On the number of distinct exponents in the prime factorization of an integer
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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.
CARLO SANNA^{1} ^{}
Volume 130, 2020
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