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

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      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

       

      CARLO SANNA1

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

       
  • Proceedings – Mathematical Sciences | News

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