On Schemmel Nontotient Numbers

2014/12/09 by Colin Defant, Defant, Colin
Computer Science · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Computability, Logic, AI Algorithms #math.NT #msc:11A25 #msc:11N64

paper · pdf · doi:10.48550/arxiv.1412.3089

10 pages, 0 figures

arxiv created 2014/12/09 · arxiv updated 2014/12/10

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)=\begincases 0, if p≤ r;
p^α-1(p-r), if p>r \endcases for all primes p and positive integers α. The function S₁ 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 Zsigmondy's Theorem in order to generalize a result due to Mendelsohn.

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