cp's OEIS Frontend

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

%I A072364 #54 Dec 30 2024 11:24:20
%S A072364 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,
%T A072364 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,
%U A072364 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
%N A072364 Decimal expansion of (1/e)^(1/e).
%C A072364 Minimum value of x^x for real x>0.
%C A072364 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
%C A072364 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
%D A072364 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.
%H A072364 G. C. Greubel, <a href="/A072364/b072364.txt">Table of n, a(n) for n = 0..10000</a>
%H A072364 Alex Chin et al., <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.424">Pick a Tree-Any Tree</a>, The American Mathematical Monthly, 122.5 (2015): 424-432.
%H A072364 Plouffe's Inverter entry for <a href="https://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=.69220062755&amp;lookup_type=simple">.69220062755</a>.
%H A072364 Jonathan Sondow and Diego Marques, <a href="http://arxiv.org/abs/1108.6096">Algebraic and transcendental solutions of some exponential equations</a>, arXiv:1108.6096 [math.NT], 2011; Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 3 in the link.
%F A072364 From _Amiram Eldar_, Aug 19 2020: (Start)
%F A072364 Equals Sum_{k>=0} (-1)^k/(exp(k)*k!).
%F A072364 Equals Product_{k>=0} exp((-1)^(k+1)/k!). (End)
%e A072364 0.69220062755534635386...
%p A072364 evalf(exp(-1/exp(1)), 120);  # _Alois P. Heinz_, Oct 26 2021
%t A072364 RealDigits[E^(-1/E), 10, 111][[1]]
%o A072364 (PARI) (1/exp(1))^(1/exp(1))
%o A072364 (PARI) exp(-1/exp(1)) \\ _Charles R Greathouse IV_, Sep 01 2011
%o A072364 (Magma) (Exp(-1))^(Exp(-1)); // _G. C. Greubel_, May 29 2018
%Y A072364 Cf. A068985 (1/e), A001113 (e), A072365 ((1/3)^(1/3)), A073229 (e^(1/e)), A073230 ((1/e)^e).
%Y A072364 Cf. also A258707.
%K A072364 cons,nonn
%O A072364 0,1
%A A072364 _Rick L. Shepherd_, Jul 18 2002