

A073226


Decimal expansion of e^e.


38



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, 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, 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
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OFFSET

2,2


COMMENTS

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).
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
A weak form of Schanuel's Conjecture implies that e^e is transcendentalsee Marques and Sondow (2012).


LINKS

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, Annales Mathematicae et Informaticae 37 (2010) 151164.
Simon Plouffe, exp(E) to 2000 places
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations
A. Vernescu, About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e, Gen. Math., Vol. 15, No. 1 (2007), 7580.


FORMULA

Equals Sum_{n>=0} e^n/n!.  Richard R. Forberg, Dec 29 2013
Equals Product_{n>=0} e^(1/n!).  Amiram Eldar, Jun 29 2020


EXAMPLE

15.15426224147926418976043027262991190552854853685613976914...


MATHEMATICA

RealDigits[ E^E, 10, 110] [[1]]


PROG

(PARI) exp(exp(1))
(PARI) { default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(xd)*10; write("b073226.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009
(MAGMA) Exp(Exp(1)); // G. C. Greubel, May 29 2018


CROSSREFS

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).
Sequence in context: A122002 A322602 A228639 * A309282 A021198 A275976
Adjacent sequences: A073223 A073224 A073225 * A073227 A073228 A073229


KEYWORD

cons,nonn


AUTHOR

Rick L. Shepherd, Jul 21 2002


STATUS

approved



