A051000 Sum of cubes of odd divisors of n.

1, 1, 28, 1, 126, 28, 344, 1, 757, 126, 1332, 28, 2198, 344, 3528, 1, 4914, 757, 6860, 126, 9632, 1332, 12168, 28, 15751, 2198, 20440, 344, 24390, 3528, 29792, 1, 37296, 4914, 43344, 757, 50654, 6860, 61544, 126, 68922, 9632, 79508, 1332, 95382, 12168, 103824, 28

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• mult

Comments

The sum of cubes of even divisors of 2*k equals 8*A001158(k), and the sum of cubes of even divisors of 2*k-1 vanishes, for k >= 1. - Wolfdieter Lang, Jan 07 2017

Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 21 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• John A. Ewell, On a relation between two divisor functions, JP Journal of Algebra, Number Theory and Applications, Vol. 7, No. 2 (2007), pp. 241-243.
• 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. A000203, A000593, A001227, A001158, A006519, A050999, A051001, A051002.

Programs

Haskell

a051000 = sum . map (^ 3) . a182469_row
-- Reinhard Zumkeller, May 01 2012

Mathematica

Table[Total[Select[Divisors[n],OddQ]^3],{n,50}] (* Harvey P. Dale, Jun 28 2012 *)
f[2, e_] := 1; f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

PARI

a(n) = sumdiv(n, d, (d%2)*d^3); \\ Michel Marcus, Jan 04 2017

Python

from sympy import divisor_sigma
def A051000(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),3)) # Chai Wah Wu, Jul 16 2022

Formula

Dirichlet g.f.: (1-2^(3-s))*zeta(s)*zeta(s-3). Dirichlet convolution of (-1)^n*A176415(n) and A000578. - R. J. Mathar, Apr 06 2011

a(n) = Sum_{k=1..A001227(n)} A182469(n,k)^3. - Reinhard Zumkeller, May 01 2012

G.f.: Sum_{k>=1} (2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 720. - Vaclav Kotesovec, Jan 31 2019

Multiplicative with a(2^e) = 1 and a(p^e) = (p^(3*e+3)-1)/(p^3-1) for p > 2. - Amiram Eldar, Sep 14 2020

For k>=0, a(2^k) = 1. - Vaclav Kotesovec, Sep 21 2020

G.f.: Sum_{n >= 1} x^n*(1 + 23*x^(2*n) + 23*x^(4*n) + x^(6*n))/(1 - x^(2*n))^4. See row 4 of A060187. - Peter Bala, Dec 20 2021

a(n) = Sum_{k=0..n-1} A000203(2*n-2*k-1)*A000203(2*k+1)/A006519(n)^3 (Ewell, 2007). - Amiram Eldar, Feb 24 2024