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arXiv · 2308.09149

On the tower factorization of integers

Jean-Marie De Koninck·William Verreault
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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.

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Jean-Marie De Koninck, William Verreault. 2023-08-17. On the tower factorization of integers. https://arxiv.org/abs/2308.09149

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