Ap_0^{2-\delta}+3mp_0$. With this choice of $p_0$, we may write $\beta(p_0-m)^2>\beta p_0^2-2\beta mp_0\geq \beta p_0^2-2mp_0>Ap_0^{2-\delta}+mp_0$. But $\beta (p_0-m)=L_m(p_0-m)$ because $p(p_0-m)>m$. Thus, \begin{equation}\label{5} (p_0-m)L_m(p_0-m)>Ap_0^{2-\delta}+mp_0. \end{equation} Let $\alpha$ be an integer, and, for now, assume $\alpha\geq 2$. Rearranging and multiplying the inequality \eqref{5} by $p_0^{\alpha-2}$, we have $-mp_0^{\alpha-1}+p_0^{\alpha-1}L_m(p_0-m)>mp_0^{\alpha-2}L_m(p_0-m)+Ap_0^{\alpha-\delta}$. After further algebraic manipulation, we find $p_0^{\alpha-1}(p_0-m)+p_0^{\alpha-2}(p_0-m)L_m(p_0-m)>p_0^{\alpha}+Ap_0^{\alpha-\delta}$. Noticing that the left-hand side of the preceding inequality is simply $L_m(p_0^{\alpha})+L_m^{(2)}(p_0^{\alpha})$, we have $L_m(p_0^{\alpha})+L_m^{(2)}(p_0^{\alpha})>p_0^{\alpha}+Ap_0^{\alpha-\delta}$. This is the desired result for $\alpha\geq 2$. To show that the result holds when $\alpha=1$, it suffices to show that $L_m(p_0)+L_m^{(2)}(p_0)>\displaystyle{\frac{L_m(p_0^2)+L_m^{(2)}(p_0^2)}{p_0}}$. This reduces to $p_0-m+L_m(p_0-m)>p_0-m+\displaystyle{\frac{p_0-m}{p_0}L_m(p_0-m)}$, which is obviously true. \end{proof} \begin{corollary} \label{Cor2.1} For any positive even integer $m$, there exist infinitely many $D_m$-abundant numbers. \end{corollary} We conclude this section with a remark about $D_m$-perfect numbers. Using \textit{Mathematica}, one may check that for $m\in \{2,4,6\}$, the only $D_m$-perfect number less than $100,000$ is $37,147$, which is $D_2$-perfect. Unfortunately, this data is too scarce to make any reasonable conjecture about the nature or distribution of $D_m$-perfect numbers for positive even integers $m$. \section{Numerical analysis and concluding remarks} In 1943, H. Shapiro investigated a function $C$, which counts the number of iterations of the $\phi$ function needed to reach $2$ \cite{shapiro43}. Shapiro showed that the function $C$ is additive, and he established bounds for its values. In this paper, we have not gone into much detail exploring the functions $R_m$ because they prove, in general, to be either completely uninteresting or very difficult to handle. For example, for any integer $n>1$, \[R_3(n)=\begin{cases} 1, & \mbox{if } n \not\equiv 1,5 \pmod{6}; \\ 2, & \mbox{if } n\equiv 1,5 \pmod{6}. \end{cases}\] On the other hand, the function $R_4$ does not seem to obey any nice pattern or exhibit any sort of nice additive behavior. There seems to be some hope in analyzing the function $R_2$, so we make the following conjecture. \begin{conjecture} If $x>3$ is an odd integer, then \[R_2(x)\geq \frac{\log \left(\frac{49}{15}x\right)}{\log 7}.\] \end{conjecture} We note that it is not difficult to prove, using Lemma \ref{Lem2.1} and a bit of case work, that $R_2(x)\leq 3\displaystyle{\frac{\log(x+2)}{\log3} -3}$ for all integers $x>1$ (with equality only at $x=7$). However, as Figure~\ref{one} shows, this is a very weak upper bound (at least for relatively small $x$). It is tempting to think, based on the figure, that $R_2(x)\leq 3+\displaystyle{\frac{\log x}{\log 3}}$ for all positive integers $x$. However, setting $x=480,314,203$ yields a counterexample because $3+\displaystyle{\frac{\log 480,314,203}{\log 3}\approx 21.196<22=R_2(480,314,203)}$. \begin{figure} \begin{center} \epsfysize=7cm \epsfbox{JISFIG.eps} \caption{A plot of the first $300,000$ values of the function $R_2$, as well as some important curves. Note that the black streaks in the figure are, in actuality, several overlapping dots.