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Decimal expansion of the solution to x^x = 2.
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%I #56 Feb 13 2024 12:44:36

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

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

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

%N Decimal expansion of the solution to x^x = 2.

%C The constant 1.559610469462... is transcendental. - Nick Hobson, Nov 29 2006

%H G. C. Greubel, <a href="/A030798/b030798.txt">Table of n, a(n) for n = 1..10000</a>

%H Nick Hobson, <a href="https://web.archive.org/web/20160414002427/http://www.qbyte.org/puzzles/p029s.html#remark">Solution to puzzle 29: x^x. Remark: x^x = 2</a>.

%H Gianni Sarcone, <a href="http://www.archimedes-lab.org/numbers/Num1_69.html">Zoo of Numbers: Numbers NaN to 6</a>, Archimedes Lab, Genoa, Italy.

%H 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 top of p. 4 in the link.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals log(2)/LambertW(log(2)). - _Simon Plouffe_, Mar 23 2005

%F Equals 1/A104748.

%e 1.559610469462369349970388768765002993284883511843091424719594569...

%t RealDigits[ Log[2]/ProductLog[Log[2]], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 23 2005 *)

%t RealDigits[x/.FindRoot[x^x==2,{x,1},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, May 27 2020 *)

%o (PARI) solve(x=1, 2, x^x-2) \\ _Michel Marcus_, Jan 14 2015

%o (PARI) log(2)/lambertw(log(2)) \\ _Charles R Greathouse IV_, May 14 2019

%Y Cf. A153510 (continued fraction), A199550 (x^x^x = 2).

%K nonn,cons

%O 1,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E Definition clarified by _Jonathan Sondow_, Sep 02 2011