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 A227158 Second-order term in the asymptotic expansion of B(x), the count of numbers up to x which are the sum of two squares. 7
 5, 8, 1, 9, 4, 8, 6, 5, 9, 3, 1, 7, 2, 9, 0, 7, 9, 7, 9, 2, 8, 1, 4, 9, 8, 8, 4, 5, 0, 2, 3, 6, 7, 5, 5, 9, 3, 0, 4, 8, 3, 2, 8, 7, 3, 0, 7, 1, 7, 7, 2, 5, 2, 1, 8, 2, 3, 4, 2, 1, 2, 9, 9, 2, 6, 5, 2, 5, 1, 2, 3, 1, 5, 5, 5, 9, 5, 0, 3, 4, 6, 1, 4, 3, 0, 1, 2, 3, 6, 1, 3, 1, 4, 9, 2, 4, 1, 3, 4, 9, 6 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS K = A064533, the Landau-Ramanujan constant, is the first-order term. This constant is c = lim_{x->infinity} (B(x)*sqrt(log x)/(K*x) - 1)*log x. [Corrected by Alessandro Languasco, Sep 14 2022] REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99. LINKS Table of n, a(n) for n=0..100. Alexandru Ciolan, Alessandro Languasco and Pieter Moree, Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms, section 10, 47500 digits are obtained, Journal of Mathematical Analysis and Applications, 2022; see also preprint on arXiv, arXiv:2109.03288 [math.NT], 2021. David Hare, Landau-Ramanujan Constant, second order obtained about 5000 digits, 1996. D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86. Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant EXAMPLE 0.58194865931729079777136487517474826173838317235153574360562.... MATHEMATICA digits = 101; 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]; RealDigits[f[m], 10, digits] // First (* Jean-François Alcover, May 27 2014 *) PROG (PARI) L(s)=sumalt(k=0, (-1)^k/(2*k+1)^s) LL(s)=L'(s)/L(s) ZZ(s)=zeta'(s)/zeta(s) sm(x)=my(s); forprime(q=2, x, if(q%4==3, s+=log(q)/(q^8-1))); s+1/49/x^7+log(x)/7/x^7 1/2+log(2)/4-Euler/4-LL(1)/4-ZZ(2)/4+LL(2)/4-log(2)/12-ZZ(4)/4+LL(4)/4-log(2)/60+sm(1e5)/2 CROSSREFS Cf. A064533, A001481. Sequence in context: A171709 A093157 A122998 * A098881 A185393 A073333 Adjacent sequences: A227155 A227156 A227157 * A227159 A227160 A227161 KEYWORD nonn,cons AUTHOR Charles R Greathouse IV, Jul 03 2013 EXTENSIONS Corrected and extended by Jean-François Alcover, Mar 19 2014 and again May 27 2014 STATUS approved

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Last modified June 2 13:01 EDT 2023. Contains 363097 sequences. (Running on oeis4.)