This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340711 #62 Sep 27 2024 07:48:51
%S A340711 1,2,7,3,9,8,6,6,1,3,2,0,6,8,3,3,9,2,5,1,5,8,1,6,8,3,8,2,1,3,8,9,4,7,
%T A340711 2,7,3,4,7,6,2,7,4,4,4,6,7,6,7,3,5,7,8,9,4,0,0,2,9,6,8,1,4,4,0,9,8,7,
%U A340711 4,8,6,6,8,1,5,3,7,7,6,0,6,9,5,5,6,2,0,1,2,2,8,5,4,3,8,1,1,4,6,6,0,7,3,0,5,9,2,7,4,0,5,9,2,2,4,4,6,8,1,3
%N A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).
%H A340711 Vaclav Kotesovec, <a href="/A340711/b340711.txt">Table of n, a(n) for n = 1..500</a>
%H A340711 Salma Ettahri, Olivier Ramaré, and Léon Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler product</a>, arXiv:1908.06808 [math.NT], 2019, p. 20.
%H A340711 Steven Finch, <a href="https://arxiv.org/abs/0907.4894">Quartic and Octic Characters Modulo n</a>, arXiv:1008.2547 [math.NT], 2007-2010 p. 11 (formula on kappa(5) and kappa(-5)).
%H A340711 Steven Finch, Greg Martin and Pascal Sebah, <a href="http://www.math.ubc.ca/~gerg/papers/downloads/RUNM.pdf">Roots of unity and nullity modulo n</a>, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
%H A340711 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 (formulas).
%H A340711 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://doi.org/10.1016/j.jnt.2006.12.015">A note on Mertens' formula for arithmetic progressions</a>, Journal of Number Theory Volume 127, Issue 1, (2007), 37-46.
%H A340711 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://doi.org/10.1090/S0025-5718-08-02148-0">On the constant in the Mertens product for arithmetic progressions. II: Numerical values</a>, Math. Comp. 78 (2009), 315-326.
%H A340711 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://projecteuclid.org/euclid.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 A340711 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://www.dei.unipd.it/~languasco/MCcomput/MCfinalresults.pdf">Computation of the Mertens constants - more than 100 correct digits</a>, (2007), 1-134 (digital data).
%H A340711 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 A340711 Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%H A340711 G. Niklasch, <a href="http://guests.mpim-bonn.mpg.de/moree/Moree.en.html">Some number-theoretical constants arising as products of rational functions of p over primes</a>
%H A340711 Doug S. Phillips and Peter Zvengrowski, <a href="https://www.researchgate.net/publication/321953140_CONVERGENCE_OF_DIRICHLET_SERIES_AND_EULER_PRODUCTS">Convergence of Dirichlet Series and Euler Products</a>, Contributions Section of Natural Mathematical and Biotechnical Sciences 38(2):153 (2017).
%F A340711 D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
%F A340711 E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
%F A340711 F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = A340710.
%F A340711 G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = this constant.
%F A340711 H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
%F A340711 D*E*F*G*H = 5/2.
%F A340711 E*F*G*H = 30/13.
%F A340711 D*E*H = sqrt(5)/2.
%F A340711 D*F*G = 13*sqrt(5)/12.
%F A340711 F*G = sqrt(5).
%F A340711 E*H = 6*sqrt(5)/13.
%F A340711 Equals Sum_{q in A004617} 2^A001221(q)/q^2. - _R. J. Mathar_, Jan 27 2021
%e A340711 1.273986613206833925158...
%t A340711 (* Using Vaclav Kotesovec's function Z from A301430. *)
%t A340711 $MaxExtraPrecision = 1000; digits = 121;
%t A340711 digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
%t A340711 digitize[1/(Z[5, 3, 4]/Z[5, 3, 2]^2)]
%Y A340711 Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340710.
%K A340711 nonn,cons
%O A340711 1,2
%A A340711 _Artur Jasinski_, Jan 16 2021