login
Inverse Euler transform of primes.
11

%I #26 Oct 09 2019 07:51:37

%S 2,0,1,0,2,-3,2,-4,4,-3,4,-5,10,-21,20,-18,34,-46,64,-99,126,-182,258,

%T -319,464,-685,936,-1352,1888,-2570,3690,-5188,7292,-10501,14742,

%U -20766,29610,-41650,59052,-84338,119602,-170279,242256,-343356,489550,-698073

%N Inverse Euler transform of primes.

%H Seiichi Manyama, <a href="/A030010/b030010.txt">Table of n, a(n) for n = 1..5000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Product_{k>=1} 1/(1-x^k)^{a(k)} = 1 + Sum_{n>=1} prime(n) * x^n.

%F From _Vaclav Kotesovec_, Oct 09 2019: (Start)

%F a(n) ~ -(-1)^n * A072508^n / n.

%F a(n) ~ -(-1)^n / (n * A088751^n). (End)

%e (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 pp = Prime[Range[n = 100]]; s = {};

%t For[i = 1, i <= n, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i-d]], {d, i-1}]]];

%t 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 Cf. A000040, A030011, A072508, A088751.

%K sign

%O 1,1

%A _N. J. A. Sloane_