On the Tower Factorization of Integers

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa; factoring each exponent a as a product of prime powers qb, and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of "floors" in its tower factorization. In particular, given a fixed integer k≥1, we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.