A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3*n)/(3*n), where P(x) is the prime zeta function.
0, 5, 5, 6, 1, 3, 2, 6, 2, 5, 9, 6, 2, 7, 7, 7, 1, 0, 1, 7, 8, 7, 4, 7, 4, 6, 3, 4, 5, 3, 0, 5, 1, 5, 2, 9, 0, 1, 8, 0, 3, 7, 2, 6, 6, 1, 0, 0, 2, 8, 8, 4, 3, 8, 7, 4, 6, 5, 0, 4, 0, 1, 0, 3, 6, 2, 5, 6, 6, 5, 4, 5, 0, 3, 2, 6, 4, 2, 2, 6, 7, 3, 7, 0, 8, 3, 9, 0, 9, 7, 7, 2, 4, 7, 4, 5, 8, 2, 7, 3, 5, 8, 9, 3, 3, 5
Offset: 0
Examples
0.055613262596277710178747463453...
Crossrefs
Cf. A387293.
Programs
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Mathematica
RealDigits[Log[Zeta[3]/Zeta[6]]/3, 10, 105, -1][[1]]
Formula
Equals log(zeta(3)/zeta(6))/3.
Equals log(3*(35*zeta(3))^(1/3)/Pi^2).
Sum_{p prime} Sum_{n>=1} (-1)^(n+1)/p^(3*n)/(3*n) = Sum_{p prime} log((1+1/p^3))/3 = log(Product_{p prime} (1+1/p^3))/3 = log(zeta(3)/zeta(6))/3. - Amiram Eldar, Aug 25 2025