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

%I A104748 #34 Jul 21 2025 08:33:21
%S A104748 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,
%T A104748 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,
%U A104748 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
%N A104748 Decimal expansion of solution to x*2^x = 1.
%C A104748 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
%C A104748 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
%H A104748 Stanislav Sykora, <a href="/A104748/b104748.txt">Table of n, a(n) for n = 0..2000</a>
%H A104748 J. Sondow and D. Marques, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from151to164.pdf">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 160.
%H A104748 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%H A104748 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A104748 x = 0.641185744504985984486200482114823666562820957191101... = (1/2)^(1/2)^(1/2)^...
%t A104748 RealDigits[ ProductLog[ Log[2]]/Log[2], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 23 2005 *)
%t A104748 RealDigits[x/.FindRoot[x 2^x==1,{x,.6},WorkingPrecision->100]][[1]] (* _Harvey P. Dale_, Apr 17 2019 *)
%o A104748 (PARI) lambertw(log(2))/log(2) \\ _Stanislav Sykora_, Nov 03 2013
%Y A104748 Equals 1/A030798.
%Y A104748 Cf. A073084.
%K A104748 nonn,cons
%O A104748 0,1
%A A104748 _Zak Seidov_, Mar 23 2005
%E A104748 More terms from _Robert G. Wilson v_, Mar 23 2005
%E A104748 Offset corrected by _R. J. Mathar_, Feb 05 2009