On Sparsely Schemmel Totient Numbers

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by $$S_r(p^{\alpha })=\{\begin{array}{ll}0,& ifp\le r;\\ p^{\alpha -1}(p-r),& ifp>r\end{array}$$ for all primes $p$ and positive integers $\alpha$. The function $S_1$ is simply Euler’s totient function $\varphi$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $\varphi (n)<\varphi (m)$ 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.