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%I #26 Oct 03 2023 13:14:29
%S 1,19,85,311,631,1615,2409,4991,6898,11989,14653,26435,28575,45771,
%T 53635,79887,83539,131062,130341,196241,204765,278407,279865,424235,
%U 394406,542925,558778,749199,707311,1019065,923553,1278255,1245505,1587241,1520079,2145278,1874199,2476479,2428875,3149321,2825803,3890535,3418845,4557083,4352638,5317435
%N a(n) = Sum_{ d|n } sigma(n/d)*d^4.
%H Seiichi Manyama, <a href="/A027848/b027848.txt">Table of n, a(n) for n = 1..1000</a>
%F Dirichlet g.f.: zeta(x-1)*zeta(x-4).
%F Multiplicative with a(p^e) = (p^(4e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2+p+1)/(p^7 - (p^3+p^2+p+1)*p + p^2+p+1). - _Mitch Harris_, Jun 27 2005
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma_3(k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 23 2018
%F Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 450. - _Vaclav Kotesovec_, Feb 16 2020
%t f[p_, e_] := (1 + p + p^2 - p^(e+1) - p^(e+2) - p^(e+3) - p^(e+4) + p^(4*e+7))/(1 - p^3 - p^4 + p^7); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Oct 03 2023 *)
%o (PARI)N=66; x='x+O('x^N); /* that many terms */
%o c=sum(j=1,N,j*x^j);
%o t=log(1/prod(j=1,N, eta(x^(j))^(j^3)));
%o Vec(serconvol(t,c)) /* show terms */
%o /* _Joerg Arndt_, May 03 2008 */
%Y Cf. A001001, A027847, A288391.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_
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