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%I #46 Mar 11 2021 03:02:27
%S 5,7,3,8,1,3,8,6,2,6,1,2,0,7,0,5,9,9,0,4,7,8,8,6,3,9,3,4,5,7,9,0,6,3,
%T 2,7,6,6,4,7,7,6,1,0,9,5,5,8,6,8,7,3,8,6,2,4,8,7,0,9,3,8,7,1,4,6,2,2,
%U 4,3,8,8,5,7,6,7,0,1,3,6,8,1,9,2,8,5,4,5,7,7,5,2,8,5,2,0,6,3,0
%N Decimal expansion of Product_{p prime > 2} 1-1/(p^2-3p+3), a constant related to I. M. Vinogradov's proof of the "ternary" Goldbach conjecture.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 88.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach's Conjecture</a>.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/VinogradovsTheorem.html">Vinogradov's theorem</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Vinogradov's_theorem">Vinogradov's theorem</a>.
%F Equals A005597 / A271951.
%e 0.5738138626120705990478863934579063276647761095586873862487...
%t $MaxExtraPrecision = 600; digits = 99; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = LinearRecurrence[{6, -14, 15, -6}, {0, 0, -2, -9}, terms + 10]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First
%o (PARI) prodeulerrat(1-1/(p^2-3*p+3), 1, 3) \\ _Amiram Eldar_, Mar 11 2021
%Y Cf. A005597, A271951.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Apr 17 2016
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