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On Quotients of Values of Euler's Function on Factorials
- Source :
- Bulletin of the Australian Mathematical Society (2021)
- Publication Year :
- 2021
-
Abstract
- Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function $\phi$ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers $a$ and $b$, the least positive integer $c=c(a,b)$ such that the quotient $\phi(c!)/\phi(a!)\phi(b!)$ is an integer. We derive results on the limit of the ratio $c(a,b)/(a+b)$ as $a$ and $b$ tend to infinity. Furthermore, we show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ with an exception of a set of density zero.<br />Comment: 13 pages, 2 figures; to appear in Bulletin of Australian Mathematical Society
- Subjects :
- Mathematics - Number Theory
Primary: 11A25, Secondary: 11B65, 11N37
Subjects
Details
- Database :
- arXiv
- Journal :
- Bulletin of the Australian Mathematical Society (2021)
- Publication Type :
- Report
- Accession number :
- edsarx.2110.09875
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1017/S0004972721000939