A307794 a(1) = 1; a(n+1) = Sum_{d|n} sigma(d)*a(d), where sigma = sum of divisors (A000203).

1, 1, 4, 17, 123, 739, 8888, 71105, 1066698, 13867091, 249608380, 2995300561, 83868424715, 1174157946011, 28179790775372, 676314978609683, 20965764337966871, 377383758083403679, 14717966565266619443, 294359331305332388861, 12363091914824209940661, 395618941274374718172273

Offset: 1

Author

Ilya Gutkovskiy, Apr 29 2019

Keywords

• nonn

Crossrefs

Cf. A000203, A001001, A307793.

Programs

Mathematica

a[n_] := a[n] = Sum[DivisorSigma[1, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 22}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[1, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 22}]

PARI

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

Formula

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

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