A338460 Decimal expansion of the largest real root of e^(x-1) = Gamma(x+1).

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

Offset: 1

Author

Michael P. May, Jan 31 2021

Keywords

• nonn
• cons

Comments

Decimal expansion of the constant value for which the average and the minimum prime gaps are equal for the prime number sequences of higher order P', P'', P''', and P'''' as represented by A333242, A262275, A333243 and A333244.

x-1 is the smallest average gap size for the set of all prime numbers P. - Michael P. May, Jan 26 2025

Examples

			3.61479370319252544738653662560345463353151659694750...
		

Crossrefs

Cf. A333242, A262275, A333243, A333244.

Cf. A001113, A078335, A336763.

Programs

Maple

Digits:= 155:
fsolve(exp(x-1)=GAMMA(x+1), x=3..4);  # Alois P. Heinz, Feb 01 2021

Mathematica

RealDigits[x /. FindRoot[LogGamma[x + 1] - x + 1, {x, 3}, WorkingPrecision -> 110], 10, 105][[1]] (* Amiram Eldar, Feb 01 2021 *)

PARI

solve(x=3,4,lngamma(x+1)-x+1) \\ Hugo Pfoertner, Feb 01 2021

Formula

x| (log(x!))^n * (log(x!) + 1) = x * (x-1)^n, for n >= 0