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A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6*p^2-4*p+1)/(p-1)^4).

%I A065419 #26 Feb 12 2025 11:10:43
%S A065419 3,0,7,4,9,4,8,7,8,7,5,8,3,2,7,0,9,3,1,2,3,3,5,4,4,8,6,0,7,1,0,7,6,8,
%T A065419 5,3,0,2,2,1,7,8,5,1,9,9,5,0,6,6,3,9,2,8,2,9,8,3,0,8,3,9,6,2,6,0,8,8,
%U A065419 8,7,6,7,2,9,6,6,9,2,9,9,4,8,1,3,8,4,0,2,6,4,6,8,1,7,1,4,9,3,8
%N A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6*p^2-4*p+1)/(p-1)^4).
%C A065419 For comparison: Product_{n>=5} (1-(6n^2-4n+1)/(n-1)^4) = 3/32. - _R. J. Mathar_, Feb 25 2009
%D A065419 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.
%H A065419 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^(4).
%H A065419 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 A065419 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%e A065419 0.30749487875832709312335448607107685302...
%t A065419 $MaxExtraPrecision = 1000; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{5, -4}, {-12, -60}, 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 A065419 (PARI) prodeulerrat(1-(6*p^2-4*p+1)/(p-1)^4, 1, 5) \\ _Amiram Eldar_, Mar 10 2021
%Y A065419 Cf. A065418, A027377, A065431.
%K A065419 cons,nonn
%O A065419 0,1
%A A065419 _N. J. A. Sloane_, Nov 15 2001
%E A065419 A sign in the definition corrected by _R. J. Mathar_, Feb 25 2009