Year: 2014
Type: preprint
Abstract: For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $ϕ(n)n$. We define a sparsely Schemmel totient number of order $r$ to be a positive integer $n$ such that $S_r(n)>0$ and $S_r(n)n$ with $S_r(m)>0$. We then generalize some of the results of Masser and Shiu.
Source: arXiv (Cornell University)
Author: Defant, Colin
Institution: University of Florida
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W107805703