A307817 a(1) = 1; a(n+1) = Sum_{d|n} sigma(n/d)*a(d), where sigma = sum of divisors (A000203).

1, 1, 4, 8, 18, 24, 52, 60, 106, 135, 213, 225, 397, 411, 599, 719, 1001, 1019, 1533, 1553, 2192, 2464, 3151, 3175, 4502, 4641, 5888, 6404, 8145, 8175, 11040, 11072, 13863, 14811, 17886, 18390, 23723, 23761, 28440, 30140, 36650, 36692, 45952, 45996, 55095, 58535, 68084, 68132, 83720, 84193

Offset: 1

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

• nonn

Crossrefs

Cf. A000203, A007557, A307794.

Programs

Mathematica

a[n_] := a[n] = Sum[DivisorSigma[1, (n - 1)/d] a[d] , {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 50}]
terms = 50; A[] = 0; Do[A[x] = x (1 + Sum[DivisorSigma[1, k]  A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[Sum[DivisorSigma[1, i] a[j] x^(i j), {j, 1, n - 1}], {i, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 50}]

PARI

lista(nn) = { my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sumdiv(n-1, d, sigma((n-1)/d)*va[d])); va;} \\ Michel Marcus, Apr 30 2019

Formula

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{k>=1} sigma(k)*A(x^k)).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} sigma(i)*a(j)*x^(i*j)).