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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

Table of n, a(n) for n=0..103.

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: A021928 A185111 A086058 * A013662 A222225 A140244

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 August 21 00:45 EDT 2017. Contains 290855 sequences.