cp's OEIS Frontend

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

%I A377302 #40 Aug 27 2025 00:43:33
%S A377302 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,
%T A377302 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,
%U A377302 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
%N A377302 Decimal expansion of the smallest positive real solution to Gamma(1+z) = Gamma(1-z).
%C A377302 This is the second smallest solution, the smallest one being purely imaginary A377297.
%C A377302 When expressed in terms of Gauss's Pi function, it is the smallest real solution to Pi(z) = Pi(-z).
%F A377302 Gamma(1+2.3611910871634166...) = Gamma(1-2.3611910871634166...) = 2.8607322727573070...
%F A377302 Smallest positive real root of the equation x*sin(Pi*x)*Gamma(x)^2 = Pi. - _Vaclav Kotesovec_, Oct 25 2024
%F A377302 Smallest positive real root of the equation Sum_{n>=1} x/n - arctanh(x/n) = x*euler_gamma - i*Pi. - _Jwalin Bhatt_, Aug 23 2025
%e A377302 2.36119108716341663449734103963240374354852871572581359610190443169213741...
%p A377302 Digits:= 120:
%p A377302 fsolve(GAMMA(1+z)=GAMMA(1-z), z=1..3);  # _Alois P. Heinz_, Oct 25 2024
%t A377302 RealDigits[x /. FindRoot[Gamma[1 + x] == Gamma[1 - x], {x, 5/2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 23 2024 *)
%t A377302 RealDigits[x /. FindRoot[x*Sin[Pi*x]*Gamma[x]^2 == Pi, {x, 2}, WorkingPrecision -> 120]][[1]](* _Vaclav Kotesovec_, Oct 25 2024 *)
%o A377302 (Python)
%o A377302 from mpmath import mp, nstr, factorial, findroot
%o A377302 mp.dps = 120
%o A377302 root = findroot(lambda z: factorial(z)-factorial(-z), 2.4)
%o A377302 A377302 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']
%Y A377302 Cf. A000796, A001620, A377297.
%K A377302 nonn,cons,changed
%O A377302 1,1
%A A377302 _Jwalin Bhatt_, Oct 23 2024