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%I #6 Jun 18 2020 08:02:07
%S 2,3,5,1,0,9,7,1,4,0,7,7,8,7,6,6,2,8,3,2,3,4,1,6,6,0,8,5,2,3,3,7,7,1,
%T 2,7,8,6,3,0,3,8,4,5,2,1,8,8,5,9,6,0,2,7,4,3,4,3,3,3,2,7,7,7,1,8,6,9,
%U 1,8,0,2,0,4,5,5,1,6,8,5,5,3,0,7,2,9,6,3,5,0,1,9,1,0,9,1,9,8,3,0,5,2,7,2,4,5
%N Decimal expansion of Sum_{primes p} 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2 / (p^3 + p - 2)^2.
%e 0.23510971407787662832341660852337712786303845218859602743433...
%t ratfun = 2*p*(2*p^3 - 9*p^2 - 1) / (p^3 + p - 2)^2; zetas = 0; ratab = Table[konfun = Together[Simplify[ratfun - c*(p^power/(p^power - 1)^2)]]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(-Zeta'[power]^2/Zeta[power]^2 + Zeta''[power]/Zeta[power]) /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]^2*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}]
%Y Cf. A208133, A335705.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Jun 18 2020
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