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A241017
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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.
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2
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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
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OFFSET
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0,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
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FORMULA
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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 = 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)).
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EXAMPLE
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0.822825249678847032995328716261464949475693118894850218393815613...
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MATHEMATICA
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S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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