A354709 - OEIS

%I #16 Jun 06 2022 02:49:14

%S 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,

%T 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,

%U 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

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

%C 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.

%e 2.52906617358092992925958712930189459230009223994439976118899256270135780066...

%t Block[{$MaxExtraPrecision = 1000},

%t Do[CC = Join[{0},

%t     Series[(3 (-1 + 2 p))/(1 - 3 p + p^2 + p^3) //. p -> 1/x, {x, 0,

%t        t}][[3]]];

%t   Print[N[-Sum[

%t         CC[[k]]*(PrimeZetaP'[k] + Log[2]/2^k), {k, 1, Length[CC]}] + (

%t       3 (-1 + 2 p) Log[p])/(1 - 3 p + p^2 + p^3) //. p -> 2, 75]], {t,

%t     1000, 1500, 100}]]

%t 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 *)

%Y Cf. A256392.

%K nonn,cons

%O 1,1

%A _David Nguyen_, Jun 03 2022

%E More digits from _Vaclav Kotesovec_, Jun 04 2022