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Decimal expansion of the y-coordinate for the largest solution to e^x = Gamma(x+1).
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%I #16 May 31 2021 03:25:15

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

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

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

%N Decimal expansion of the y-coordinate for the largest solution to e^x = Gamma(x+1).

%C This number is the y-coordinate of the point at which the factorial function, Gamma(x+1), begins to exceed the exponential function.

%H Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers-2.html">Notable Properties of Specific Numbers</a>.

%F Equals exp(A078335).

%e x = 5.29031609311977071072...

%e y = 198.40406130311276776911...

%t RealDigits[x /. FindRoot[Gamma[Log[x] + 1] == x, {x, 200}, WorkingPrecision -> 120], 10, 115][[1]] (* _Amiram Eldar_, May 31 2021 *)

%o (PARI) \p200

%o exp(solve (x=5,6,exp(x)-gamma(x+1))) \\ _Hugo Pfoertner_, Dec 12 2019

%Y Cf. A078335.

%K nonn,cons

%O 3,2

%A _Eliora Ben-Gurion_, Dec 12 2019