Exponential factorial

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

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

The exponential factorial can also be defined with the recurrence relation

Using the recurrence relation, the first exponential factorials are:^[1]

The exponential factorials were introduced by Dee David Smith, in his 1974 book Orthogonic Taxonomy of Fundamental Concepts. Smith used a notation for the th exponential factorial that enclosed the number 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.^[2]

The exponential factorials exhibit growth equivalent to tetration (in the sense that, for any base , the exponential factorial is bounded above and below by expressions of the form with constant ). The exponential factorial of 6 (a number with approximately 5 × 10^183 230 decimal digits) is larger than a googolplex.

The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number: 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.^[3]