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arXiv · 2110.09875

On Quotients of Values of Euler's Function on Factorials

Ayan Nath·Abhishek Jha
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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.

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BibTeXRIS

Ayan Nath, Abhishek Jha. 2021-10-19. On Quotients of Values of Euler's Function on Factorials. https://doi.org/10.1017/s0004972721000939

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