Exponential factorial

In mathematics, the exponential factorial of a positive integer ${\displaystyle n}$ is a quickly growing function of ${\displaystyle n}$ 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 ${\displaystyle n}$ raised to the power of ${\displaystyle n-1}$, which in turn is raised to the power of ${\displaystyle n-2}$, and so on in a right-grouping manner. That is,

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

The exponential factorial can also be defined with the recurrence relation

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

Using the recurrence relation, the first exponential factorials are:

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 ${\displaystyle n}$th exponential factorial that enclosed the number ${\displaystyle n}$ 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 ${\displaystyle b>e^{1/e}}$, the exponential factorial is bounded above and below by expressions of the form ${\displaystyle b\uparrow \uparrow (n+c)}$ with constant ${\displaystyle c}$). 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: $${\displaystyle {\begin{aligned}&{\frac {1}{1}}+{\frac {1}{2^{1}}}+{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}+{\frac {1}{6^{5^{4^{3^{2^{1}}}}}}}+\ldots \\&=1.611114925808376736\underbrace {111111111111\ldots 111111111111} _{183212}272243682859\ldots \end{aligned}}}$$ 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.

References