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%I #28 Mar 12 2023 10:45:27
%S 2,0,1,6,6,2,1,5,4,5,7,3,3,4,0,8,1,1,5,2,6,2,7,9,6,8,5,9,7,1,5,1,1,5,
%T 4,2,6,4,5,0,1,8,4,1,7,7,5,2,3,6,4,7,4,8,0,6,1,0,9,1,9,2,8,3,4,4,7,8,
%U 1,4,3,4,1,6,1,6,1,8,2,7,8,7,2,5,5,4,1,3,5,1,3,9,8,3,0,6,1,8,0,4
%N Decimal expansion of Sierpiński's S~ (S "tilde" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.
%C From _Vaclav Kotesovec_, Mar 10 2023: (Start)
%C Sum_{k=1..n} A002654(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =
%C 4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...
%C The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576268... is wrong! (End)
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's constant, p. 122.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, Section 2.10 p. 17.
%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84, section (K), formula (22).
%F S_tilde = 2*S - 12/Pi^2*zeta'(2) + log(2)/3 - 1, where S = A086058 - 1 = A062089 / Pi.
%e 2.01662154573340811526279685971511542645018417752364748061...
%t S = 2*EulerGamma + 2*Log[2] + 3*Log[Pi] - 4* Log[Gamma[1/4]]; (* S~ *) St = 2*S - 12/Pi^2*Zeta'[2] + Log[2]/3 - 1; RealDigits[St, 10, 100] // First
%Y Cf. A062089, A002654, A086058, A241009.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Aug 07 2014
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