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On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions
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Colin Defant

Department of Mathematics

University of Florida

Gainesville, FL 32611-8105

USA

**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.

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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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