A391182 - OEIS

%I #11 Dec 04 2025 04:11:54

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

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

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

%N Decimal expansion of a constant related to the asymptotic counting function of numbers of the form x^2 + 2*y^2 (A002479).

%C Analogous to Landau-Ramanujan constant (A064533) as sums of 2 squares (A001481) are analogous to A002479.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3.1, p. 99.

%H David Brink, Pieter Moree, and Robert Osburn, <a href="https://archive.mpim-bonn.mpg.de/id/eprint/2552/1/preprint_2010_22.pdf">On computations of Shanks and Schmid</a>, 2010.

%H Salma Ettahri, Olivier Ramaré, and Léon Surel, <a href="https://doi.org/10.1090/mcom/3630">Fast multi-precision computation of some Euler products</a>, Mathematics of Computation, Vol. 90, No. 331 (2021), pp. 2247-2265; <a href="https://ramare-olivier.github.io/Maths/mcom3630.pdf">alternative link</a>.

%H Pieter Moree and Robert Osburn, <a href="https://doi.org/10.5169/seals-2239">Two-dimensional lattices with few distances</a>, L'Enseignement Mathématique, Vol. 52, No. 3-4 (2006), pp. 361-380; <a href="https://arxiv.org/abs/math/0604163">arXiv preprint</a>, arXiv:math/0604163 [math.NT], 2006; <a href="http://hdl.handle.net/10197/7958">alternative link</a>.

%H Daniel Shanks and Larry P. Schmid, <a href="https://doi.org/10.1090/S0025-5718-1966-0210678-1">Variations on a theorem of Landau. Part I</a>, Mathematics of Computation, Vol. 20, No. 96 (1966), pp. 551-569.

%F Equals lim_{n->oo} (sqrt(log(n))/n) * Sum_{k=1..n} A391183(k).

%F Equals 2^(-1/4) * Product_{p prime == 5 or 7 (mod 8)} (1 - 1/p^2)^(-1/2).

%e 0.872887558130914612920063683587300963485686018866808...

%t $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[2^(-1/4) * Sqrt[Z[8, 5, 2] * Z[8, 7, 2]], digits]], 10, digits - 1][[1]] (* using _Vaclav Kotesovec_'s code at A175646 *)

%Y Cf. A001481, A002479, A003628, A064533, A301430, A391183.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Dec 02 2025