Exponential factorial

In mathematics, the exponential factorial of a positive integer $n$ is a quickly growing function of $n$ defined by iterating the process of exponentiation, analogously to the way that the factorial can be defined by iterated multiplication.

Definition

The exponential factorial is a positive integer $n$ raised to the power of $n-1$, which in turn is raised to the power of $n-2$, and so on in a right-grouping manner. That is,

$$n^{\left((n-1)^{\left(n-2)^{\cdot ^{\cdot ^{\cdot }}}\right)}\right)}$$

The exponential factorial can also be defined with the recurrence relation

$$a_{1}=1,\quad a_{n}=n^{a_{n-1}}.$$

Using the recurrence relation, the first exponential factorials are:

1

2¹ = 2

3² = 9

4⁹ = 262144

5²⁶²¹⁴⁴ = 6206069878...8212890625 (183,231 digits)

Growth rate

The exponential factorials were introduced by Dee David Smith, in his 1974 book Orthogonic Taxonomy of Fundamental Concepts. Smith used a notation for the $n$th exponential factorial that enclosed the number $n$ in a three-sided box (leaving the left side open). He asked "whether or not there are other methods that give larger numbers". The answer is yes: these numbers grow quickly, but not more quickly than other known methods.

The exponential factorials exhibit growth equivalent to tetration (in the sense that, for any base $b>e^{1/e}$, the exponential factorial is bounded above and below by expressions of the form $b\uparrow \uparrow (n+c)$ with constant $c$). The exponential factorial of 6 (a number with approximately 5 × 10^183 230 decimal digits) is larger than a googolplex.

Reciprocal sum

The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number: $${\begin{aligned}&{\frac {1}{1}}+{\frac {1}{2^{1}}}+{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}+{\frac {1}{6^{5^{4^{3^{2^{1}}}}}}}+\ldots \\&=1.611114925808376736\underbrace {111111111111\ldots 111111111111} _{183212}272243682859\ldots \end{aligned}}$$ This sum is transcendental because it is a Liouville number: its partial sums provide a sequence of rational numbers that approximate it more accurately, relative to their denominators, than would be possible for an algebraic number.