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

0,1

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

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.

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

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

Cf. also A258707.

Sequence in context: A195403 A021595 A197696 * A087016 A161480 A309819

Adjacent sequences: A072361 A072362 A072363 * A072365 A072366 A072367

Keyword

cons,nonn

Author

Rick L. Shepherd, Jul 18 2002

Status

approved