OFFSET
0,1
COMMENTS
Writing the equation as (1/2)^x = x, the solution is the value of the infinite power tower function h(t) = t^t^t^... at t = 1/2. The solution is a transcendental number. - Jonathan Sondow, Aug 29 2011
Equals LambertW(log(2))/log(2) since, for 1/E^E <= c < 1, c^c^c^...= LambertW(log(1/c))/log(1/c). - Stanislav Sykora, Nov 03 2013
LINKS
Stanislav Sykora, Table of n, a(n) for n = 0..2000
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 160.
Wikipedia, Lambert W function
EXAMPLE
x = 0.641185744504985984486200482114823666562820957191101... = (1/2)^(1/2)^(1/2)^...
MATHEMATICA
RealDigits[ ProductLog[ Log[2]]/Log[2], 10, 111][[1]] (* Robert G. Wilson v, Mar 23 2005 *)
RealDigits[x/.FindRoot[x 2^x==1, {x, .6}, WorkingPrecision->100]][[1]] (* Harvey P. Dale, Apr 17 2019 *)
PROG
(PARI) lambertw(log(2))/log(2) \\ Stanislav Sykora, Nov 03 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Zak Seidov, Mar 23 2005
EXTENSIONS
More terms from Robert G. Wilson v, Mar 23 2005
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved