On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions
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.
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
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.
Full version: pdf,
(Concerned with sequences
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.
Journal of Integer Sequences home page