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%I #16 Mar 02 2022 12:02:39
%S 0,1,1,6,1,120,1,840,10080,30240,1,3659040,1,17297280,3632428800,
%T 259459200,1,1490852563200,1,845092863014400,3379030566912000,
%U 28158588057600,1,2565331847811532800,1077167364120207360000,64764752532480000,5001134190558105600000
%N a(n) = Sum_{p|n, p prime} n!/(p!*(n/p)!).
%F E.g.f.: Sum_{p prime} (exp(x^p) - 1)/p!.
%F a(n) = 1 if and only if n is prime.
%t a[1] = 0; a[n_] := Plus @@ (n!/((p=FactorInteger[n][[;;,1]])!*(n/p)!)); Array[a, 30] (* _Amiram Eldar_, Mar 02 2022 *)
%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, n!/(f[k, 1]!*(n/f[k, 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. A121860, A352012, A352059, A352060.
%K nonn
%O 1,4
%A _Seiichi Manyama_, Mar 02 2022
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