On Sparsely Schemmel Totient Numbers

2014/12/09 by Colin Defant, Defant, Colin
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Limits and Structures in Graph Theory #math.NT #msc:11A25

paper · pdf · doi:10.48550/arxiv.1412.3080

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

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. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers n satisfying ϕ(n)<ϕ(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.

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