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%I #17 Aug 14 2020 01:47:18
%S 1,1,1,7,8,6,4,1,5,1,1,8,9,9,4,4,9,7,3,1,4,0,4,0,9,9,6,2,0,2,6,5,6,5,
%T 4,4,4,1,6,3,1,1,5,5,1,2,0,4,1,2,8,8,4,2,6,5,0,6,2,8,6,5,1,4,0,1,6,0,
%U 5,4,5,5,1,8,4,2,0,3,8,5,9,1,8,1,4,8,8,0,0,7,3,5,6,5,0,0,5,2,7,1,2,9,1,2,7
%N Decimal expansion of the Goh-Schmutz constant.
%C This constant is the limit of the logarithm expected order of a random permutation of length n, divided by sqrt(n/log n). In other words, log(A060014(n)/n!) ~ c sqrt(n/log n) where c is this constant. Stong improves the error term to O(sqrt(n) log log n/log n). - _Charles R Greathouse IV_, Nov 06 2014
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 287.
%H P. Erdős and P. Turán, <a href="http://ftp.bolyai.hu/~p_erdos/1965-08.pdf">On some problems of a statistical group theory, IV</a>, Acta Math. Acad. Sci. Hungar. 19 (1968), pp. 413-435. <a href="https://www.renyi.hu/~p_erdos/1968-11.pdf">[alternate link]</a>
%H William M. Y. Goh and Eric Schmutz, <a href="http://www.pages.drexel.edu/~schmutze/PAPERS/musn.pdf">The expected order of a random permutation</a>, Bulletin of the London Mathematical Society 23:1 (1991), pp. 34-42.
%H Richard Stong, <a href="https://doi.org/10.37236/1379">The average order of a permutation</a>, Electronic Journal of Combinatorics 5 (1998), 6 pp.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Goh-SchmutzConstant.html">Goh-Schmutz Constant</a>
%F From _Amiram Eldar_, Aug 13 2020: (Start)
%F Equals Integral_{x=0..oo} log(x+1)/(exp(x) - 1) dx.
%F Equals Integral_{x=0..oo} log(1 - log(1 - exp(-x))) dx.
%F Equals Integral_{x=0..oo} x*exp(-x)/((1 - exp(-x)) * (1 - log(1 - exp(-x)))) dx.
%F Equals -Sum_{k>=1} exp(k) * Ei(-k)/k, where Ei is the exponential integral. (End)
%e 1.1178641511899449731...
%t RealDigits[ N[ Integrate[Log[1 + t]/(E^t - 1), {t, 0, Infinity}], 105]][[1]] (* _Jean-François Alcover_, Oct 26 2012 *)
%o (PARI) intnum(t=0,[oo,1],log(1+t)/(exp(t)-1)) \\ _Charles R Greathouse IV_, Nov 05 2014
%K nonn,cons
%O 1,4
%A _Eric W. Weisstein_, Aug 05 2008
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