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 $$ \#\{n \leq x : f(n) = k\} \sim A_k x $$ 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.