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%I #44 Jun 12 2021 09:00:13
%S 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,
%T 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,
%U 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
%N Decimal expansion of -log(log(2)).
%C 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
%C 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
%D Donald Knuth, The Art of Computer Programming, 3rd Edition, Volume 1. Boston: Addison-Wesley Professional (1997): 619, Table 1 of Appendix A.
%H Vincenzo Librandi, <a href="/A074785/b074785.txt">Table of n, a(n) for n = 0..1000</a>
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">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)</a>, Journal of Mathematical Analysis and Applications, Vol. 442, No. 2 (2016), pp. 404-434.
%H Dmitrii Kouznetsov and Henryk Trappmann, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02342-2">Portrait of the four regular super-exponentials to base sqrt(2)</a>, Math. Comp., Vol. 79, No. 271 (2010), pp. 1727-1756, eq. (3.2).
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap64.html">log(log(2))</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GumbelDistribution.html">Gumbel Distribution</a>.
%F 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
%e log(log(2)) = -0.36651292058166432701243915823266946945...
%t RealDigits[-Log[Log[2]], 10, 120][[1]] (* _Harvey P. Dale_, Nov 24 2013 *)
%o (PARI) -log(log(2)) \\ _Charles R Greathouse IV_, Jan 04 2016
%Y Cf. A001620, A059200.
%K cons,easy,nonn
%O 0,1
%A _Benoit Cloitre_, Sep 07 2002
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