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A365797
Decimal expansion of smallest positive number x such that Gamma(x) = 2.
0
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
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
Sequence in context: A151966 A010778 A202322 * A232523 A376878 A224821
KEYWORD
nonn,cons
AUTHOR
David Ulgenes, Sep 19 2023
STATUS
approved