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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/130/0027

• # Keywords

Prime factorization; squarefree numbers; powerful number

• # Abstract

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 (Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430) about numbers having mutually distinct exponents in their prime factorization.

• # Author Affiliations

1. Department of Mathematics, Università degli Studi di Genova, Genoa, Italy

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019