On Schemmel Nontotient Numbers

TL;DR

This paper explores Schemmel nontotient numbers, generalizing known results about nontotients by analyzing the properties of the Schemmel totient functions and applying Zsigmondy's Theorem.

Contribution

It extends existing theories of nontotient numbers to the Schemmel totient functions, providing new generalizations and proofs.

Findings

Many results about nontotient numbers generalize to Schemmel nontotients.

Modified proofs demonstrate the broader applicability of known results.

Zsigmondy's Theorem is used to extend classical results to this new context.

Abstract

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $\alpha$. The function $S_1$ is simply Euler's totient function $\phi$. 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.