1,3

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.

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.)

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

Number of even divisors of n = A000005(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2003

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

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

a(n)=1 for n=A000079. - Lekraj Beedassy (boodhiman(AT)hotmail.com), Apr 12 2005

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

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

Also the number of factors of the n-th Lucas polynomial. - T. D. Noe (noe(AT)sspectra.com), Mar 09 2006

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.

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.

Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

K. S. Brown's Mathpages, Partitions into Consecutive Integers

A. Heiligenbrunner, Sum of adjacent numbers (in German).

N. J. A. Sloane, Transforms

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Binomial Number

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).

a(n) =A000005(n)/(A007814(n)+1) =A000005(n)/A001511(n).

Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f.: Sum_{n>=1} x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 16 2002

a(n)=A000005(A000265(n)). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 07 2005

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

Moebius transform is period 2 sequence [1, 0, ...] = A000035.

a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2006

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

Dirichlet g.f.: zeta(s)^2*(1-1/2^s). - Ralf Stephan, Jun 17 2007

Number of occurrences of n in A049777. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2005

Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 06 2007

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 + ...

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:

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Aug 27 2004)

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d%2))} /* Michael Somos Oct 06 2007 */

(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 */

Cf. A000005, A000593, A050999, A051000, A051001, A051002, A054844, A069283.

Cf. A109814, A118235, A118236, A115369.

A113414(2*n) = a(n).

Cf. A051731.

Sequence in context: A035228 A035164 A023588 this_sequence A060764 A105149 A068307

Adjacent sequences: A001224 A001225 A001226 this_sequence A001228 A001229 A001230

nonn,easy,nice,mult,core

N. J. A. Sloane (njas(AT)research.att.com).