

A073229


Decimal expansion of e^(1/e).


30



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, 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, 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
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OFFSET

1,2


COMMENTS

e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
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
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
For large n, A234604(n)/A234604(n1) converges to e^(1/e).  Richard R. Forberg, Dec 28 2013
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
Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e).  Yuval Paz, Dec 29 2018
The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (17961863) in 1850.  Amiram Eldar, Jun 17 2021


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ovidiu Furdui, Problem 11982, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; A Limit of a Power of a Sum, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
Simon Plouffe, exp(1/e).
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151164; see Definition 4.1 on p. 158.
Jacob Steiner, Über das größte Product der Theile oder Summanden jeder Zahl, Crelle, Vol. 40 (1850), pp. 208; alternative link.
Eric Weisstein's World of Mathematics, Steiner's Problem.


FORMULA

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
Equals Sum_{k>=0} exp(k)/k!.  Amiram Eldar, Aug 13 2020
Equals lim_{x>oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017).  Amiram Eldar, Mar 26 2022


EXAMPLE

1.44466786100976613365833910859...


MAPLE

evalf[110](exp(exp(1))); # Muniru A Asiru, Dec 29 2018


MATHEMATICA

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


PROG

(PARI) exp(1)^exp(1)


CROSSREFS

Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
Cf. A093157, A103476.
Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.
Sequence in context: A114742 A273909 A098013 * A116446 A102126 A277433
Adjacent sequences: A073226 A073227 A073228 * A073230 A073231 A073232


KEYWORD

cons,nonn


AUTHOR

Rick L. Shepherd, Jul 22 2002


STATUS

approved



