OFFSET
1,3
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.
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
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
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 *)
PROG
(Haskell)
a051000 = sum . map (^ 3) . a182469_row
-- Reinhard Zumkeller, May 01 2012
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
STATUS
approved