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%I #15 Sep 05 2019 15:48:27
%S 1,3,4,16,6,36,8,84,29,54,12,264,14,72,72,440,18,357,20,396,96,108,24,
%T 1776,67,126,208,528,30,936,32,2304,144,162,144,3290,38,180,168,2664,
%U 42,1248,44,792,714,216,48,11360,121,819,216,924,54,3264,216,3552,240,270,60,8880
%N a(1) = 1; a(n) = Sum_{d|n, d<n} sigma(n/d) * a(d), where sigma = A000203.
%F G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} sigma(k) * A(x^k).
%F a(p) = p + 1, where p is prime.
%t a[n_] := If[n == 1, n, Sum[If[d < n, DivisorSigma[1, n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
%t nmax = 60; A[_] = 0; Do[A[x_] = x + Sum[DivisorSigma[1, k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%o (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n] = sumdiv(n, d, v[d]*sigma(n/d))); v} \\ _Andrew Howroyd_, Sep 05 2019
%Y Cf. A000203, A046692, A307817, A325211.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Sep 05 2019