A386787 a(n) = n^4*sigma_7(n).
0, 1, 2064, 177228, 4227328, 48828750, 365798592, 1977329144, 8657571840, 31395415077, 100782540000, 285311685252, 749200886784, 1792160422598, 4081207353216, 8653821705000, 17730707193856, 34271896391154, 64800136718928, 116490259028540, 206415142080000, 350438089532832
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..7500
Crossrefs
Programs
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Magma
[0] cat [n^4*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
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Mathematica
Table[n^4*DivisorSigma[7, n], {n, 0, 30}] (* or *) nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12, {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[(33*E2[x]^4*E4[x]^2 + 110*E2[x]^2*E4[x]^3 + 13*E4[x]^4 - 132*E2[x]^3*E4[x]*E6[x] - 132*E2[x]*E4[x]^2*E6[x] + 88*E2[x]^2*E6[x]^2 + 20*E4[x]*E6[x]^2)/41472, {x, 0, terms}], x]
Formula
G.f.: Sum_{k>=1} k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12.
a(n) = (33*A386815(n) + 110*A386816(n) + 13*A282012(n) - 132*A386817(n) - 132*A282596(n) + 88*A386818(n) + 20*A282287(n))/41472.
a(n) = n^4*A013955(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-11). - R. J. Mathar, Aug 03 2025