\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/ ~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf A Note On Perfect Totient Numbers } \vskip 1cm \large Moujie Deng\footnote{Supported by the Natural Science Foundation of Hainan province, Grant No.\ 808101.}\\ Department of Applied Mathematics \\ College of Information Science and Technology \\ Hainan University\\ Haikou 570228\\ P. R. China\\ \href{mailto:dmj2002@hotmail.com}{\tt dmj2002@hotmail.com} \\ \end{center} \vskip .2 in \begin{abstract} In this note we prove that there are no perfect totient numbers of the form $3^kp$, $k\ge4$, where $s=2^a3^b+1$, $r=2^c3^ds+1$, $q=2^e3^fr+1$, and $p=2^g3^hq+1$ are primes with $a,c,e,g\ge1$, and $b,d,f,h\ge0$. \end{abstract} \newtheorem{theorem}{\sc Theorem} \newtheorem{lemma}{\sc Lemma} \section{Introduction} Let $\phi$ denote Euler's totient function. Define $\phi^1(n)=\phi(n)$ and $\phi^k(n)=\phi(\phi^{k-1}(n))$ for all integers $n>2$ , $k\ge2$. Let $c$ be the smallest positive integer such that $\phi^c(n)=1$. Define the arithmetic function $S$ by \[ S(n)=\sum_{k=1}^{c} \phi^{k}(n). \] We say that $n$ is a {\it perfect totient number\/} (or PTN for short) if $S(n)=n$. There are infinitely many PTNs, since it is easy to show that $3^k$ is a PTN for all positive integers $k$. Perez Cacho~\cite{cacho} proved that $3p$, for an odd prime $p$, is a PTN if and only if $p=4n+1$, where $n$ is a PTN. Mohan and Suryanarayana~\cite{ms} proved that $3p$, for an odd prime $p$, is not a PTN if $p\equiv3\pmod4$. Thus PTNs of the form $3p$ have been completely characterized. D. E. Iannucci, the author and G. L. Cohen~\cite{ian} investigated PTNs of the form $3^kp$ in the following cases: \begin{enumerate} \item $k\ge2$, $p=2^c3^dq+1$ and $q=2^a3^b+1$ are primes with $a$,$c\ge1$ and $b$, $d\ge0$; \item $k\ge2$, $p=2^e3^fq+1$, $q=2^c3^dr+1$ and $r=2^a3^b+1$ are all primes with $a$, $c$, $e\ge1$ and $b$, $d$, $f\ge0$; \item $k\ge3$, $p=2^g3^hq+1$, $q=2^e3^fr+1$, $r=2^c3^ds+1$ and $s=2^a3^b+1$, are all primes with $a,c,e,g\ge1$, $b,d,f,h\ge0.$ \end{enumerate} In the first case, they determined all PTNs for $k=2,3$ and proved that there are no PTNs of the form $3^kp$ for $k\ge4$ by solving the related Diophantine equations. In the remaining cases, they only found several PTNs by computer searches. The author (\cite{deng1,deng2}) gave all solutions to the Diophantine equations $2^x-2^y3^z-2\cdot3^u=9^k+1,$ and $2^x-2^y3^z-4\cdot3^w=3\cdot9^k+1,$ which shows that there are no PTNs of the form $3^kp$ for $k\ge4$ in the second case mentioned above. In general, let${\cal M}$ be the set of all perfect totients, I.~E.~Shparlinski~\cite{Shparlinski} has shown that ${\cal M}$ is of asymptotic density zero, and F. Luca~\cite{Luca} showed that $\sum_{m\in {\cal M}}\frac{1}{m}$ converges. The purpose of this note is to prove that, in the third case mentioned above, there are no PTNs of the form $3^kp$ for $k\ge4$. \section{Lemmas} We first deduce related Diophantine equations. Let $k\ge3$, $n=3^kp$. Suppose all of $s=2^a3^b+1$, $r=2^c3^ds+1$, $q=2^e3^fr+1$, and $p=2^g3^hq+1$ are prime with $a,c,e,g\ge1$, $b,d,f,h\ge0$. If $n$ is a PTN , then $S(n)=n$ by definition, which implies the diophantine equation \begin{equation}\label{eq1} 2^g(2^e(2^c(2^a-3^{d+f+h+k-3})-3^{f+h+k-2})-3^{h+k-1})=3^k+1. \end{equation} Apparently, $g=1$ or 2 for $k$ even or odd, respectively. Next, according to $k=2k_{1}$ or $k=2k_{1}+1$, we consider more general Diophantine equations \begin{equation}\label{eq2} 2^x-2^y3^z-2^u3^v-2\cdot3^w=9^{k_{1}}+1, \end{equation} with $x\ge4$, $y,u,w>0$, $z,v\ge0$, $k_{1}\ge2$, and \begin{equation}\label{eq3} 2^x-2^y3^z-2^u3^v-4\cdot3^w=3\cdot9^{k_{1}}+1, \end{equation} with $x\ge4$, $y,u,w>0$, $z,v\ge0$, $k_{1}\ge1$, respectively. Since the terms $2^y3^z$ and $2^u3^v$ have symmetry in (\ref{eq2}) and (\ref{eq3}), we need only