On Sparsely Schemmel Totient Numbers

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 $\hat{I}\pm$. The function $S_1$ is simply Euler's totient function $\ddot{I}•$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $\ddot{I}•(n)<\ddot{I}•(m)$ 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.

14 pages, 0 figures, Supported by National Science Foundation grant no. 1262930