A357296 Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).
1, 3, 7, 31, 121, 851, 5041, 43261, 369601, 3748249, 39916801, 490801081, 6227020801, 87861842641, 1310800947457, 21018206008801, 355687428096001, 6419518510204801, 121645100408832001, 2435836129700029057, 51102829650622464001, 1124549558817839481601
Offset: 1
Keywords
Programs
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Mathematica
a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k))))) -
PARI
a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));
Formula
a(n) = n! * Sum_{d|n} 1 / (d^(n/d-1) * d!).
If p is prime, a(p) = 1 + p! = A038507(p).