider the presentation of the digits of the natural numbers in a triangular form for successive bases, b. Now examine the main diagonal of these triangles and note the first occurrence of the n digits (0 through b-1). This is its own triangle presented here.
%C A104620 See A104606 through A104613, A091425, A104614 through A104619 as examples in the OEIS data base for triangular forms to base n>1.
%C A104620 t(n,2)=1, t(n,4)=2, t(n,7)=3, t(n,11)=4, t(n,16)=5 and t(n,1+i(i+1)/2)=i.
%C A104620 1
%C A104620 2 1
%C A104620 4 1 9
%C A104620 6 1 8 2
%C A104620 3 1 4 2 19
%C A104620 10 1 7 2 5 31
%t A104620 f[n_] := If[n == 1, 0, Block[{t = Flatten[ IntegerDigits[ Range[ 2000], n]]}, u = t[[ Table[ i(i + 1)/2, {i, 100}]]]; Table[ Position[u, i, 1, 1], {i, 0, n - 1}]]]; Flatten[ Table[ f[n], {n, 13}]]
%Y A104620 Cf. A104606, A104607, A104608, A104609, A104610, A104611, A104612, A104613, A091425, A104614, A104615, A104616, A104617, A104618, A104619.
%Y A104620 Adjacent sequences: A104617 A104618 A104619 this_sequence A104621 A104622 A104623
%Y A104620 Sequence in context: A112481 A134851 A038001 this_sequence A024539 A128271 A092891
%K A104620 base,nonn,tabl
%O A104620 1,2
%A A104620 Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 17 2005

%I A024539
%S A024539 2,1,4,1,14,2,1,3,1,7,1,1,2,1,4,1,24,2,1,3,1,8,1,1,2,1,5,1,82,2,1,3,1,12,2,
%T A024539 1,3,1,6,1,1,2,1,4,1,18,2,1,3,1,8,1,1,2,1,5,1,41,2,1,3,1,10,1,1,2,1,6,1,1,2,
%U A024539 1,4,1,15,2,1,3,1,7,1,1,2,1,4,1,27,2,1,3,1,9,1,1,2,1,5,1,140,2
%N A024539 a(n) = [ 1/{n*sqrt(2)} ], where {x} := x - [ x ].
%Y A024539 Adjacent sequences: A024536 A024537 A024538 this_sequence A024540 A024541 A024542
%Y A024539 Sequence in context: A134851 A038001 A104620 this_sequence A128271 A092891 A080212
%K A024539 nonn
%O A024539 1,1
%A A024539 Clark Kimberling (ck6(AT)evansville.edu)

%I A128271
%S A128271 1,1,2,1,4,1,16,1,64,3,32,27,256,27,1024,243,1024,243,512,27,1024,27,
%T A128271 1024,243,8192,243,16384,243,4096,243,7168,81,12544,243,15680,27,39200,
%U A128271 27,62720,243,313600,243,1568000,27,31360000,27,17920000,27,31360000,27
%N A128271 a(n) = the denominator of b(n): {b(n)} is such that the continued fraction (of rational terms) [b(1);b(2),...,b(n)] equals the n-th prime, for every positive integer n.
%H A128271 Diana Mecum, <a href="http://www.research.att.com/~njas/sequences/b128271.txt">Table of n, a(n) for n = 1..500</a>
%e A128271 b(n): 2, 1, -3/2, 4, -3/4, 12, -3/16,...
%e A128271 The 4th prime, 7, equals [b(1);b(2),b(3),b(4)] = 2 +1/(1 +1/(-3/2 +1/4)).
%e A128271 The 5th prime, 11, equals [b(1);b(2),b(3),b(4),b(5)] = 2 +1/(1 +1/(-3/2 +1/(4 -4/3))).
%Y A128271 Cf. A128270.
%Y A128271 Adjacent sequences: A128268 A128269 A128270 this_sequence A128272 A128273 A128274
%Y A128271 Sequence in context: A038001 A104620 A024539 this_sequence A092891 A080212 A009832
%K A128271 frac,nonn
%O A128271 1,3
%A A128271 Leroy Quet (qq-quet(AT)mindspring.com), Feb 22 2007
%E A128271 More terms from Diana Mecum (diana.mecum(AT)gmail.com), Jun 24 2007

