proposed
approved
editing
Seiichi Manyama, <a href="/A354888/b354888.txt">Table of n, a(n) for n = 1..386</a>
reviewed
a[n_] := n! * DivisorSum[n, #^#/#! &]; Array[a, 18] (* Amiram Eldar, Jun 10 2022 *)
If p is prime, a(p) = p^p + p! = A053042(p).
Cf. A327579, A354843, A354889.