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%I #17 Jul 31 2023 02:25:01
%S 1,6,33,376,3245,67716,828583,22050176,420850809,12580687900,
%T 285351587411,11736333558720,302881333613053,13450914411140584,
%U 463402585399165875,22345557703564558336,827240617573764860177,48442529220731147887020
%N a(n) = n! * Sum_{d|n} d^n / d!^(n/d).
%F E.g.f.: Sum_{k>0} (k * x)^k/(k! - (k * x)^k).
%F If p is prime, a(p) = p^p + p!.
%t a[n_] := n! * DivisorSum[n, #^n / #!^(n/#) &]; Array[a, 20] (* _Amiram Eldar_, Jul 31 2023 *)
%o (PARI) a(n) = n!*sumdiv(n, d, d^n/d!^(n/d));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (k*x)^k/(k!-(k*x)^k))))
%Y Cf. A354890, A358593.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Feb 23 2023
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