A356459 a(n) = n! * Sum_{k=1..n} Sum_{d|k} d/(k/d)!.

1, 7, 40, 281, 2006, 17677, 159020, 1678721, 18555850, 230978981, 2979853592, 43323807265, 644160764846, 10543905398405, 178896116995276, 3284281839169217, 61879477543508690, 1264313089711322821, 26333205612282941600, 588074615109602665601

Offset: 1

Author

Seiichi Manyama, Aug 08 2022

Keywords

• nonn

Crossrefs

Cf. A354863, A355886, A356009.

Programs

Mathematica

Table[n! * Sum[Sum[d/(k/d)!, {d,Divisors[k]}], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 11 2025 *)

PARI

a(n) = n!*sum(k=1, n, sumdiv(k, d, d/(k/d)!));

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*(exp(x^k)-1))/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A354863(k)/k!.

E.g.f.: (1/(1-x)) * Sum_{k>0} k * (exp(x^k) - 1).

Conjecture: a(n) ~ c * n! * n^2, where c = 0.5732... - Vaclav Kotesovec, Aug 12 2025