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A030010 Inverse Euler transform of primes.

%I A030010 #26 Oct 09 2019 07:51:37
%S A030010 2,0,1,0,2,-3,2,-4,4,-3,4,-5,10,-21,20,-18,34,-46,64,-99,126,-182,258,
%T A030010 -319,464,-685,936,-1352,1888,-2570,3690,-5188,7292,-10501,14742,
%U A030010 -20766,29610,-41650,59052,-84338,119602,-170279,242256,-343356,489550,-698073
%N A030010 Inverse Euler transform of primes.
%H A030010 Seiichi Manyama, <a href="/A030010/b030010.txt">Table of n, a(n) for n = 1..5000</a>
%H A030010 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A030010 Product_{k>=1} 1/(1-x^k)^{a(k)} = 1 + Sum_{n>=1} prime(n) * x^n.
%F A030010 From _Vaclav Kotesovec_, Oct 09 2019: (Start)
%F A030010 a(n) ~ -(-1)^n * A072508^n / n.
%F A030010 a(n) ~ -(-1)^n / (n * A088751^n). (End)
%e A030010 (1-x)^(-2) * (1-x^3)^(-1) * (1-x^5)^(-2) * (1-x^6)^3 * (1-x^7)^(-2) * ... = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + ... .
%t A030010 pp = Prime[Range[n = 100]]; s = {};
%t A030010 For[i = 1, i <= n, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i-d]], {d, i-1}]]];
%t A030010 Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* _Jean-François Alcover_, May 10 2019 *)
%Y A030010 Cf. A000040, A030011, A072508, A088751.
%K A030010 sign
%O A030010 1,1
%A A030010 _N. J. A. Sloane_