} \label{one} \end{center} \end{figure} The author has found that investigating bounds of the function $R_2$ naturally leads to a question about the infinitude of twin primes, which hints at the potential difficulty of the problem. Indeed, Harrington and Jones \cite{harrington10} have arrived at the same conclusion while studying the function $C_2(x):=R_2(x)-1$, and they conjecture that the values of $C_2(x)+C_2(y)-C_2(xy)$ can be arbitrarily large. To avoid the unpredictability of the values of the function $C_2$, Harrington and Jones have restricted the domain of $C_2$ to the set $D$ of positive integers $k$ with the property that none of the numbers in the set $\{k,L_2(k),L_2^{(2)}(k),\ldots\}$ has a prime factor that is congruent to $1$ modulo $3$. With this restriction of the domain of $C_2$, these two authors have established results analogous to those that Shapiro gave for the function $C$ mentioned earlier. In fact, we speculate that methods analogous to those that Harrington and Jones have used could easily generalize to allow for analogous results concerning functions $C_m(x):=R_m(x)-1$ if one is willing to use a sufficiently restricted domain of $C_m$. We next remark that, in Theorem \ref{Thm2.4}, the requirement that $m\not=4$ is essential. For example, write $p_1=306,167$, $p_2=4+p_1^2$, $p_3=4+p_2^2$, $p_4=4+p_3^2$, and $p_5=4+p_4^2$. Then the number $5p_5$ is a $D_4$-abundant multiple of 5. Lastly, we have not spent much effort analyzing the ``sizes" of the functions $D_m$ or searching for $D_m$-perfect numbers. We might inquire about the average order or possible upper and lower bounds for $D_m$ for a general positive even integer $m$. In addition, it is natural to ask if there even are any $D_m$-perfect numbers other than $37,147$ for even positive integers $m$. \section{Acknowledgments and dedications} Dedicated to my parents Marc and Susan, my brother Jack, and my sister Juliette. Also dedicated to Mr.~Jacob Ford, who wrote a program to find values of the functions $L_m$, $R_m$, and $H_m$. Mr.~Ford and my father also sparked my interest in computer programming, which I used to analyze the functions $D_m$. Finally, I would like to thank the unknown referee for taking the time to read carefully through my work and for his or her valuable suggestions. \begin{thebibliography}{9} \bibitem{harrington10} J. Harrington and L. Jones, On the iteration of a function related to Euler's $\phi$-function. \emph{Integers} {\bf 10} (2010), 497--515. \bibitem{herzog92} J. Herzog and P. R. Smith, Lower bounds for a certain class of error functions. \emph{Acta Arith.} {\bf 60} (1992), 289--305. \bibitem{iannucci03} D. Iannucci, M. Deng, and G. Cohen, On perfect totient numbers. \emph{J. Integer Seq.} {\bf 6} (2003), \href{https://cs.uwaterloo.ca/journals/JIS/VOL6/Cohen2/cohen50.html}{Article 03.4.5}. \bibitem{luca06} F. Luca, On the distribution of perfect totients. \emph{J. Integer Seq.} {\bf 9} (2006), \href{https://cs.uwaterloo.ca/journals/JIS/VOL9/Luca/luca66.html}{Article 06.4.4}. \bibitem{pillai29} S. S. Pillai, On a function connected with $\phi(n)$. \emph{Bull. Amer. Math. Soc.} {\bf 35} (1929), 837--841. \bibitem{rosser62} J. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. \emph{Illinois J. Math.} {\bf 6} (1962), 64--94. \bibitem{schemmel69} V. Schemmel, \"{U}ber relative Primzahlen, \emph{J. Reine Angew. Math.}, {\bf 70} (1869), 191--192. \bibitem{shapiro43} H. Shapiro, An arithmetic function arising from the $\phi$ function. \emph{Amer. Math. Monthly} {\bf 50} (1943), 18--30. \bibitem{white62} G. K. White, Iterations of generalized Euler functions. \emph{Pacific J. Math.} {\bf{12}} (1962), 777--783. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11N64; Secondary 11B83. \noindent \emph{Keywords: } Schemmel totient function, iterated arithmetic function, summatory function, perfect totient number. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000010}, \seqnum{A003434}, \seqnum{A058026}, \seqnum{A092693}, \seqnum{A123565}, \seqnum{A241663}, \seqnum{A241664}, \seqnum{A241665}, \seqnum{A241666}, \seqnum{A241667}, and \seqnum{A241668}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 26 2014; revised versions received October 12 2014; November 7 2014; January 8 2015. Published in {\it Journal of Integer Sequences}, January 13 2015. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}