A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.

1, 5, 22, 125, 746, 5677, 44780, 420401, 4206970, 47543141, 562891352, 7573655905, 104684547566, 1596368400005, 25482043382476, 439969180782017, 7835163501390290, 151712475696833221, 3004182138648663200, 63854641556089628801, 1400563708969910620822

Offset: 1

Author

Seiichi Manyama, Jul 20 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..448
• Vaclav Kotesovec, Plot of a(n)/(n!*n) for n = 1..10000

Crossrefs

Cf. A006218, A057625, A356010, A356458, A229837.

Cf. A356009, A356459.

Programs

Mathematica

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

PARI

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

PARI

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

PARI

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

PARI

a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ Seiichi Manyama, Aug 08 2022

Formula

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

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

a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/d! = n! * Sum_{k=1..n} A057625(k)/k!. - Seiichi Manyama, Aug 08 2022

a(n) ~ A229837 * n! * n. - Vaclav Kotesovec, Aug 11 2025