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Inverse Euler transform of {1, primes}.
7

%I #19 May 10 2019 10:35:31

%S 1,1,1,1,1,1,-1,-2,-3,-2,4,-1,5,3,-4,-5,-9,3,-3,15,19,0,6,-39,-27,-22,

%T 5,57,50,107,-49,-96,-142,-213,83,138,468,365,0,-327,-1215,-618,-526,

%U 957,2572,1831,1673,-2820,-4516,-6155,-3880,5998,9282,18213,7414

%N Inverse Euler transform of {1, primes}.

%H Seiichi Manyama, <a href="/A030011/b030011.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 + x + Sum_{n>=1} prime(n) * x^(n + 1).

%e (1-x)^(-1) * (1-x^2)^(-1) * (1-x^3)^(-1) * (1-x^4)^(-1) * (1-x^5)^(-1) * (1-x^6)^(-1) * (1-x^7) * (1-x^8)^2 * ... = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + ... .

%t pp = Prepend[Prime[Range[n = 100]], 1]; s = {};

%t For[i = 1, i <= n + 1, 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,

%t n + 1}] (* _Jean-François Alcover_, May 10 2019 *)

%Y Cf. A000040, A030010.

%K sign

%O 1,8

%A _N. J. A. Sloane_