A321140 a(n) = Sum_{d|n} sigma_3(d).

1, 10, 29, 83, 127, 290, 345, 668, 786, 1270, 1333, 2407, 2199, 3450, 3683, 5349, 4915, 7860, 6861, 10541, 10005, 13330, 12169, 19372, 15878, 21990, 21226, 28635, 24391, 36830, 29793, 42798, 38657, 49150, 43815, 65238, 50655, 68610, 63771, 84836, 68923, 100050, 79509, 110639, 99822

Offset: 1

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

• nonn
• mult
• easy

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

Crossrefs

Cf. A001158, A007429, A007433, A027848.

Programs

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 *)

PARI

a(n) = sumdiv(n, d, sigma(d, 3)); \\ Michel Marcus, Oct 28 2018

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