On Schemmel Nontotient Numbers

摘要

For each positive integer r, let S_r denote the r~(th) Schemmel totient function, the multiplicative arithmetic function defined by S_r(p~α)={0, if p ≤ r; p~(α-1) (p-r), if p＞r 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 Bang's theorem in order to generalize a result due to Mendelsohn.