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A001227 Number of odd divisors of n. 136
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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. [Tom Verhoeff (Tom.Verhoeff(AT)acm.org)]

Also number of partitions into consecutive positive integers including the trivial partition of length 1 (e.g. 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) [Henry Bottomley, Apr 13 2000]

a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003

a(n)=1 for n=A000079. - Lekraj Beedassy, 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, Mar 07 2006

Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006

Lengths of rows of triangle A182469;

a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. [Reinhard Zumkeller, May 01 2012]

Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013

Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014

REFERENCES

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.

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).

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.

LINKS

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).

M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

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, Odd Divisor Function

Eric Weisstein's World of Mathematics, Binomial Number

FORMULA

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, Aug 01, 2001.

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

a(n)=A000005(A000265(n)). - Lekraj Beedassy, 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, which means a(n) is the Dirichlet convolution of A000035 and A057427.

a(n) = A113414(2*n). N. J. A. Sloane, Jan 24 2006 (corrected in Nov 10 2007)

a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, 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, Apr 15 2007

Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005

Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007

G.f.: x/(1-x) + sum(n=1,infinity, x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = sum(n>=1, x^n/(1-x^n) ). [Joerg Arndt, Nov 06 2010]

a(n) = A000005(n) - A183063(n).

a(n) = d(n) if n is odd, else d(n) - d(n/2). (See the Weisstein page). - Gary W. Adamson, Mar 15 2011

Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011

EXAMPLE

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 + ...

MAPLE

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:

MATHEMATICA

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

Table[Total[Mod[Divisors[n], 2]], {n, 105}] (* Zak Seidov, Apr 16 2010 *)

f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)

a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}] (* Michael Somos, May 17 2013 *)

a[ n_] := DivisorSum[ n, Mod[ #, 2] &] (* Michael Somos, May 17 2013 *)

PROG

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

(PARI) {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]} /* Michael Somos, Oct 06 2007 */

(PARI) a(n)=numdiv(n>>valuation(n, 2)) \\ Charles R Greathouse IV, Mar 16, 2011

(PARI) a(n)=sum(k=1, round(solve(x=1, n, x*(x+1)/2-n)), (k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

(PARI) a(n)=sumdivmult(n, d, d%2) \\ Charles R Greathouse IV, Aug 29 2013

(Haskell)

a001227 = length . a182469_row

-- Reinhard Zumkeller, May 01 2012, Jul 25 2011

(Sage)

def A001227(n) : return len(filter(is_odd, divisors(n)))

[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

CROSSREFS

Cf. A000005, A000593, A050999, A051000, A051001, A051002, A054844, A069283, A109814, A118235, A118236, A115369, A051731, A183063, A183064, A136655, A125911, A183063.

Sequence in context: A023588 A175242 A225843 * A060764 A105149 A068307

Adjacent sequences:  A001224 A001225 A001226 * A001228 A001229 A001230

KEYWORD

nonn,easy,nice,mult,core

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 21 10:10 EDT 2014. Contains 245845 sequences.