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A073229 Decimal expansion of e^(1/e). 28

%I

%S 1,4,4,4,6,6,7,8,6,1,0,0,9,7,6,6,1,3,3,6,5,8,3,3,9,1,0,8,5,9,6,4,3,0,

%T 2,2,3,0,5,8,5,9,5,4,5,3,2,4,2,2,5,3,1,6,5,8,2,0,5,2,2,6,6,4,3,0,3,8,

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

%N Decimal expansion of e^(1/e).

%C e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).

%C Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - _Benoit Cloitre_, Aug 06 2002; corrected by _Robert FERREOL_, Jun 12 2015

%C x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - _Amarnath Murthy_, Nov 26 2002

%C For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - _Richard R. Forberg_, Dec 28 2013

%C Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - _Stanislav Sykora_, May 25 2015

%C Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - _Yuval Paz_, Dec 29 2018

%C The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - _Amiram Eldar_, Jun 17 2021

%H Alois P. Heinz, <a href="/A073229/b073229.txt">Table of n, a(n) for n = 1..10000</a>

%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap19.html">exp(1/e)</a>.

%H Jonathan Sondow and Diego 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, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.

%H Jacob Steiner, <a href="https://doi.org/10.1515/crll.1850.40.208">Über das größte Product der Theile oder Summanden jeder Zahl</a>, Crelle, Vol. 40 (1850), pp. 208; <a href="https://eudml.org/doc/147456">alternative link</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SteinersProblem.html">Steiner's Problem</a>.

%F Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - _Peter Bala_, Oct 30 2019

%F Equals Sum_{k>=0} exp(-k)/k!. - _Amiram Eldar_, Aug 13 2020

%e 1.44466786100976613365833910859...

%p evalf[110](exp(exp(-1))); # _Muniru A Asiru_, Dec 29 2018

%t RealDigits[ E^(1/E), 10, 110] [[1]]

%o (PARI) exp(1)^exp(-1)

%Y Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).

%Y Cf. A093157, A103476.

%Y Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.

%K cons,nonn

%O 1,2

%A _Rick L. Shepherd_, Jul 22 2002

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Last modified July 31 04:46 EDT 2021. Contains 346367 sequences. (Running on oeis4.)