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A001227
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Number of odd divisors of n.
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135
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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
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice,mult,core
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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