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A340127 Decimal expansion of Product_{primes p == 4 (mod 5)} p^2/(p^2-1).

%I A340127 #129 Dec 04 2024 09:47:00
%S A340127 1,0,0,4,9,6,0,3,2,3,9,2,2,2,9,7,5,5,8,9,9,3,7,4,9,6,2,4,8,1,0,2,5,2,
%T A340127 1,8,4,7,9,5,5,1,0,2,9,4,1,8,8,0,2,2,8,8,0,1,9,9,5,2,8,3,7,8,5,2,1,5,
%U A340127 0,7,1,2,7,7,0,0,7,0,0,7,6,9,8,8,5,4,3,2,4,9,1,3,6,1,1,8,0,0,6,1,9
%N A340127 Decimal expansion of Product_{primes p == 4 (mod 5)} p^2/(p^2-1).
%H A340127 Vaclav Kotesovec, <a href="/A340127/b340127.txt">Table of n, a(n) for n = 1..501</a>
%H A340127 S. Ettahri, O. Ramare, L. Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arxiv:1908.06808 (2019), Section 9.
%H A340127 Steven Finch and Pascal Sebah, <a href="https://arxiv.org/abs/0912.3677">Residue of a Mod 5 Euler Product</a>, arXiv:0912.3677 [math.NT], 2009 (C(5,n) = mu(n,5) formulas p.2).
%H A340127 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://www.dei.unipd.it/~languasco/MCcomput/MertensConstantsfinal.gp">Computation of the Mertens constants mod q; 3 <= q <= 100</a>, (2007) (GP-PARI procedure 100 digits accuracy).
%H A340127 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://doi.org/10.7169/facm/1269437065">On the constant in the Mertens product for arithmetic progressions. I. Identities</a>, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
%H A340127 For other links see A340711.
%F A340127 Equals (1/C(5,4))*Pi*sqrt(3*C(5,1)*C(5,2)*C(5,3)/(5*C(5,4)*log(2+sqrt(5)))).
%F A340127 for definitions of Mertens constants C(5,n) see A. Languasco and A. Zaccagnini 2010.
%F A340127 for high precision numerical values C(5,n) see A. Languasco and A. Zaccagnini 2007.
%F A340127 C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
%F A340127 C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
%F A340127 C(5,3)=0.8059510404482678640573768602784309320812881149390108979348...
%F A340127 C(5,4)=1.29936454791497798816084001496426590950257497040832966201678...
%F A340127 Equals (1/C(5,4)^2)*Pi*sqrt(3*exp(-gamma)/(4*log(2 + sqrt(5)))), where gamma is the Euler-Mascheroni constant A001620.
%F A340127 Equals Sum_{k>=1} 1/A004618(k)^2. - _Amiram Eldar_, Jan 24 2021
%e A340127 1.0049603239222975589937496248102521847955102941880228801995283785215071277...
%t A340127 (* Using Vaclav Kotesovec's function Z from A301430. *)
%t A340127 $MaxExtraPrecision = 1000; digits = 121;
%t A340127 digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
%t A340127 digitize[Z[5, 4, 2]]
%Y A340127 Cf. A004618, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004.
%K A340127 nonn,cons
%O A340127 1,4
%A A340127 _Artur Jasinski_, Jan 15 2021