A242433 Decimal expansion of one of the Pell-Stevenhagen constants.

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

Offset

0,1

Comments

P. Stevenhagen conjectured that the asymptotic counting function of the squarefree integers for which the negative Pell equation x^2 - n*y^2 = -1 has an integer solution, was f(n) ~ (6/Pi^2)*P*K*n/sqrt(log(n)).

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 119.

Links

Table of n, a(n) for n=0..99.

Eric Weisstein's MathWorld, Landau-Ramanujan Constant

Eric Weisstein's MathWorld, Pell Constant

Formula

(6/Pi^2)*P*K where P is the Pell constant 0.5805775582... and K the Landau-Ramanujan constant 0.7642236535...

Example

0.26973184619696337738212710674891...

Mathematica

(* After Victor Adamchik *) LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}]/Sqrt[2], n]]; K = LandauRamanujan[100]; P = 1 - QPochhammer[1/2, 1/4]; RealDigits[6/Pi^2*P*K, 10, 100] // First

Crossrefs

Cf. A064533, A141848.

Sequence in context: A155678 A134946 A175575 * A011046 A246828 A136701

Adjacent sequences: A242430 A242431 A242432 * A242434 A242435 A242436

Keyword

nonn,cons

Author

Jean-François Alcover, May 14 2014

Status

approved