A305619 - OEIS

%I #6 Mar 27 2019 03:52:54

%S 0,1,2,-3,4,-10,636,-1078,-18416,-131976,5035920,5333592,187347744,

%T -4079616528,-14669908512,-140154110640,28743506893056,

%U -92449999037568,2738959517576448,-52969092379214976,34211286306178560,-16812071564735736576,1407763084021569335808

%N Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)).

%e E.g.f.: A(x) = x^2/2! + 2*x^3/3! - 3*x^4/4! + 4*x^5/5! - 10*x^6/6! + ...

%e exp(A(x)) = 1 + x^2/2 + x^3/3 + x^5/5 + x^7/7 + ... + x^A000040(k)/A000040(k) + ...

%e exp(exp(A(x))-1) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + ... + A218002(k)*x^k/k! + ...

%p a:=series(log(1+add(x^ithprime(k)/ithprime(k),k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 26 2019

%t nmax = 23; 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]] (n - 1)! - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]

%Y Cf. A000040, A002120, A007447, A218002, A305618.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, Jun 06 2018