Search: id:A001227
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%I A001227
%S A001227 1,1,2,1,2,2,2,1,3,2,2,2,2,2,4,1,2,3,2,2,4,2,2,2,3,2,4,2,2,4,2,1,4,2,4,
%T A001227 3,2,2,4,2,2,4,2,2,6,2,2,2,3,3,4,2,2,4,4,2,4,2,2,4,2,2,6,1,4,4,2,2,4,4,
%U A001227 2,3,2,2,6,2,4,4,2,2,5,2,2,4,4,2,4,2,2,6,4,2,4,2,4,2,2,3,6,3,2,4,2,2,8
%N A001227 Number of odd divisors of n.
%C A001227 Comment from Tom Verhoeff (Tom.Verhoeff(AT)acm.org): also (1) number 
               of ways to write n as difference of two triangular numbers (A000217); 
               (2) number of ways to arrange n identical objects in a trapezoid.
%C A001227 Comment from Henry Bottomley (se16(AT)btinternet.com), Apr 13 2000: Also 
               number of sums of sequences of consecutive positive integers including 
               sequences of length 1 (e.g. 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful 
               for cribbage players.)
%C A001227 a(n) is also the number of factors in the factorization of the Chebyshev 
               polynomial of thee first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), 
               Aug 28 2003
%C A001227 Number of even divisors of n = A000005(2*n) * (1 - n mod 2). - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2003
%C A001227 Number of ways to present n as sum of consecutive integers. The trivial 
               solution n=n is also counted. Equals 1 + A069283. - Alfred Heiligenbrunner 
               (alfred.heiligenbrunner(AT)gmx.at), Jun 07 2004
%C A001227 Number of factors in the factorization of the polynomial x^n+1 over the 
               integers. See also A000005. - T. D. Noe (noe(AT)sspectra.com), Apr 
               16 2003
%C A001227 a(n)=1 for n=A000079. - Lekraj Beedassy (boodhiman(AT)hotmail.com), Apr 
               12 2005
%C A001227 For n odd, n is prime iff the n-th term of the sequence is 2. - George 
               J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
%C A001227 Also number of partitions of n such that if k is the largest part, then 
               each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 
               because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 07 2006
%C A001227 Also the number of factors of the n-th Lucas polynomial. - T. D. Noe 
               (noe(AT)sspectra.com), Mar 09 2006
%D A001227 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 
               Entry 47.
%D A001227 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 1, p. 306.
%D A001227 Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 
               1994), see exercise 2.30 on p. 65.
%D A001227 P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and 
               New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.
%H A001227 N. J. A. Sloane, <a href="b001227.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A001227 K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath107.htm">
               Partitions into Consecutive Integers</a>
%H A001227 A. Heiligenbrunner, <a href="http://www.heiligenbrunner.at/main/ahsummen.htm">
               Sum of adjacent numbers (in German)</a>.
%H A001227 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A001227 T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, 
               Vol. 2, 1999, #99.1.6.
%H A001227 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OddDivisorFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A001227 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinomialNumber.html">Binomial Number</a>
%F A001227 Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
%F A001227 a(n) =A000005(n)/(A007814(n)+1) =A000005(n)/A001511(n).
%F A001227 Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001227 G.f.: Sum_{n>=1} x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 16 2002
%F A001227 a(n)=A000005(A000265(n)). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 
               07 2005
%F A001227 G.f.: Sum_{k>0} x^(2k-1)/(1-x^(2k-1)) = Sum_{k>0} x^(k(k+1)/2)/(1-x^k). 
               - Michael Somos Oct 30 2005
%F A001227 Moebius transform is period 2 sequence [1, 0, ...] = A000035.
%F A001227 a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 18 2006
%F A001227 Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular 
               matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4,...]. 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%F A001227 Dirichlet g.f.: zeta(s)^2*(1-1/2^s). - Ralf Stephan, Jun 17 2007
%F A001227 Number of occurrences of n in A049777. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jun 19 2005
%F A001227 Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius 
               transform. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 06 2007
%e A001227 q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 
               + ...
%p A001227 for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod 
               d = 0 then s := s+1: fi: od: print(s); od:
%t A001227 f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], 
               {n, 105}] (from Robert G. Wilson v Aug 27 2004)
%o A001227 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d%2))} /* Michael Somos Oct 06 
               2007 */
%o A001227 (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 
               4, p) * X))[n])} /* Michael Somos Oct 06 2007 */
%Y A001227 Cf. A000005, A000593, A050999, A051000, A051001, A051002, A054844, A069283.
%Y A001227 Cf. A109814, A118235, A118236, A115369.
%Y A001227 A113414(2*n) = a(n).
%Y A001227 Cf. A051731.
%Y A001227 Sequence in context: A035228 A035164 A023588 this_sequence A060764 A105149 
               A068307
%Y A001227 Adjacent sequences: A001224 A001225 A001226 this_sequence A001228 A001229 
               A001230
%K A001227 nonn,easy,nice,mult,core
%O A001227 1,3
%A A001227 N. J. A. Sloane (njas(AT)research.att.com).

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