A377154 Expansion of e.g.f. exp(Sum_{k>=1} A000082(k)*x^k/k).

1, 1, 7, 43, 385, 3721, 47911, 612067, 9559873, 157478545, 2910837511, 56866891291, 1224263236417, 27618866777113, 673173639519655, 17237263465417171, 469017851840595841, 13367670808113197857, 401964392506370969863, 12604372518766870306315, 414278024498330114803201

Offset: 0

Author

Vaclav Kotesovec, Oct 31 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..428

Crossrefs

Cf. A000082, A001615, A308462.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[Sum[k * Product[1 + 1/p, {p, Select[Divisors[k], PrimeQ]}] * x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) ~ n! * 5^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) / (6^(1/3) * Pi^(5/6) * n^(2/3)).

a(n) ~ 10^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) * n^(n - 1/6) / (3^(1/3) * Pi^(1/3) * exp(n)).

E.g.f.: exp(Sum_{k>=1} A001615(k)*x^k).