On the numbers $n$ relatively prime to $\Omega (n)-\omega (n)$
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- by Yuchen Ding
- Proc. Amer. Math. Soc. 150 (2022), 3811-3819
- DOI: https://doi.org/10.1090/proc/15953
- Published electronically: April 29, 2022
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Abstract:
Let $\Omega (n)$ and $\omega (n)$ be the number of all prime factors and different prime factors of $n$, respectively. An elegant result of Alladi showed that the probability of $(n,\Omega (n))=1$ or $(n,\omega (n))=1$ is $6/\pi ^2$. In the other aspect, denoting by $d_k$ the density of the numbers $n$ with $\Omega (n)-\omega (n)=k$, then another elegant formula of Rényi states that $d_k\sim c_0/2^k$ as $k$ tends to infinity, where $c_0=(1/4)\prod _{p>2}(1-(p-1)^{-2})^{-1}$. In this paper, we investigate the numbers $n$ which are relatively prime to $\Omega (n)-\omega (n)$. Let $\mathscr {D}$ denote the set of such numbers. It is proved that there exists a positive constant $c_1$ such that $\mathscr {D}(x)=c_1x+O_{\varepsilon }(\sqrt {x}\log ^{1+\varepsilon }x)$ for any $\varepsilon >0$.References
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Bibliographic Information
- Yuchen Ding
- Affiliation: School of Mathematical Science, Yangzhou University, Yangzhou 225002, People’s Republic of China
- MR Author ID: 1199999
- ORCID: 0000-0001-7016-309X
- Email: ycding@yzu.edu.cn
- Received by editor(s): June 21, 2021
- Received by editor(s) in revised form: October 28, 2021, and December 11, 2021
- Published electronically: April 29, 2022
- Additional Notes: The author was supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20210784). He was also supported by the foundations of the projects “Jiangsu Provincial Double–Innovation Doctor Program” (No. JSSCBS20211023) and “Golden Pheonix of the Green City–Yang Zhou” to excellent PhD (No. YZLYJF2020PHD051)
- Communicated by: Amanda Folsom
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3811-3819
- MSC (2020): Primary 11N37, 11A25, 11A05
- DOI: https://doi.org/10.1090/proc/15953
- MathSciNet review: 4446231