Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #28 Nov 27 2024 15:53:15
%S 1,1,5,3,0,8,0,5,6,1,5,8,5,4,4,7,8,7,0,3,6,5,2,5,8,0,6,8,5,6,1,7,6,3,
%T 3,6,5,1,0,4,8,4,4,8,7,0,8,0,3,9,3,1,8,8,6,7,7,9,2,3,1,9,0,2,1,0,3,5,
%U 4,6,8,4,1,3,2,5,2,9,8,2,0,0,4,3,5,4,9,2,5,3,5,9,2,8,1,2,0,7,8,1,2
%N Decimal expansion of Pi^2/(16*K^2*G) = Product_{p prime congruent to 3 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="/A243381/b243381.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 Equals Pi^2/(16*K^2*G), 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
%e 1.1530805615854478703652580685617633651...
%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]; Pi^2/(16*LandauRamanujanK^2*Catalan) // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)
%Y Cf. A002145, A064533, A243379.
%K nonn,cons
%O 1,3
%A _Jean-François Alcover_, Jun 04 2014