A338460 Decimal expansion of 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

1,1

Comments

Decimal expansion of the constant value x 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.

Links

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

Formula

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

Example

3.61479370319252544738653662560345463353151659694750...

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 *)

Prog

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

Crossrefs

Cf. A333242, A262275, A333243, A333244.

Cf. A001113, A078335, A336763.

Sequence in context: A200358 A181099 A163327 * A175032 A078768 A307281

Adjacent sequences: A338457 A338458 A338459 * A338461 A338462 A338463

Keyword

nonn,cons

Author

Michael P. May, Jan 31 2021

Status

approved