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Seiichi Manyama, <a href="/A354888/b354888.txt">Table of n, a(n) for n = 1..386</a>

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a[n_] := n! * DivisorSum[n, #^#/#! &]; Array[a, 18] (* Amiram Eldar, Jun 10 2022 *)

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If p is prime, a(p) = p^p + p! = A053042(p).

Cf. A327579, A354843, A354889.