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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Crossrefs
Programs
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Magma
[0] cat [n^4*DivisorSigma(5, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
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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]
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) = n^4*A001160(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-9). - R. J. Mathar, Aug 03 2025