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Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).
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%I #21 Mar 10 2021 03:18:10

%S 6,3,5,1,6,6,3,5,4,6,0,4,2,7,1,2,0,7,2,0,6,6,9,6,5,9,1,2,7,2,5,2,2,4,

%T 1,7,3,4,2,0,6,5,6,8,7,3,3,2,3,7,2,4,5,0,8,9,9,7,3,4,4,6,0,4,8,6,7,8,

%U 4,6,1,3,1,1,6,1,3,9,1,8,8,2,0,8,0,2,9,1,3,8,6,7,6,4,0,4,6,1,7

%N Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

%C For comparison: Product_{n>=5} (1-(3n-1)/(n-1)^3) = 3/8 . - _R. J. Mathar_, Feb 25 2009

%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 [math.NT], 2009-2011, constant C_1^(3).

%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]

%F The constant equals Product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-A027376(n). - _Michael Somos_, Apr 05 2003

%e 0.635166354604271207206696591272522417342...

%t $MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{4, -3}, {-6, -24}, 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 (* _Jean-François Alcover_, Apr 17 2016 *)

%o (PARI) prodeulerrat(1-(3*p-1)/(p-1)^3, 1, 5) \\ _Amiram Eldar_, Mar 10 2021

%Y Cf. A066654, A065419, A027376.

%K cons,nonn

%O 0,1

%A _N. J. A. Sloane_, Nov 15 2001