A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 36, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 73, 65, 66, 67, 68, 69, 70, 71, 81, 73, 74, 75

Offset: 1

Author

Ilya Gutkovskiy, Sep 19 2019

Keywords

• nonn
• mult

Comments

Sum of divisors d of n such that n/d is a cube.

Inverse Moebius transform of A078429.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000578, A004709 (fixed points), A010057, A061704, A076752, A078429, A113061.

Programs

Mathematica

nmax = 75; CoefficientList[Series[Sum[x^(k^3)/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[(n/#)^(1/3)] &]; Table[a[n], {n, 1, 75}]
f[p_, e_] := (p^(e+3) - p^Mod[e, 3])/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

PARI

A327626(n) = sumdiv(n,d,ispower(n/d,3)*d); \\ Antti Karttunen, Sep 19 2019

Formula

a(n) = Sum_{d|n} A078429(d).

a(n) = Sum_{d|n} A010057(n/d) * d. Dirichlet convolution of A000027 and A010057.

D.g.f.: zeta(s-1)*zeta(3s). - R. J. Mathar, Jun 05 2020

Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1890. - Vaclav Kotesovec, May 20 2021

Multiplicative with a(p^e) = (p^(e+3) - p^(e mod 3))/(p^3-1). - Amiram Eldar, May 25 2025