exponential factorial

Given a positive integer $n$, the ”power tower” $n^{{(n-1)}^{(n-2)\mathrm{\dots }}}$ is the exponential factorial of $n$. The recurrence relation is $a_1=1$, $a_n=n^{a_{n-1}}$ for $n>1$.

So for example, $9=3^{2^1}$, $262144=4^{3^{2^1}}$. The exponential factorial for 5 has almost two hundred thousand base 10 digits. The ones that are small enough are listed in sequence A049384 of Sloane’s OEIS.

The sum of the reciprocals of the exponential factorials is a Liouville number.

$$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{a_i}\approx 1.6111149258083767361111111$$

Title	exponential factorial
Canonical name	ExponentialFactorial
Date of creation	2013-03-22 16:01:38
Last modified on	2013-03-22 16:01:38
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	7
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 05A10
Related topic	Factorial