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 A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi. 2
 8, 2, 2, 8, 2, 5, 2, 4, 9, 6, 7, 8, 8, 4, 7, 0, 3, 2, 9, 9, 5, 3, 2, 8, 7, 1, 6, 2, 6, 1, 4, 6, 4, 9, 4, 9, 4, 7, 5, 6, 9, 3, 1, 1, 8, 8, 9, 4, 8, 5, 0, 2, 1, 8, 3, 9, 3, 8, 1, 5, 6, 1, 3, 0, 3, 7, 0, 9, 0, 9, 5, 6, 4, 4, 6, 4, 0, 1, 6, 6, 7, 5, 7, 2, 1, 9, 5, 3, 2, 5, 7, 3, 2, 3, 4, 4, 5, 3, 2, 4, 7, 2, 1, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123. LINKS M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9. Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246 Eric Weisstein's MathWorld, Sierpiński's Constant Wikipedia, Sierpiński's Constant FORMULA S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function. S = log[Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant. S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function. S = A062089 / Pi, where A062089 is Sierpiński's K constant. S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015] S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx. Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)). EXAMPLE 0.822825249678847032995328716261464949475693118894850218393815613... MATHEMATICA S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First CROSSREFS Cf. A062089, A062539, A068466, A086058. Sequence in context: A185111 A319188 A086058 * A114314 A013662 A291362 Adjacent sequences:  A241014 A241015 A241016 * A241018 A241019 A241020 KEYWORD nonn,cons AUTHOR Jean-François Alcover, Aug 08 2014 STATUS approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)