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 $ϕ$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient numbers generalize to results concerning Schemmel nontotient numbers. We also invoke Zsigmondy's Theorem in order to generalize a result due to Mendelsohn.
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: W107528871