

A339161


Decimal expansion of least positive x such that Gamma(x) = x^2.


2



1, 5, 2, 5, 7, 9, 6, 4, 5, 7, 2, 0, 0, 6, 7, 9, 7, 7, 3, 6, 0, 1, 6, 3, 3, 2, 7, 9, 9, 6, 9, 7, 9, 0, 2, 2, 4, 3, 3, 0, 6, 8, 7, 5, 3, 4, 4, 9, 2, 0, 3, 6, 7, 2, 5, 6, 8, 7, 5, 8, 1, 0, 2, 3, 4, 9, 2, 2, 8, 4, 0, 5, 0, 9, 6, 6, 2, 9, 5, 8, 2, 3, 3, 2, 7, 2, 6, 6, 9, 7, 3, 6, 4, 7, 7, 0, 7, 4, 8, 5, 5, 3, 0, 8, 9, 7, 3, 5, 1, 2, 9, 5
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OFFSET

1,2


COMMENTS

Motivated by a question about solutions to (x1)! = x^2 in a mathematical discussion group. As can be seen from growth of the respective functions, the only solution in the positive integers is x = 1; (51)! ~ 5^2 is a near miss.
If (x1)! is replaced by the Gamma function Gamma(x), then in addition to the positive noninteger solution x = 5.0367... (see A264785) there are noninteger solutions increasingly close to each negative integer: x = 1.5259... is the largest one (also the farthest from an integer), then come 1.806544..., 3.017901..., 3.997382..., 5.000333... etc. They are to the left of odd and to the right of even negative integers, and the reciprocals of the distances 1/(x[n]+n), rounded to nearest integers, go: 2, 5, 56, 382, 3002, ... (see A339167).


LINKS

Table of n, a(n) for n=1..112.


EXAMPLE

The largest negative solution to Gamma(x) = x^2 is x =
1.525796457200679773601633279969790224330687534492036725687581023492284050966...


MATHEMATICA

RealDigits[x /. FindRoot[Gamma[x]  x^2, {x, 3/2}, WorkingPrecision > 120], 10, 100][[1]] (* Amiram Eldar, May 28 2021 *)


PROG

(PARI) localprec(5+N=100); digits(solve(x=1.7, 1.1, gamma(x)x^2)\.1^N)


CROSSREFS

Cf. A264785 (solution near 5.036...), A339167.
Sequence in context: A082571 A087300 A074455 * A142702 A236184 A201530
Adjacent sequences: A339158 A339159 A339160 * A339162 A339163 A339164


KEYWORD

nonn,cons


AUTHOR

M. F. Hasler, Nov 25 2020


STATUS

approved



