A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

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

Offset

0,1

Comments

Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.

Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - David Ulgenes, Feb 19 2024

Links

Table of n, a(n) for n=0..105.

K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.

Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Formula

Equals ((((1/2)!/2)!/2)!/2)!/2...

Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.

Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - David Ulgenes, Feb 11 2024

Example

0.4428773964847274374520325165206056717103645380663664...

Maple

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

Mathematica

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

Prog

(PARI) solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Crossrefs

Cf. A030169, A030171.

Sequence in context: A151966 A010778 A202322 * A232523 A224821 A034933

Adjacent sequences: A365794 A365795 A365796 * A365798 A365799 A365800

Keyword

nonn,cons

Author

David Ulgenes, Sep 19 2023

Status

approved