A218802 - OEIS

A218802

Decimal expansion of largest x such that x^2 = Gamma(x+1).

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

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OFFSET

1,1

COMMENTS

In other words, the largest number whose square and factorial coincide.

As one knows from the famous illustration in Jahnke and Emde (p. 13) (or Abramowitz and Stegun, p. 255), there are infinitely many solutions to x^2 = x!. For example, there is another solution near -1.8065. - N. J. A. Sloane, Dec 24 2012

Decimal expansion of greatest real fixed point of Gamma(x). (The only other positive fixed point is 1.) - Rick L. Shepherd, Feb 24 2014

The interval (c, x) = (0.2541970697269031..., 3.562382285390897691415...) with Gamma(c) = x = Gamma(x) is the interval of convergence to 1 of the iteration x->Gamma(x). - Andrea Pinos, Jul 06 2023

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 255.

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 13.

LINKS

Table of n, a(n) for n=1..107.

EXAMPLE

3.562382285390897691415...

MAPLE

Digits:= 150:

s:= convert(fsolve(x^2 = GAMMA(x+1), x=7/2)/10, string):

seq(parse(s[n+1]), n=1..120); # Alois P. Heinz, Dec 26 2012

MATHEMATICA

RealDigits[x /. FindRoot[x^2 == Gamma[x + 1], {x, 3}, WorkingPrecision -> 100]][[1]] (* Bruno Berselli, Dec 24 2012 *)

PROG