On Sparsely Schemmel Totient Numbers

TL;DR

This paper introduces and analyzes sparsely Schemmel totient numbers, generalizing the concept of sparsely totient numbers by using Schemmel totient functions, and extends previous results in this area.

Contribution

It defines sparsely Schemmel totient numbers of order r and generalizes existing results on sparsely totient numbers to this broader context.

Findings

Established properties of sparsely Schemmel totient numbers.

Generalized results from Masser and Shiu to the Schemmel totient setting.

Identified conditions for minimality among positive integers with positive Schemmel totient values.

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 $\phi$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $\phi(n)<\phi(m)$ for all integers $m>n$. We define a sparsely Schemmel totient number of order $r$ to be a positive integer $n$ such that $S_r(n)>0$ and $S_r(n)<S_r(m)$ for all $m>n$ with $S_r(m)>0$. We then generalize some of the results of Masser and Shiu.