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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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Motivated by a question about solutions to (x-1)! = 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; (5-1)! ~ 5^2 is a near miss.

If (x-1)! is replaced by the Gamma function Gamma(x), then in addition to the positive non-integer solution x = 5.0367... (see A264785) there are non-integer 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

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Last modified November 30 13:12 EST 2021. Contains 349419 sequences. (Running on oeis4.)