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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.04574 |
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Table of Contents:
- Given an integer $k\ge2$, let $ω_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $ω_k(n)=m$ for every integer $m\ge0$. We also show that the generating function $\sum_{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod_{p} \bigl(1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum_{j=2}^\infty a_j ω_j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $Ω(n)-ω(n)$.