paper

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

arXiv:1506.05426

Abstract

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_{m}$ , we define two new functions, denoted $R_{m}$ and $H_{m}$ , that arise from iterating $L_{m}$ . Roughly speaking, $R_{m}$ counts the number of iterations of $L_{m}$ needed to reach either $0$ or $1$ , and $H_{m}$ takes the value (either $0$ or $1$ ) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer $m$ , the function $H_{m}$ is completely multiplicative. We then introduce an iterate summatory function, denoted $D_{m}$ , and define the terms $D_{m}$ -deficient, $D_{m}$ -perfect, and $D_{m}$ -abundant. We proceed to prove several results related to these definitions, culminating in a proof that, for all positive even integers $m$ , there are infinitely many $D_{m}$ -abundant numbers. Many open problems arise from the introduction of these functions and terms, and we mention a few of them, as well as some numerical results.

16 pages, 1 figure