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 A143300 Decimal expansion of the Goh-Schmutz constant. 2
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 287. LINKS P. Erdős and P. Turán, On some problems of a statistical group theory, IV, Acta Math. Acad. Sci. Hungar. 19 (1968), pp. 413-435. [alternate link] William M. Y. Goh and Eric Schmutz, The expected order of a random permutation, Bulletin of the London Mathematical Society 23:1 (1991), pp. 34-42. Richard Stong, The average order of a permutation, Electronic Journal of Combinatorics 5 (1998), 6 pp. Eric Weisstein's World of Mathematics, Goh-Schmutz Constant FORMULA From Amiram Eldar, Aug 13 2020: (Start) Equals Integral_{x=0..oo} log(x+1)/(exp(x) - 1) dx. Equals Integral_{x=0..oo} log(1 - log(1 - exp(-x))) dx. Equals Integral_{x=0..oo} x*exp(-x)/((1 - exp(-x)) * (1 - log(1 - exp(-x)))) dx. Equals -Sum_{k>=1} exp(k) * Ei(-k)/k, where Ei is the exponential integral. (End) EXAMPLE 1.1178641511899449731... MATHEMATICA RealDigits[ N[ Integrate[Log[1 + t]/(E^t - 1), {t, 0, Infinity}], 105]][[1]] (* Jean-François Alcover, Oct 26 2012 *) PROG (PARI) intnum(t=0, [oo, 1], log(1+t)/(exp(t)-1)) \\ Charles R Greathouse IV, Nov 05 2014 CROSSREFS Sequence in context: A197762 A181624 A277683 * A303985 A242816 A333566 Adjacent sequences:  A143297 A143298 A143299 * A143301 A143302 A143303 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Aug 05 2008 STATUS approved

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Last modified September 24 21:35 EDT 2020. Contains 337322 sequences. (Running on oeis4.)