A305187 Decimal expansion of the solution to x^x^x = 3.

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

Offset

1,2

Comments

Let x(m) be the solution to the equation x^x^x^...^x = m, where x appears m times on the left hand side; e.g.,

decimal

m equation solution x(m) expansion

==== ==================== ============= =============

1 x = 1 1.00000000... A000007

2 x^x = 2 1.55961046... A030798

3 x^x^x = 3 1.63507847... this sequence

4 x^x^x^x = 4 1.62036995...

5 x^x^x^x^x = 5 1.59340881...

6 x^x^x^x^x^x = 6 1.56864406...

7 x^x^x^x^x^x^x = 7 1.54828598...

.

10 x^x^x^x^...^x = 10 1.50849792...

.

100 x^x^x^x^...^x = 100 1.44567285...

.

1000 x^x^x^x^...^x = 1000 1.44467831...

.

Then x(1) < x(m) < x(3) for all m >= 4.

Let y(k/2) be the solution to the equation y^y^y^...^y = (k/2)*y^y, where y appears k times on the left hand side; e.g.,

decimal

k equation solution y(k/2) expansion

= ========================= =============== =========

1 y = (1/2)*y^y 2 A000038

2 y^y = (2/2)*y^y indeterminate

3 y^y^y = (3/2)*y^y 1.6998419085...

4 y^y^y^y = (4/2)*y^y 1.6396207046...

5 y^y^y^y^y = (5/2)*y^y 1.5987769216...

6 y^y^y^y^y^y = (6/2)*y^y 1.5694666408...

7 y^y^y^y^y^y^y = (7/2)*y^y 1.5476452822...

.

What is lim_{k -> infinity} y(k/2)?

Lim_{m -> infinity} x(m) = e^(1/e). - Jon E. Schoenfield, Jul 23 2018

Lim_{k -> infinity} y(k/2) = e^(1/e). - Jon E. Schoenfield, Aug 01 2018

Links

Table of n, a(n) for n=1..104.

Example

1.635078474636375245899757198787500888...

Mathematica

RealDigits[ FindRoot[ x^x^x == 3, {x, 1}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

Prog

(PARI) default(realprecision, 333);
solve(x=1.6, 1.7, x^x^x-3) \\ Joerg Arndt, May 27 2018

Crossrefs

Cf. A000007, A000038, A030798.

Sequence in context: A019165 A195490 A195471 * A065418 A228725 A286982

Adjacent sequences: A305184 A305185 A305186 * A305188 A305189 A305190

Keyword

nonn,cons

Author

Juri-Stepan Gerasimov, May 27 2018

Extensions

More digits from Michel Marcus, Joerg Arndt, May 27 2018

Status

approved