A073229 - OEIS

A073229

Decimal expansion of e^(1/e).

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(n-1) 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 (1796-1863) in 1850. - Amiram Eldar, Jun 17 2021

REFERENCES

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 35.

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. 151-164; 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]]