A143300 Decimal expansion of the Goh-Schmutz constant.

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

Offset: 1

Author

Eric W. Weisstein, Aug 05 2008

Keywords

• nonn
• cons

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

Examples

			1.1178641511899449731...
		

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

Programs

Mathematica

RealDigits[ N[ Integrate[Log[1 + t]/(E^t - 1), {t, 0, Infinity}], 105]][[1]] (* Jean-François Alcover, Oct 26 2012 *)

PARI

intnum(t=0,[oo,1],log(1+t)/(exp(t)-1)) \\ Charles R Greathouse IV, Nov 05 2014

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)