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%I #20 Nov 05 2021 09:29:43
%S 4,0,9,8,7,4,8,8,5,0,8,8,2,3,6,4,7,4,4,7,8,7,8,1,2,1,2,3,3,7,9,5,5,2,
%T 7,7,8,9,6,3,5,8,0,1,3,2,5,4,9,4,5,4,6,9,8,2,6,3,3,6,3,9,8,8,2,2,6,4,
%U 8,2,3,6,1,7,3,9,6,5,9,6,5,1,5,4,6,0,8,4,5,4,4,9,9,6,2,0,2,8,1
%N Decimal expansion of Hardy-Littlewood constant C_5 = Product_{p prime > 5} 1/(1-1/p)^5 (1-5/p).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.
%e 0.4098748850882364744787812123379552778963580132549454698263363988...
%t $MaxExtraPrecision = 800; digits = 99; terms = 800; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{6, -5}, {-20, -120}, 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/(1-1/p)^5*(1-5/p), 1, 7) \\ _Amiram Eldar_, Mar 11 2021
%Y Cf. A005597, A065418, A065419, A001692.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Apr 17 2016
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