A355888 a(n) = Sum_{k=1..n} k! * floor(n/k).

1, 4, 11, 38, 159, 888, 5929, 46276, 409163, 4038086, 43954887, 522957240, 6749978041, 93928274284, 1401602642411, 22324392570758, 378011820666759, 6780385526758368, 128425485935590369, 2561327494115859316, 53652269665825304363, 1177652997443472901166

Offset: 1

Author

Seiichi Manyama, Jul 20 2022

Keywords

• nonn

Crossrefs

Partial sums of A062363.

Cf. A006218, A024916.

Programs

Mathematica

Table[Sum[k!*Floor[n/k], {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)

PARI

a(n) = sum(k=1, n, n\k*k!);

PARI

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

PARI

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

Python

from math import factorial
def A355888(n): return factorial(n)+n+sum(factorial(k)*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

Formula

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

G.f.: (1/(1-x)) * Sum_{k>0} k! * x^k/(1 - x^k).

a(n) ~ n!. - Vaclav Kotesovec, Aug 11 2025