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a(n) = n! * Sum_{d|n} (n/d) / d!.
4

%I #20 Aug 30 2023 02:00:32

%S 1,5,19,121,601,5641,35281,406561,3447361,45420481,439084801,

%T 7565564161,80951270401,1525654690561,20737536019201,421943967244801,

%U 6046686277632001,150482493928166401,2311256907767808001,61410502863943833601,1132546296081328128001

%N a(n) = n! * Sum_{d|n} (n/d) / d!.

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

%F If p is prime, a(p) = 1 + p * p!.

%t a[n_] := n! * DivisorSum[n, (n/#) / #! &]; Array[a, 21] (* _Amiram Eldar_, Aug 30 2023 *)

%o (PARI) a(n) = n!*sumdiv(n, d, n/d/d!);

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

%o (Python)

%o from math import factorial

%o from sympy import divisors

%o def A354863(n):

%o f = factorial(n)

%o return sum(f*n//d//factorial(d) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 09 2022

%Y Cf. A057625, A354843, A354862.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jun 09 2022