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

Author

Jean-François Alcover, May 14 2014

Keywords

• nonn
• cons

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)).

Examples

			0.26973184619696337738212710674891...
		

References

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

Links

• Eric Weisstein's MathWorld, Landau-Ramanujan Constant
• Eric Weisstein's MathWorld, Pell Constant

Crossrefs

Cf. A064533, A141848.

Programs

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

Formula

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