cp's OEIS Frontend

A065418 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

%I A065418 #27 Feb 12 2025 11:09:48
%S A065418 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 A065418 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 A065418 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 A065418 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).
%C A065418 For comparison: Product_{n>=5} (1-(3n-1)/(n-1)^3) = 3/8 . - _R. J. Mathar_, Feb 25 2009
%D A065418 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.
%H A065418 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 A065418 B. H. Mayoh, <a href="https://doi.org/10.1007/BF01939334">The 2nd Goldbach conjecture revisited</a>, BIT 8 (1968) 128-133 Table 5.
%H A065418 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%F A065418 The constant equals Product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-A027376(n). - _Michael Somos_, Apr 05 2003
%e A065418 0.635166354604271207206696591272522417342...
%t A065418 $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 A065418 (PARI) prodeulerrat(1-(3*p-1)/(p-1)^3, 1, 5) \\ _Amiram Eldar_, Mar 10 2021
%Y A065418 Cf. A066654, A065419, A027376.
%K A065418 cons,nonn
%O A065418 0,1
%A A065418 _N. J. A. Sloane_, Nov 15 2001