OFFSET
1,2
COMMENTS
Inverse Möbius transform applied twice to cubes.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms
FORMULA
G.f.: Sum_{k>=1} sigma_3(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^3*tau(n/d).
From Jianing Song, Oct 28 2018: (Start)
Multiplicative with a(p^e) = (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2.
Dirichlet g.f.: zeta(s)^2*zeta(s-3). (End)
Sum_{k=1..n} a(k) ~ Pi^8 * n^4 / 32400. - Vaclav Kotesovec, Nov 08 2018
MAPLE
with(numtheory): seq(coeff(series(add(sigma[3](k)*x^k/(1-x^k), k=1..n), x, n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 28 2018
MATHEMATICA
Table[Sum[DivisorSigma[3, d], {d, Divisors[n]}] , {n, 45}]
nmax = 45; Rest[CoefficientList[Series[Sum[DivisorSigma[3, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
PROG
(PARI) a(n) = sumdiv(n, d, sigma(d, 3)); \\ Michel Marcus, Oct 28 2018
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Oct 28 2018
STATUS
approved