# On the number of distinct exponents in the prime factorization of an integer

## TL;DR

This paper studies the distribution of the number of distinct exponents in the prime factorization of integers, providing asymptotic formulas for their counts and extending previous results on numbers with distinct exponents.

## Contribution

It establishes new asymptotic formulas for the distribution of the number of distinct exponents in prime factorizations, generalizing prior work by Aktaş and Ram Murty.

## Key findings

- Asymptotic count of integers with a fixed number of distinct exponents
- Asymptotic count of integers with a specific relation to total prime factors
- Extension of previous results on numbers with distinct exponents

## Abstract

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n) = k\} \sim A_k x $$ and $$ \#\{n \leq x : f(n) = \omega(n) - k\} \sim \frac{B x (\log \log x)^k}{k! \log x} , $$ as $x \to +\infty$, where $\omega(n)$ is the number of prime factors of $n$ and $A_k, B > 0$ are some explicit constants. The latter asymptotic extends a result of Akta\c{s} and Ram Murty about numbers having mutually distinct exponents in their prime factorization.