A264785 Decimal expansion of largest x such that x^2 = Gamma(x).

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, 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, 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

Offset

1,1

Comments

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

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

Links

Table of n, a(n) for n=1..120.

Example

5.03672257058871109516917896...

Mathematica

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

Prog

(PARI) solve(x=5, 6, gamma(x+1)-x^3)
(PARI) solve(x=5, 6, gamma(x)-x^2)

Crossrefs

Cf. A218802, A339161, A339167.

Sequence in context: A161485 A326054 A062526 * A197573 A019947 A193182

Adjacent sequences: A264782 A264783 A264784 * A264786 A264787 A264788

Keyword

nonn,cons

Author

Franklin T. Adams-Watters, Nov 24 2015

Status

approved