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A001227
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Number of odd divisors of n.
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100
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 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.) [Henry Bottomley, Apr 13 2000]
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 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, 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
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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.
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
| 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, Odd Divisor Function
Eric Weisstein's World of Mathematics, Binomial Number
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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 (vladeta(AT)eunet.rs), 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) = 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 DELEHAM, 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). [From 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
| 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
| 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
| f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Aug 27 2004)
Table[Total[Mod[Divisors[n], 2]], {n, 105}] [From Zak Seidov, Apr 16 2010]
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* RGWv *)
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PROG
| (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 */
(PARI) a(n)=numdiv(n>>valuation(n, 2)) \\ Charles R Greathouse IV, Mar 16, 2011
(Haskell)
a001227 n = length $ filter ((== 0) . mod n) [1, 3..n]
-- Reinhard Zumkeller, 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
| Cf. A000005, A000593, A050999, A051000, A051001, A051002, A054844, A069283, A109814, A118235, A118236, A115369, A051731, A183063, A183064, A136655, A125911.
A113414(2*n) = a(n).
Cf. A183063.
Sequence in context: A035164 A023588 A175242 * A060764 A105149 A068307
Adjacent sequences: A001224 A001225 A001226 * A001228 A001229 A001230
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KEYWORD
| nonn,easy,nice,mult,core,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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