Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.1

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

Colin Defant
Department of Mathematics
University of Florida
Gainesville, FL 32611-8105


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 Lm, we define two new functions, denoted Rm and Hm, that arise from iterating Lm. Roughly speaking, Rm counts the number of iterations of Lm needed to reach either 0 or 1, and Hm 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 Hm is completely multiplicative. We then introduce an iterate summatory function, denoted Dm, and define the terms Dm-deficient, Dm-perfect, and Dm-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 Dm-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.

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(Concerned with sequences A000010 A003434 A058026 A092693 A123565 A241663 A241664 A241665 A241666 A241667 A241668.)

Received April 26 2014; revised versions received October 12 2014; November 7 2014; January 8 2015. Published in Journal of Integer Sequences, January 13 2015.

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