

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
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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.
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 MasserGramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 12351246
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 MasserGramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(x)*log(x)/(1exp(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

JeanFrançois Alcover, Aug 08 2014


STATUS

approved