determine the solutions $(x, y, z, u, v, w, k)$ to (\ref{eq2}) and (\ref{eq3}) such that $y>u$ or $y=u,z\ge v$. Let $ (x, y, z, u, v, w, k_{1})$ be any solution to the equation (\ref{eq2}) (or (\ref{eq3})), and let \begin {center} $ (x, y, z, u, v, w, k_{1})\equiv(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu, )\pmod{36, 36, 36, 36, 36, 36, 18}$ \end {center} denote $x\equiv \alpha\pmod{36}, y\equiv \beta\pmod{36}, z\equiv \gamma\pmod{36}, u\equiv \delta\pmod{36}, v\equiv \lambda\pmod{36}$, $w\equiv \mu\pmod{36}$, and $k_{1}\equiv \nu\pmod{18}$. In solving equation (\ref{eq2}) and equation (\ref{eq3}), we first determine all the $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$. \begin {lemma} Let $(x, y, z, u, v, w, k_{1})$ be any solution to the equation $(2)$, and let \begin {center} $(x, y, z, u, v, w, k_{1})\equiv(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)\pmod{36, 36, 36, 36, 36, 36, 18}$. \end {center} Then all the possible $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ with $36\ge \alpha,\beta,\delta,\lambda\ge 1, 35\ge \gamma,\lambda\ge 0$, $19\ge \nu\ge 2$, $\beta>\delta$ or $\beta=\delta$ and $\gamma\ge \lambda$ are listed in Table $1$ and Table $1'$. \end {lemma} \noindent{\sc Proof:} Since \begin {center} $2^{36}\equiv 1\pmod{5\cdot7\cdot13\cdot19\cdot37\cdot73}, 3^{36}\equiv 1\pmod{5\cdot7\cdot13\cdot19\cdot37\cdot73}$, \end {center} $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ must satisfy \begin{equation}\label{eq4} 2^\alpha-2^\beta3^\gamma-2^\delta3^\lambda-2\cdot3^\mu\equiv 9^\nu+1 \pmod{5\cdot7\cdot13\cdot19\cdot37\cdot73} . \end{equation} But note that $2^x\equiv 0\pmod{2^4},9^{k_{1}}\equiv 0\pmod{3^3},2^{36}\equiv 1\pmod{3^3},3^{36}\equiv 1\pmod{2^4}$; $M=36l+m$ implies $2^M\equiv 0\,$ or $\,2^m\pmod{2^4}$ and $3^M\equiv 0\,$ or $\,3^m\pmod{3^3}$. Hence $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ must satisfy one of the $4$ congruences \begin{equation}\label{eq5} -2^\beta\cdot B\cdot3^\gamma-2^\delta\cdot D\cdot3^\lambda-2\cdot3^\mu\equiv 9^\nu+1 \pmod{2^4} \end{equation} and one of the $8$ congruences \begin{equation}\label{eq6} 2^\alpha-2^\beta3^\gamma\cdot C-2^\delta3^\lambda\cdot E-2\cdot3^\mu\cdot F\equiv 1 \pmod{3^3}, \end{equation} where $B, C, D, E, F$ take value $0,1$ independently. The congruences (\ref{eq4}), (\ref{eq5}) and (\ref{eq6}) were tested on a computer with a program written in UBASIC. All the $(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)$ that satisfy (\ref{eq4}), (\ref{eq5}) and (\ref{eq6}) are divided into two parts: those listed in Table 1 are in fact solutions to equation (\ref{eq2}), and the remainder, listed in Table $1'$, are not.\quad$\square$ Similarly, we have \begin {lemma} Let $(x, y, z, u, v, w, k_{1})$ be any solution to the equation $(3)$, and let \begin {center} $(x, y, z, u, v, w, k)\equiv(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)\pmod{36, 36, 36, 36, 36, 36, 18}$. \end {center} Then all the possible $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ with $36\ge \alpha,\beta,\delta,\mu\ge 1, 35\ge \gamma,\lambda\ge 0$, $18\ge \nu\ge 1$, $\beta>\delta$ or $\beta=\delta$ and $\gamma\ge \lambda$ are listed in Table 2 and Table $2'$. \end {lemma} \begin {lemma} Let $(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)$ be any solution to equation $(2)$ or $(3)$ that is listed in Table 1 or Table 2, and suppose \begin{enumerate} \item $\alpha>\beta>\delta$ ; or \item $\alpha>\beta+2$, $\beta=\delta$ ; \end{enumerate} holds. Then there is no other solution $(x, y, \gamma, u, \lambda, \mu, \nu)$ to equation $(2)$ or $(3)$ that satisfies $(x, y, u)\equiv(\alpha, \beta, \delta)\pmod{36, 36, 36}$ \end {lemma} \noindent{\sc Proof:} Let $x=\alpha+36i, y=\beta+36j, u=\delta+36l.