A353992 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.

1, 7, 41, 314, 2194, 22764, 195348, 2374224, 27940176, 384636960, 4673720160, 95522440320, 1323221996160, 23481816503040, 489968947641600, 10853692580505600, 190580382936115200, 5408424680491929600, 105077728210820198400, 3399507785578641408000

Offset: 1

Author

Seiichi Manyama, Aug 06 2022

Keywords

• nonn

Crossrefs

Cf. A078308, A353993, A356010, A356298, A356406.

Programs

Mathematica

a[n_] := n! * Sum[DivisorSum[k, #^(k/# + 1) &]/k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 06 2022 *)

PARI

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

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=1..n} A078308(k)/k.

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

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