A307793 a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).

1, 1, 3, 7, 24, 49, 205, 411, 1668, 5011, 20095, 40191, 241372, 482745, 1931393, 7725627, 38629803, 77259607, 463562851, 927125703, 5562774334, 22251097753, 89004431205, 178008862411, 1424071142304, 4272213426961, 17088854190591, 68355416767375, 410132502535664, 820265005071329

Offset: 1

Author

Ilya Gutkovskiy, Apr 29 2019

Keywords

• nonn

Crossrefs

Cf. A000005, A007557, A060640, A307794, A319133.

Programs

Mathematica

a[n_] := a[n] = Sum[DivisorSigma[0, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[0, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 30}]

PARI

a(n) = if (n==1, 1, sumdiv(n-1, d, numdiv(d)*a(d))); \\ Michel Marcus, Apr 29 2019

Formula

G.f.: x * (1 + Sum_{n>=1} tau(n)*a(n)*x^n/(1 - x^n)).

L.g.f.: -log(Product_{i>=1, j>=1} (1 - x^(i*j))^(a(i*j)/(i*j))) = Sum_{n>=1} a(n+1)*x^n/n.