%I A092891
%S A092891 2,1,4,1,16,2,1,1,1,1,9,2,1,1,2,1,4,2,2,3,2,8,17,2,1,8,19,7,1,2,4,1,1,
%T A092891 14,1,1,9,11,4,5,1,6,4,65,15,13,1,1,5,1,1,1,79,11,14,4,13,1,2,1,7,14,1,
%U A092891 20,4,8,1,29,23,4,1,11,26,26,1,1,5,22,5,75,2,1,1,1,3,4,2,43,1,11,11,4,5
%N A092891 Greatest common divisor of quadruples a,b,c,d such that a < b < c < d, (a*b*c) mod (a+b+c) = d, (a*b*d) mod (a+b+d) = c, (a*c*d) mod (a+c+d) = b, (b*c*d) mod (b+c+d) = a. The quadruples are ordered according to sum of first three components, secondary by first component, thirdly by second component.
%C A092891 First, second, third and fourth component of the quadruples are resp. in A092887, A092888, A092889, A092890.
%e A092891 The third quadruple is 12, 60, 128, 160, hence a(3) = gcd(4*3,4*3*5,4*32,4*8*5) = 4.
%o A092891 (PARI) {m=1760;for(n=6,m, for(a=1,(n-3)\3, for(b=a+1,(n-a-1)\2,c=n-a-b;d=a*b*c%(a+b+c); if(c<d,if(a*b*d%(a+b+d) == c,if(a*c*d%(a+c+d) == b,if(b*c*d%(b+c+d) == a,print1(gcd(a,gcd(b,gcd(c,d))), ","))))))))}
%Y A092891 Cf. A092887, A092888, A092889, A092890.
%Y A092891 Adjacent sequences: A092888 A092889 A092890 this_sequence A092892 A092893 A092894
%Y A092891 Sequence in context: A104620 A024539 A128271 this_sequence A080212 A009832 A016445
%K A092891 nonn
%O A092891 1,1
%A A092891 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 12 2004

%I A080212
%S A080212 1,2,1,4,1,20,1,8,84,252,1,220,1,3432,455,16,1,816,1,15504,1330,705432,
%T A080212 1,2024,53130,10400600,2925,1184040,1,4060,1,32,5456,2333606220,324632,
%U A080212 7140,1,35345263800,9139,658008,1,11480,1,7669339132,14190
%N A080212 Binomial(n, smallest odd prime factor of n).
%C A080212 For n>1: a(n)=1 iff n is prime.
%Y A080212 Cf. A078701, A080211, A080213.
%Y A080212 Adjacent sequences: A080209 A080210 A080211 this_sequence A080213 A080214 A080215
%Y A080212 Sequence in context: A024539 A128271 A092891 this_sequence A009832 A016445 A130544
%K A080212 nonn
%O A080212 1,2
%A A080212 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 06 2003

%I A009832
%S A009832 0,1,2,1,4,1,62,1,1384,1,50522,1,2702764,1,199360982,1,19391512144,1,
%T A009832 2404879675442,1,370371188237524,1,69348874393137902,1,
%U A009832 15514534163557086904,1,4087072509293123892362,1
%V A009832 0,1,2,1,-4,1,62,1,-1384,1,50522,1,-2702764,1,199360982,1,-19391512144,1,
%W A009832 2404879675442,1,-370371188237524,1,69348874393137902,1,
%X A009832 -15514534163557086904,1,4087072509293123892362,1
%N A009832 Expansion of tanh(x).exp(x).
%t A009832 Tanh[ x ]*Exp[ x ]
%Y A009832 Adjacent sequences: A009829 A009830 A009831 this_sequence A009833 A009834 A009835
%Y A009832 Sequence in context: A128271 A092891 A080212 this_sequence A016445 A130544 A019441
%K A009832 sign,easy
%O A009832 0,3
%A A009832 R. H. Hardin (rhh(AT)cadence.com)
%E A009832 Extended with signs Mar 15 1997 by Olivier Gerard.

%I A016445
%S A016445 2,1,4,1,230,1,2,50,1,11,2,2,1,1,7,1,8,1,90,1,3,2,5,5,1,
%T A016445 2,3,2,4,1,17,1,401,13,2,1,22,3,1,8,1,22,1,4,1,1,3,1,2,
%U A016445 3,3,2,7,32,1,14,2,1,1,1,3,1,6,58,1,1,1,1,1,1,3,2,6,1,3
%N A016445 Continued fraction for ln(17).
%Y A016445 Adjacent sequences: A016442 A016443 A016444 this_sequence A016446 A016447 A016448
%Y A016445 Sequence in context: A092891 A080212 A009832 this_sequence A130544 A019441 A007739
%K A016445 nonn,cofr
%O A016445 1,1
%A A016445 njas

%I A130544
%S A130544 0,0,0,0,0,1,2,1,4,2,1,1,1,3,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,
%T A130544 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A130544 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A130544 Multiplicative persistence of n!!.
%C A130544 From 24!! on all the numbers have same digits equal to zero thus the persistence is equal to 1.
%e A130544 6!!= 6*4*2= 48 --> 4*8=32 --> 3*2= 6 --> Persistence=2
%e A130544 13!!=135135 --> 1*3*5*1*3*5=225 -->2*2*5=20 --> 2*0=0 --> Persistence=3
%p A130544 P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; w:=1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);
%Y A130544 Cf. A031346, A130543.
%Y A130544 Adjacent sequences: A130541 A130542 A130543 this_sequence A130545 A130546 A130547
%Y A130544 Sequence in context: A080212 A009832 A016445 this_sequence A019441 A007739 A031424
%K A130544 easy,nonn
%O A130544 0,7
%A A130544 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), Jun 04 2007

