A104748 Decimal expansion of solution to x*2^x = 1.

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

Offset: 0

Author

Zak Seidov, Mar 23 2005

Keywords

• nonn
• cons

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

Examples

			x = 0.641185744504985984486200482114823666562820957191101... = (1/2)^(1/2)^(1/2)^...
		

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
• Index entries for transcendental numbers

Crossrefs

Equals 1/A030798.

Cf. A073084.

Programs

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 *)

PARI

lambertw(log(2))/log(2) \\ Stanislav Sykora, Nov 03 2013

Extensions

More terms from Robert G. Wilson v, Mar 23 2005

Offset corrected by R. J. Mathar, Feb 05 2009