On arithmetic functions related to iterates of the Schemmel totient functions

Defant, Colin

Journal of integer sequences, Tome 18 (2015) no. 2

Résumé

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.

Détail

Classification : 11N64, 11B83
Keywords: schemmel totient function, iterated arithmetic function, summatory function, perfect totient number