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a(n) = Sum_{p|n, p prime} (n-1)!/(n/p-1)!.
2

%I #13 Mar 02 2022 12:03:01

%S 0,1,2,6,24,180,720,840,20160,378000,3628800,6985440,479001600,

%T 6235669440,47221574400,259459200,20922789888000,2972883513600,

%U 6402373705728000,20274518622758400,1219830034655232000,51090956251003468800,1124000727777607680000

%N a(n) = Sum_{p|n, p prime} (n-1)!/(n/p-1)!.

%F E.g.f.: Sum_{p prime} (exp(x^p) - 1)/p.

%t a[1] = 0; a[n_] := Plus @@ ((n-1)!/(n/FactorInteger[n][[;;,1]] - 1)!); Array[a, 25] (* _Amiram Eldar_, Mar 02 2022 *)

%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, (n-1)!/(n/f[k, 1]-1)!);

%o (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*(exp(x^k)-1)/k))))

%Y Cf. A087906, A352012, A352058, A352060.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Mar 02 2022