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

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 $$\#\{n\le x:f(n)=k\}\sim A_{k}x$$ and $$\#\{n\le x:f(n)=\omega (n)-k\}\sim \frac{Bx(\log \log x{)}^k}{k!\log x},$$ 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 Aktaş and Ram Murty about numbers having mutually distinct exponents in their prime factorization.