%I #24 Aug 21 2022 09:50:46

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

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

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

%N Decimal expansion of 192*K^2*G/Pi^4 = Product_{p prime congruent to 1 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

%H G. C. Greubel, <a href="/A243380/b243380.txt">Table of n, a(n) for n = 1..2500</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">Ramanujan constant</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a>

%F 192*K^2*G/Pi^4, where K is the Landau-Ramanujan constant (A064533) and G Catalan's constant (A006752).

%F A243380 * A243381 = 12/Pi^2. - _Vaclav Kotesovec_, Apr 30 2020

%F Equals A175647 / 1.001652229636651... both constants from p 26 of arXiv:1008.2537v2. - _R. J. Mathar_, Aug 21 2022

%e 1.0544399447999484896488194648267179483...

%t digits = 101; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 192*LandauRamanujanK^2*Catalan/Pi^4 // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

%Y Cf. A002144, A064533, A088539.

%K cons,nonn

%O 1,3

%A _Jean-François Alcover_, Jun 04 2014