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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Aug 05 2008
STATUS
approved