A072364 Decimal expansion of (1/e)^(1/e).

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

Offset: 0

Author

Rick L. Shepherd, Jul 18 2002

Keywords

• cons
• nonn

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

Examples

			0.69220062755534635386...
		

References

• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.

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.
• Jonathan Sondow and Diego 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.

Crossrefs

Cf. A068985 (1/e), A001113 (e), A072365 ((1/3)^(1/3)), A073229 (e^(1/e)), A073230 ((1/e)^e).

Cf. also A258707.

Programs

Magma

(Exp(-1))^(Exp(-1)); // G. C. Greubel, May 29 2018

Maple

evalf(exp(-1/exp(1)), 120);  # Alois P. Heinz, Oct 26 2021

Mathematica

RealDigits[E^(-1/E), 10, 111][[1]]

PARI

(1/exp(1))^(1/exp(1))

PARI

exp(-1/exp(1)) \\ Charles R Greathouse IV, Sep 01 2011

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)