A386788 a(n) = n^4*sigma_8(n).

0, 1, 4112, 531522, 16843008, 244141250, 2185618464, 13841289602, 68988964864, 282472589763, 1003908820000, 3138428391362, 8952429298176, 23298085151042, 56915382843424, 129766445482500, 282578800148480, 582622237313282, 1161527289105456, 2213314919196482, 4112073026880000

Offset: 0

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..7500 from Vincenzo Librandi)

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386787.

Cf. A013956, A386751, A386778, A386782.

Programs

Magma

[0] cat [n^4*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Mathematica

Table[n^4*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13, {k, 1, nmax}], {x, 0, nmax}], x]

PARI

a(n) = if (n, n^4*sigma(n,8), 0); \\ Michel Marcus, Aug 03 2025

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13.

a(n) = n^4*A013956(n).

Dirichlet g.f.: zeta(s-4)*zeta(s-12). - R. J. Mathar, Aug 03 2025