A391182 Decimal expansion of a constant related to the asymptotic counting function of numbers of the form x^2 + 2*y^2 (A002479).

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, 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, 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

Offset

0,1

Comments

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

References

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

Links

Table of n, a(n) for n=0..105.

David Brink, Pieter Moree, and Robert Osburn, On computations of Shanks and Schmid, 2010.

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation, Vol. 90, No. 331 (2021), pp. 2247-2265; alternative link.

Pieter Moree and Robert Osburn, Two-dimensional lattices with few distances, L'Enseignement Mathématique, Vol. 52, No. 3-4 (2006), pp. 361-380; arXiv preprint, arXiv:math/0604163 [math.NT], 2006; alternative link.

Daniel Shanks and Larry P. Schmid, Variations on a theorem of Landau. Part I, Mathematics of Computation, Vol. 20, No. 96 (1966), pp. 551-569.

Formula

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

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

Example

0.872887558130914612920063683587300963485686018866808...

Mathematica

$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 *)

Crossrefs

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

Sequence in context: A255696 A144750 A198928 * A155068 A244839 A329450

Adjacent sequences: A391179 A391180 A391181 * A391183 A391184 A391185

Keyword

nonn,cons

Author

Amiram Eldar, Dec 02 2025

Status

approved