A051002 Sum of 5th powers of odd divisors of n.

1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, 161052, 244, 371294, 16808, 762744, 1, 1419858, 59293, 2476100, 3126, 4101152, 161052, 6436344, 244, 9768751, 371294, 14408200, 16808, 20511150, 762744, 28629152, 1, 39296688, 1419858, 52541808, 59293, 69343958

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• mult

Comments

The Apostol exercise F(x) is the g.f. of a(n)*(-1)^(n+1). - Michael Somos, Jul 05 2021

Examples

			G.f. = x + x^2 + 244*x^3 + x^4 + 3126*x^5 + 244*x^6 + 16808*x^7 + x^8 + ... - _Michael Somos_, Jul 05 2021
		

References

• T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• 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 and p. 8).
• Eric Weisstein's World of Mathematics, Odd Divisor Function.
• Index entries for sequences mentioned by Glaisher

Crossrefs

Cf. A000593, A001227, A050999, A051000, A051001, A001160.

Programs

Mathematica

a[n_] := Select[ Divisors[n], OddQ]^5 // Total; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Oct 25 2012 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
a[ n_] := If[n == 0, 0, DivisorSigma[5, n / 2^IntegerExponent[n, 2]]]; (* Michael Somos, Jul 05 2021 *)

PARI

a(n) = sumdiv(n , d, (d%2)*d^5); \\ Michel Marcus, Jan 14 2014

PARI

a(n)=sumdiv(n>>valuation(n,2), d, d^5) \\ Charles R Greathouse IV, Jul 05 2021

Python

from sympy import divisor_sigma
def A051002(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),5)) # Chai Wah Wu, Jul 16 2022

Formula

Dirichlet g.f.: (1-2^(5-s))*zeta(s)*zeta(s-5). - R. J. Mathar, Apr 06 2011

G.f.: Sum_{k>=1} (2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017

The preceding g.f. is also 34*sigma_5(x^2) - 64*sigma_5(x^4) - sigma_5(-x), with sigma_5 the g.f. of A001160. Compare this with the Apostol reference which gives the g.f. of a(n)*(-1)^(n+1). - Wolfdieter Lang, Jan 31 2017

Multiplicative with a(2^e) = 1 and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2. - Amiram Eldar, Sep 14 2020

Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 11340. - Vaclav Kotesovec, Sep 24 2020

G.f.: Sum_{n >= 1} x^n*R(5,x^(2*n))/(1 - x^(2*n))^6, where R(5,x) = 1 + 237*x + 1682*x^2 + 1682*x^3 + 237*x^4 + x^5 is the fifth row polynomial of A060187. - Peter Bala, Dec 20 2021