A264785 - OEIS

%I #21 May 28 2021 06:14:22

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

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

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

%N Decimal expansion of largest x such that x^2 = Gamma(x).

%C Also, largest x such that Gamma(x+1) = x^3. In other words, the largest number whose cube and factorial coincide.

%C The equation Gamma(x) = x^2 has also negative solutions, one for each negative integer, increasingly closer to these integers: x[1] = -1.5259..., x[2] = -1.806544..., x[3] = -3.017901..., x[4] = -3.997382..., x[5] = -5.000333... etc. The distances from the integers show an interesting pattern, see A339167. - _M. F. Hasler_, Nov 25 2020

%e 5.03672257058871109516917896...

%t RealDigits[x /. FindRoot[Gamma[x] - x^2, {x, 5}, WorkingPrecision -> 120], 10, 100][[1]] (* _Amiram Eldar_, May 28 2021 *)

%o (PARI) solve(x=5,6,gamma(x+1)-x^3)

%o (PARI) solve(x=5,6,gamma(x)-x^2)

%Y Cf. A218802, A339161, A339167.

%K nonn,cons

%O 1,1

%A _Franklin T. Adams-Watters_, Nov 24 2015