proposed
approved
editing
Seiichi Manyama, <a href="/A354843/b354843.txt">Table of n, a(n) for n = 1..447</a>
a[n_] := n! * DivisorSum[n, (n/#)^#/#! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)
1, 5, 19, 145, 601, 8521, 35281, 672001, 4898881, 82615681, 439084801, 21138606721, 80951270401, 3358578263041, 49506372115201, 1227603183206401, 6046686277632001, 611515751899852801, 2311256907767808001, 254421414038266675201, 4015778465971464192001
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(k*x^k)-1)))
If p is prime, a(p) = 1 + p * p!.
