A377302 Decimal expansion of the smallest positive real solution to Gamma(1+z) = Gamma(1-z).

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

Offset

1,1

Comments

This is the second smallest solution, the smallest one being purely imaginary A377297.

When expressed in terms of Gauss's Pi function, it is the smallest real solution to Pi(z) = Pi(-z).

Links

Table of n, a(n) for n=1..105.

Formula

Gamma(1+2.3611910871634166...) = Gamma(1-2.3611910871634166...) = 2.8607322727573070...

Smallest positive real root of the equation x*Sin(Pi*x)*Gamma(x)^2 = Pi. - Vaclav Kotesovec, Oct 25 2024

Example

2.36119108716341663449734103963240374354852871572581359610190443169213741...

Maple

Digits:= 120:
fsolve(GAMMA(1+z)=GAMMA(1-z), z=1..3);  # Alois P. Heinz, Oct 25 2024

Mathematica

RealDigits[x /. FindRoot[Gamma[1 + x] == Gamma[1 - x], {x, 5/2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 23 2024 *)
RealDigits[x /. FindRoot[x*Sin[Pi*x]*Gamma[x]^2 == Pi, {x, 2}, WorkingPrecision -> 120]][[1]](* Vaclav Kotesovec, Oct 25 2024 *)

Prog

(Python)
from mpmath import mp, nstr, factorial, findroot
mp.dps = 120
root = findroot(lambda z: factorial(z)-factorial(-z), 2.4)
A377302 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']

Crossrefs

Cf. A377297.

Sequence in context: A241293 A107409 A268603 * A226871 A178483 A133031

Adjacent sequences: A377299 A377300 A377301 * A377303 A377304 A377305

Keyword

nonn,cons

Author

Jwalin Bhatt, Oct 23 2024

Status

approved