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%I #16 Mar 11 2021 10:48:20
%S 1,8,6,6,1,4,2,9,7,3,5,8,3,5,8,3,9,6,6,5,6,9,2,4,8,4,7,9,4,4,1,8,8,3,
%T 3,7,8,4,0,0,7,3,9,4,4,9,4,5,5,8,9,3,0,4,8,7,1,7,2,6,6,9,1,8,3,8,9,8,
%U 0,7,4,4,9,2,4,3,8,0,8,1,9,6,2,7,0,6,2,6,1,9,0,3,2,8,0,6,3,1,0
%N Decimal expansion of Hardy-Littlewood constant C_6 = Product_{p prime > 6} 1/(1-1/p)^6 (1-6/p).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.
%e 0.18661429735835839665692484794418833784007394494558930487172669...
%t $MaxExtraPrecision = 1600; digits = 99; terms = 1600; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{7, -6}, {-30, -210}, 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)^6*(1-6/p), 1, 7) \\ _Amiram Eldar_, Mar 11 2021
%Y Cf. A005597, A065418, A065419, A269843.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Apr 17 2016
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