A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

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

Offset: 0

Author

Jean-François Alcover, Jul 04 2014

Keywords

• nonn
• cons

Examples

			0.70475345170594788412255819759189881852...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Links

• Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18 (1964), pp. 75-86.
• Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.
• Eric Weisstein's MathWorld, Lemniscate Constant

Crossrefs

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

Programs

Mathematica

digits = 100; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

Formula

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.