A386013 a(n) = n^4*tau(n).
1, 32, 162, 768, 1250, 5184, 4802, 16384, 19683, 40000, 29282, 124416, 57122, 153664, 202500, 327680, 167042, 629856, 260642, 960000, 777924, 937024, 559682, 2654208, 1171875, 1827904, 2125764, 3687936, 1414562, 6480000, 1847042, 6291456, 4743684, 5345344, 6002500, 15116544, 3748322, 8340544, 9253764, 20480000
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
seq( n^4*numtheory[tau](n),n=1..100) ;
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Mathematica
a[n_]:=n^4*DivisorSigma[0,n]; Array[a,40] (* Stefano Spezia, Jul 14 2025 *) nmax = 40; Rest[CoefficientList[Series[Sum[k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 03 2025 *)
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PARI
a(n) = n^4 * numdiv(n); \\ Amiram Eldar, Jul 15 2025
Formula
Dirichlet g.f.: zeta^2(s-4).
From Amiram Eldar, Jul 15 2025 (Start)
Multiplicative with a(p^e) = p^(4*e) * (e+1).
Sum_{k=1..n} a(k) ~ (n^5/5) * (log(n) + 2*gamma - 1/5), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5. - Vaclav Kotesovec, Aug 03 2025
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