A354709 Decimal expansion of Sum_{p prime} 3*(2p-1)*log(p)/(p^3 + p^2 - 3p + 1).

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

Offset

1,1

Comments

Also logarithmic derivative of A(s,w) at (0,0), where A(s,w) = Product_{p prime} (1 - (1 - (p*(1 - p^(-1-s))^3)/(-1+p))*(1 - (p*(1 - p^(-1-w))^3)/(-1+p))), with A(0,0) = A256392.

Links

Table of n, a(n) for n=1..105.

Example

2.52906617358092992925958712930189459230009223994439976118899256270135780066...

Mathematica

Block[{$MaxExtraPrecision = 1000},
Do[CC = Join[{0},
    Series[(3 (-1 + 2 p))/(1 - 3 p + p^2 + p^3) //. p -> 1/x, {x, 0,
       t}][[3]]];
  Print[N[-Sum[
        CC[[k]]*(PrimeZetaP'[k] + Log[2]/2^k), {k, 1, Length[CC]}] + (
      3 (-1 + 2 p) Log[p])/(1 - 3 p + p^2 + p^3) //. p -> 2, 75]], {t,
    1000, 1500, 100}]]
ratfun = 3*(2*p - 1)/(p^3 + p^2 - 3*p + 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, 120]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 04 2022 *)

Crossrefs

Cf. A256392.

Sequence in context: A029621 A240223 A134349 * A131711 A131201 A070633

Adjacent sequences: A354706 A354707 A354708 * A354710 A354711 A354712

Keyword

nonn,cons

Author

David Nguyen, Jun 03 2022

Extensions

More digits from Vaclav Kotesovec, Jun 04 2022

Status

approved