On the tower factorization of integers

TL;DR

This paper introduces the concept of tower factorization of integers, analyzes the distribution of their heights, and provides formulas for the density of integers with a given height, revealing properties of integer factorizations.

Contribution

It defines the tower factorization and height, derives formulas for the density of integers with a specific height, and proves the existence of arbitrarily long sequences of integers with large heights.

Findings

Formulas for the density of integers with a given tower height

Average number of floors in tower factorizations

Existence of long sequences of integers with arbitrarily large heights

Abstract

Under the fundamental theorem of arithmetic, any integer $n>1$ can be uniquely written as a product of prime powers $p^a$; factoring each exponent $a$ as a product of prime powers $q^b$, 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 $k\geq 1$, 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.