%I A019441
%S A019441 1,1,2,1,4,2,1,1,2,4,10,2,4,1,4,1,8,2,6,4,2,10,11,2,20,4,6,1,28,4,5,1,10
%N A019441 Duplicate of A007739.
%Y A019441 Adjacent sequences: A019438 A019439 A019440 this_sequence A019442 A019443 A019444
%Y A019441 Sequence in context: A009832 A016445 A130544 this_sequence A007739 A031424 A013942
%K A019441 dead
%O A019441 1,3

%I A007739
%S A007739 1,1,2,1,4,2,1,1,2,4,10,2,4,1,4,1,8,2,6,4,2,10,11,2,20,4,6,1,28,4,5,1,10,
%T A007739 8,4,2,12,6,4,4,20,2,14,10,4,11,23,2,7,20,8,4,52,6,20,1,6,28,58,4,20,5,2,
%U A007739 1,4,10,22,8,22,4,35,2,3,12,20,6,10,4,13,4,18,20,82,2,8,14,28,10,11,4,4
%N A007739 Period of repeating digits of 1/n in base 8.
%H A007739 <a href="http://www.research.att.com/~njas/sequences/Sindx_1.html#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%Y A007739 Adjacent sequences: A007736 A007737 A007738 this_sequence A007740 A007741 A007742
%Y A007739 Sequence in context: A016445 A130544 A019441 this_sequence A031424 A013942 A088423
%K A007739 nonn
%O A007739 1,3
%A A007739 njas, Hal Sampson [ hals(AT)easynet.com ]
%E A007739 More terms from David W. Wilson (davidwwilson(AT)comcast.net)

%I A031424
%S A031424 2,1,4,2,1,1,6,3,2,1,1,1,8,4,1,2,1,1,1,1,10,5,2,1,2,1,1,1,1,1,12,6,
%T A031424 4,3,2,2,1,1,1,1,1,1,14,7,1,1,1,2,2,1,1,1,1,1,1,1,16,8,1,4,1,1,1,2,
%U A031424 1,1,1,1,1,1,1,1,18,9,6,1,1,3,1,2,2,1,1,1,1,1,1,1,1,1,20,10,1,5,4,1
%N A031424 Least term in period of continued fraction for sqrt(n), n square-free.
%Y A031424 Adjacent sequences: A031421 A031422 A031423 this_sequence A031425 A031426 A031427
%Y A031424 Sequence in context: A130544 A019441 A007739 this_sequence A013942 A088423 A006839
%K A031424 nonn
%O A031424 2,1
%A A031424 David W. Wilson (davidwwilson(AT)comcast.net)

%I A013942
%S A013942 2,1,4,2,1,1,6,3,2,1,1,1,8,4,2,2,1,1,1,1,10,5,3,2,2,1,1,1,1,1,12,6,
%T A013942 4,3,2,2,1,1,1,1,1,1,14,7,4,3,2,2,2,1,1,1,1,1,1,1,16,8,5,4,3,2,2,2,
%U A013942 1,1,1,1,1,1,1,1,18,9,6,4,3,3,2,2,2,1,1,1,1,1,1,1,1,20,10,6,5,4,3
%N A013942 Triangle of numbers T(n,k) = [ 2n/k ], k=1..2n, read by rows.
%C A013942 a(n) is also the leading term in period of continued fraction for n-th nonsquare.
%Y A013942 Cf. A010766.
%Y A013942 Adjacent sequences: A013939 A013940 A013941 this_sequence A013943 A013944 A013945
%Y A013942 Sequence in context: A019441 A007739 A031424 this_sequence A088423 A006839 A118235
%K A013942 nonn,tabl,easy,nice
%O A013942 1,1
%A A013942 Clark Kimberling (ck6(AT)evansville.edu)

%I A088423
%S A088423 2,1,4,2,1,2,1,1,3,2,1,1,2,1,5,2,1,3,1,1,2,1,1,3,2,1,1,2,1,6,2,1,1,2,1,
%T A088423 2,1,1,3,1,1,1,2,1,4,2,1,2,2,1,2,1,1,1,1,1,1,2,1,3,1,1,4,2,1,1,1,1,2,2,
%U A088423 1,1,2,1,1,2,1,2,1,1,2,1,1,3,2,1,1,1,1,2,2,1,4,2,1,1,1,1,1,2
%N A088423 Number of primes in arithmetic progression starting with 11 and with d=2n.
%C A088423 Arithmetic progression is stopped when next term is not prime. E.g. for n=3, a=4, that is 11,17,23,29 are prime, while n