A231098 Decimal expansion of the power tower of the ratio Pi/e.

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

Offset: 1

Author

Stanislav Sykora, Nov 06 2013

Keywords

• nonn
• cons

Comments

This limit for n->infinity of the tetration (Pi/e)^^n exists because Pi/e < e^(1/e). Its value is also the smaller one of the two solutions of x*(e/Pi)^x = 1.

Examples

			1.18752363535924990543840790...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Stanislav Sykora, PARI/GP script for Tetration Limit.
• Wikipedia, Lambert W function.
• Wikipedia, Tetration.

Crossrefs

Cf. A000796, A001113, A231097.

Programs

Mathematica

RealDigits[ProductLog[1 - Log[Pi]]/(1 - Log[Pi]), 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

PARI

See the above link

Formula

Equals LambertW(1-log(Pi))/(1-log(Pi)). - Alois P. Heinz, Jul 12 2018