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

1,4

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

Table of n, a(n) for n=1..105.

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