OFFSET
0,1
COMMENTS
Minimum value of x^x for real x>0.
Also minimum value of 1/x^(1/x) for real x>0 (occurs at e). Equals exp(Pi)/exp(1/exp(1)) * exp(-Pi). - Gerald McGarvey, Sep 21 2004
If (1/e)^(1/e) < y < 1, then x^x = y has two solutions x = a and x = b with 0 < a < 1/e < b < 1. For example, (1/e)^(1/e) < 1/sqrt(2) < 1 and (1/4)^(1/4) = (1/2)^(1/2) = 1/sqrt(2) with 1/4 < 1/e < 1/2. - Jonathan Sondow, Sep 02 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Alex Chin et al., Pick a Tree-Any Tree, The American Mathematical Monthly, 122.5 (2015): 424-432.
Plouffe's Inverter entry for .69220062755
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, arXiv:1108.6096 [math.NT], 2011; Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 3 in the link.
FORMULA
From Amiram Eldar, Aug 19 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(exp(k)*k!).
Equals Product_{k>=0} exp((-1)^(k+1)/k!). (End)
EXAMPLE
0.69220062755534635386...
MAPLE
evalf(exp(-1/exp(1)), 120); # Alois P. Heinz, Oct 26 2021
MATHEMATICA
RealDigits[E^(-1/E), 10, 111][[1]]
PROG
(PARI) (1/exp(1))^(1/exp(1))
(PARI) exp(-1/exp(1)) \\ Charles R Greathouse IV, Sep 01 2011
(Magma) (Exp(-1))^(Exp(-1)); // G. C. Greubel, May 29 2018
CROSSREFS
KEYWORD
AUTHOR
Rick L. Shepherd, Jul 18 2002
STATUS
approved