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

0,1

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

Index entries for transcendental numbers

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

Equals 1/A030798.

Cf. A073084.

Sequence in context: A060780 A199391 A106333 * A117335 A319555 A244980

Adjacent sequences: A104745 A104746 A104747 * A104749 A104750 A104751

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