  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.
