A074785 Decimal expansion of -log(log(2)).

3, 6, 6, 5, 1, 2, 9, 2, 0, 5, 8, 1, 6, 6, 4, 3, 2, 7, 0, 1, 2, 4, 3, 9, 1, 5, 8, 2, 3, 2, 6, 6, 9, 4, 6, 9, 4, 5, 4, 2, 6, 3, 4, 4, 7, 8, 3, 7, 1, 0, 5, 2, 6, 3, 0, 5, 3, 6, 7, 7, 7, 1, 3, 6, 7, 0, 5, 6, 1, 6, 1, 5, 3, 1, 9, 3, 5, 2, 7, 3, 8, 5, 4, 9, 4, 5, 5, 8, 2, 2, 8, 5, 6, 6, 9, 8, 9, 0, 8, 3, 5, 8, 3, 0

Offset: 0

Author

Benoit Cloitre, Sep 07 2002

Keywords

• cons
• easy
• nonn

Comments

The function f(p) = Integral_{x = 2..infinity} 1/(x*log(x)^p) has a minimum of -e*log(log(2)) = 0.996285... at p = 1 - 1/log(log(2)) = 3.728416... - Jean-François Alcover, May 24 2013

log(log(2)) also equals the median of the Gumbel distribution with location parameter 0 and scale parameter 1. - Jean-François Alcover, Jul 29 2014

Examples

			log(log(2)) = -0.36651292058166432701243915823266946945...
		

References

• Donald Knuth, The Art of Computer Programming, 3rd Edition, Volume 1. Boston: Addison-Wesley Professional (1997): 619, Table 1 of Appendix A.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to Pi^(-1), Journal of Mathematical Analysis and Applications, Vol. 442, No. 2 (2016), pp. 404-434.
• Dmitrii Kouznetsov and Henryk Trappmann, Portrait of the four regular super-exponentials to base sqrt(2), Math. Comp., Vol. 79, No. 271 (2010), pp. 1727-1756, eq. (3.2).
• Simon Plouffe, log(log(2)).
• Eric Weisstein's World of Mathematics, Gumbel Distribution.

Crossrefs

Cf. A001620, A059200.

Programs

Mathematica

RealDigits[-Log[Log[2]], 10, 120][[1]] (* Harvey P. Dale, Nov 24 2013 *)

PARI

-log(log(2)) \\ Charles R Greathouse IV, Jan 04 2016

Formula

Equals Sum_{n>=1} ((-1)^n/(n*n!) * (Sum_{k=1..n} abs(S1(n,k))/(k+1))), where S1(n,k) are the Stirling numbers of the first kind (Blagouchine, 2016). Without the absolute value the formula gives -gamma (= -A001620). - Amiram Eldar, Jun 12 2021