A244662 - OEIS

%I #11 Jan 05 2025 19:51:40

%S 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,

%T 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,

%U 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

%N 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.

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

%H Daniel Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0159174-9">The second-order term in the asymptotic expansion of B(x)</a>, Mathematics of Computation, Vol. 18 (1964), pp. 75-86.

%H Daniel Shanks, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/shanks.pdf">Non-hypotenuse Numbers</a>, Fib. Quart., 13:4 (1975), pp. 319-321.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>

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

%e 0.70475345170594788412255819759189881852...

%t 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

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

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jul 04 2014