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%I #52 Aug 24 2023 04:52:38
%S 1,5,1,5,4,2,6,2,2,4,1,4,7,9,2,6,4,1,8,9,7,6,0,4,3,0,2,7,2,6,2,9,9,1,
%T 1,9,0,5,5,2,8,5,4,8,5,3,6,8,5,6,1,3,9,7,6,9,1,4,0,7,4,6,4,0,5,9,1,4,
%U 8,3,0,9,7,3,7,3,0,9,3,4,4,3,2,6,0,8,4,5,6,9,6,8,3,5,7,8,7,3,4,6,0,5,1,1,5
%N Decimal expansion of e^e.
%C Given z > 0, there exist positive real numbers x < y, with x^y = y^x = z, if and only if z > e^e. In that case, 1 < x < e < y and (x, y) = ((1 + 1/t)^t, (1 + 1/t)^(t+1)) for some t > 0. (For example, t = 1 gives 2^4 = 4^2 = 16 > e^e.) Proofs of these classical results and applications of them are in Marques and Sondow (2010).
%C e^e = lim_{n->infinity} ((n+1)/n)^((n+1)^(n+1)/n^n), n > 0 an integer; cf. [Vernescu] wherein it is also stated that the assertions of the previous comment above were proved by Alexandru Lupas in 2006. - _L. Edson Jeffery_, Sep 18 2012
%C A weak form of Schanuel's Conjecture implies that e^e is transcendental--see Marques and Sondow (2012).
%H Harry J. Smith, <a href="/A073226/b073226.txt">Table of n, a(n) for n = 2..20000</a>
%H D. Marques and J. Sondow, <a href="http://arxiv.org/abs/1212.6931">The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture</a>, arXiv:1212.6931 [math.NT], 2012-2013.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap22.html">exp(E) to 2000 places</a>
%H J. Sondow and D. Marques, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from151to164.pdf">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164.
%H A. Vernescu, <a href="http://www.emis.de/journals/GM/vol15nr1/vernescu/vernescu.pdf">About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e</a>, Gen. Math., Vol. 15, No. 1 (2007), 75-80.
%F Equals Sum_{n>=0} e^n/n!. - _Richard R. Forberg_, Dec 29 2013
%F Equals Product_{n>=0} e^(1/n!). - _Amiram Eldar_, Jun 29 2020
%e 15.15426224147926418976043027262991190552854853685613976914...
%t RealDigits[ E^E, 10, 110] [[1]]
%o (PARI) exp(exp(1))
%o (PARI) { default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073226.txt", n, " ", d)); } \\ _Harry J. Smith_, Apr 30 2009
%o (Magma) Exp(Exp(1)); // _G. C. Greubel_, May 29 2018
%Y Cf. A073233 (Pi^Pi), A049006 (i^i), A001113 (e), A073227 (e^e^e), A004002 (Benford numbers), A056072 (floor(e^e^...^e), n e's), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073229 (e^(1/e)), A073230 ((1/e)^e).
%K cons,nonn
%O 2,2
%A _Rick L. Shepherd_, Jul 21 2002
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