A386813 -id:A386813 - cp's OEIS frontend

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

Vaclav Kotesovec, Aug 02 2025

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

Cf. A280022, A386783, A280025, A386784, A386786, A386787, A386788.
Cf. A001160, A282050, A282751, A282781.
Cf. A386813, A282549, A386814, A282792, A058550, A282576.

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

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]

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

A386781 a(n) = n^3*sigma_7(n).

Original entry on oeis.org

0, 1, 1032, 59076, 1056832, 9765750, 60966432, 282475592, 1082196480, 3488379453, 10078254000, 25937425932, 62433407232, 137858494046, 291514810944, 576921447000, 1108169199616, 2015993905362, 3600007595496, 6131066264660, 10320757104000, 16687528072992, 26767423561824

Offset: 0

Vaclav Kotesovec, Aug 02 2025

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

Cf. A282211, A386746, A282213, A386748, A282781, A386780, A386782.
Cf. A013955, A282060, A282753.
Cf. A386813, A282549, A282792, A058550, A282576.

[0] cat [n^3*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^3*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {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[(3*E2[x]^3*E4[x]^2 + 5*E2[x]*E4[x]^3 - 9*E2[x]^2*E4[x]*E6[x] - 3*E4[x]^2*E6[x] + 4*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]

G.f.: Sum_{k>=1} k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = (3*A386813(n) + 5*A282549(n) - 9*A282792(n) - 3*A058550(n) + 4*A282576(n))/3456.
a(n) = n^3*A013955(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-10). - R. J. Mathar, Aug 03 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula