A386785 a(n) = n^4*sigma_5(n).

0, 1, 528, 19764, 270592, 1953750, 10435392, 40356008, 138547200, 389021373, 1031580000, 2357962332, 5347980288, 10604527934, 21307972224, 38613915000, 70936231936, 118587960018, 205403284944, 322687828100, 528669120000, 797596142112, 1245004111296, 1801152941304, 2738246860800

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10.

a(n) = (4*A386813(n) + 2*A282549(n) - A386814(n) - 6*A282792(n) - A058550(n) + 2*A282576(n))/3456.

a(n) = n^4*A001160(n).

Dirichlet g.f.: zeta(s-4)*zeta(s-9). - R. J. Mathar, Aug 03 2025

Sum_{k=0..n} a(k) ~ zeta(6) * n^10 / 10. - Amiram Eldar, Nov 11 2025

Mathematica

Table[n^4*DivisorSigma[5, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(4*E2[x]^3*E4[x]^2 + 2*E2[x]*E4[x]^3 - E2[x]^4*E6[x] - 6*E2[x]^2*E4[x]*E6[x] - E4[x]^2*E6[x] + 2*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]

Prog

(Magma) [0] cat [n^4*DivisorSigma(5, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386786, A386787, A386788.

Cf. A001160, A013664, A282050, A282751, A282781.

Cf. A386813, A282549, A386814, A282792, A058550, A282576.

Sequence in context: A304512 A037944 A282096 * A223253 A233103 A223244

Adjacent sequences: A386782 A386783 A386784 * A386786 A386787 A386788

Keyword

nonn,mult,easy

Author

Vaclav Kotesovec, Aug 02 2025

Status

approved