%I #8 Dec 04 2018 20:13:06
%S 0,1,1,-3,-9,20,190,-126,-6280,-10326,293041,1519320,-16985045,
%T -194560444,1013712777,27317463952,-19210030599,-4305097718760,
%U -17733269020226,743855089334604,7868686621862292,-132351392654695270,-2854492900112993039,20150897206881256464
%N Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).
%C Logarithmic transform of A010051.
%H Alois P. Heinz, <a href="/A305618/b305618.txt">Table of n, a(n) for n = 1..472</a>
%e E.g.f.: A(x) = x^2/2! + x^3/3! - 3*x^4/4! - 9*x^5/5! + 20*x^6/6! + ...
%e exp(A(x)) = 1 + x^2/2! + x^3/3! + x^5/5! + x^7/7! + ... + x^A000040(k)/A039716(k) + ...
%e exp(exp(A(x))-1) = 1 + x^2/2! + x^3/3! + 3*x^4/4! + 11*x^5/5! + ... + A190476(k)*x^k/k! + ...
%p a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*t(n-j)*
%p j*binomial(n, j), j=1..n-1)/n))(i-> `if`(isprime(i), 1, 0))
%p end:
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Dec 04 2018
%t nmax = 24; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
%t a[n_] := a[n] = Boole[PrimeQ[n]] - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 24}]
%Y Cf. A000040, A002120, A007447, A010051, A039716, A190476, A218002, A303073, A305619.
%K sign
%O 1,4
%A _Ilya Gutkovskiy_, Jun 06 2018