A381340 Decimal value of c > 1.5 for which H(2*c) = 2*H(c) for H = Hadamard's gamma function.

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

Offset

1,2

Comments

H(x + y) >= H(x) + H(y) (i.e., H is superadditive) for all real numbers x, y >= c.

Note: H(x) = Gamma(x) * (1 + sin(Pi*x)/(2*Pi) * (Psi(x/2) - Psi((x+1)/2))) for gamma function Gamma(x) and digamma function Psi(x).

Links

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

Horst Alzer, A superadditive property of Hadamard’s gamma function, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1 (2009), 11-23.

Wikipedia, Hadamard's gamma function.

Example

1.50317609234327640372867713768804501078769660416266...

Maple

Digits:= 150:
H:= x-> GAMMA(x)*(1+sin(Pi*x)/(2*Pi)*(Psi(x/2)-Psi((x+1)/2))):
fsolve(H(2*c)=2*H(c), c=1.5..2.0);  # Alois P. Heinz, Feb 20 2025

Mathematica

H[x_] := ResourceFunction["HadamardGamma"][x]; RealDigits[x /. FindRoot[H[2*x] == 2*H[x], {x, 3/2, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 25 2025 *)

Crossrefs

Cf. A000796.

Sequence in context: A229175 A372836 A326188 * A201524 A230438 A200399

Adjacent sequences: A381337 A381338 A381339 * A381341 A381342 A381343

Keyword

nonn,cons

Author

Lee A. Newberg, Feb 20 2025

Status

approved