A030010 Inverse Euler transform of primes.

2, 0, 1, 0, 2, -3, 2, -4, 4, -3, 4, -5, 10, -21, 20, -18, 34, -46, 64, -99, 126, -182, 258, -319, 464, -685, 936, -1352, 1888, -2570, 3690, -5188, 7292, -10501, 14742, -20766, 29610, -41650, 59052, -84338, 119602, -170279, 242256, -343356, 489550, -698073

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Examples

			(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 + ... .
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000040, A030011, A072508, A088751.

Programs

Mathematica

pp = Prime[Range[n = 100]]; s = {};
For[i = 1, i <= n, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i-d]], {d, i-1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, May 10 2019 *)

Formula

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

From Vaclav Kotesovec, Oct 09 2019: (Start)

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

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