A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)*log(p)/(p^4+2*p^3+1).

4, 0, 5, 2, 3, 7, 0, 3, 1, 4, 4, 4, 2, 2, 3, 9, 2, 5, 0, 8, 5, 9, 6, 5, 0, 9, 9, 1, 1, 2, 1, 8, 5, 2, 3, 4, 1, 0, 4, 4, 1, 4, 1, 7, 2, 4, 0, 4, 1, 9, 8, 4, 9, 2, 6, 2, 3, 4, 6, 3, 6, 2, 9, 7, 7, 5, 3, 7, 9, 8, 9, 0, 1, 8, 1, 8, 6, 4, 0, 3, 8, 0, 4, 8, 7, 4, 2, 6, 4, 6, 6, 4, 3, 9, 3, 6, 8, 4, 0, 6, 3, 7, 7, 7, 8, 4

Offset: 0

Author

Amiram Eldar, Jun 19 2018

Keywords

• nonn
• cons

Comments

The constant B that appears in the asymptotic formula for the sum of the bi-unitary divisor function (A306069).

Examples

			0.405237031444223925085965099112185234104414172404198492623463629775379...
		

Crossrefs

Cf. A085609, A306069, A335705, A335707.

Programs

Mathematica

cc = CoefficientList[Series[(p^2 - p - 1)/(p^4 + 2*p^3 + 1) /. p -> 1/x, {x, 0, 30}], x]; f = FindSequenceFunction[cc]; digits = 20; B = 2 NSum[f[n + 1 // Round]*(-PrimeZetaP'[n]), {n, 2, Infinity}, Method -> "AlternatingSigns", NSumTerms -> 10 digits, WorkingPrecision -> 5 digits]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
ratfun = 2*(p^2 - p - 1)/(p^4 + 2*p^3 + 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 17 2020 *)

Extensions

a(1)-a(20) from Jean-François Alcover, Jun 19 2018

More digits from Vaclav Kotesovec, Jun 17 2020