CARLO SANNA
Articles written in Proceedings – Mathematical Sciences
Volume 130 All articles Published: 12 March 2020 Article ID 0027 Research Article
On the number of distinct exponents in the prime factorization of an integer
Let $f (n)$ be the number of distinct exponents in the prime factorization ofthe 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
$$\{n \leq x : f (n) = \omega(n) − k\} \sim \frac{Bx(log log x)^{k}}{k! log x},$$
as $x \rightarrow +\infty$, 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 Aktas and Ram Murty (
Volume 130, 2020
All articles
Continuous Article Publishing mode
Click here for Editorial Note on CAP Mode
© 2021-2022 Indian Academy of Sciences, Bengaluru.