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.Comment: 8 pages. Accepted for publication in the Amer. Math. Monthl