$ We have \begin{equation}\label{eq7} 2^\alpha(2^{36i}-1)=2^\beta3^\gamma(2^{36j}-1)+2^\delta3^\lambda(2^{36l}-1). \end{equation} In case 1, consideration of (\ref{eq7}), modulo $2^\beta$ and $2^\alpha$ in turn gives $l=0$ and $j=0$. Hence we have $i=0$. In case 2, since $3^\gamma+3^\lambda\equiv2, 4\pmod{8}$, consideration of (\ref{eq7}), modulo $2^\alpha$, gives $j=l=0$, and therefore $i=0$. \section{Main Results} \begin {theorem} All the solutions to equation $(2)$ are given by $(x,y,z,u,v,w,k_{1})=(\alpha, \beta, \gamma, \delta, \lambda,$ $\mu, \nu)$ with $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ listed in Table $1$. \end {theorem} \noindent{\sc Proof:} Let $ (x, y, z, u, v, w, k_{1})$ be any solution to equation (\ref{eq2}), and let \begin {center} $(x, y, z, u, v, w, k_{1})\equiv(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)\pmod{36, 36, 36, 36, 36, 36, 18}$. \end {center} By Lemma $1$, all of $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ are listed in Table $1$ or Table $1'$. Put $x= \alpha+36i, y= \beta+36j, z= \gamma+36l, u= \delta+36m, v= \lambda+36n, w= \mu+36t, k_{1}= \nu+18t_{1}$. Then we must have \begin{equation}\label{eq8} 2^{\alpha+36i}-2^{\beta+36j}\cdot3^{\gamma+36l}-2^{\delta+36m}\cdot3^{\lambda+36n}-2\cdot3^{\mu+36t}\equiv 9^{\nu+36t_{1}}+1 \pmod{11\cdot31\cdot181\cdot331\cdot631}. \end{equation} For $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ appearing in Table $1$, since $2^{180}\equiv3^{360}\equiv 1\pmod{11\cdot31\cdot181\cdot331\cdot631} ,$ we first test (\ref{eq8}) within \begin {center} $4\ge i\ge 0, 4\ge j\ge 0,9\ge l\ge 0,4\ge m\ge 0,9\ge n\ge 0,9\ge t\ge 0,9\ge t_{1}\ge 0.$ \end {center} With computer assistance, it follows that $l=n=t=t_{1}=0$ in this case. Since any $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ that listed in Table $1$ satisfy the conditions of Lemma $3$, we must have $i=j=m=0$ by Lemma $3$. For $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ appearing in Table $1'$, since $(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)$ is not a solution to equation(\ref{eq2}), we have $i\ge 1$. The congruence (\ref{eq8}) was then tested on a computer within the ranges \begin {center} $5\ge i\ge 1, 4\ge j\ge 0,9\ge l\ge 0,4\ge m\ge 0,9\ge n\ge 0,9\ge t\ge 0,9\ge t_{1}\ge 0$ \end {center} with no $(i,j,l,m,n,t,t_{1})$ being found, which shows that $(x, y, z, u, v, w, k_{1})$ cannot be a solution to equation (\ref{eq2}).\quad$\square$ \begin {theorem} All the solutions to equation $(3)$ are given by $(x,y,z,u,v,w,k_{1})=(\alpha, \beta, \gamma, \delta, \lambda,$ $\mu, \nu)$ with $\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu$ listed in Table $2$ . \end {theorem} \noindent{\sc Proof:} The proof is basically the same as that for theorem $1$, with the only difference being that $(\alpha, \beta, \gamma, \delta, \lambda, \mu, \nu)=(9,7,0,7,0,1,2)$, listed in Table $2$, does not satisfy the conditions of Lemma $3$. Suppose that $x=9+36i,y=7+36j,z=36l,u=7+36m,v=36n,w=1+36t,k_{1}=2+36t_{1}$ is a solution to equation (\ref{eq3}). Then a computer test of the related congruence within $4\ge i\ge 0, 4\ge j\ge 0,9\ge l\ge 0,4\ge m\ge 0,9\ge n\ge 0,9\ge t\ge 0,9\ge t_{1}\ge 0$ gives $l=n=t=t_{1}=0$. Consideration of (\ref{eq7}) with $\alpha,\beta,\gamma,\delta,\lambda$ replaced by $9,7,0,7,0$ , modulo $2^7$, gives $j=l=0$. Therefor $i=0$. \begin{theorem} There are no PTNs of the form $3^kp$, $k\ge4$, where all of $s=2^a3^b+1$, $r=2^c3^ds+1$, $q=2^e3^fr+1$, and $p=2^g3^hq+1$ are prime with $a,c,e,g\ge1$, $b,d,f,h\ge0$. \end {theorem} \noindent{\sc Proof:} Suppose $(a,c,d,e,f,g,h,k)$ is a solution to equation (\ref{eq1}). Let $x=a+c+e+g,$\\$y=c+e+g$, $z=d+f+k-3$, $u=e+g$, $v=f+h+k-2$, $w=h+k-1$, and $k_{1}=\frac{k}{2}$ or $k_{1}=\frac{k-1}{2}$ for $k$ even or odd, respectively. Then $(x, y, z, u, v, w, k_{1})$ must be a solution to equation (\ref{eq2}) or equation (\ref{eq3}). From the first two theorems it follows that the only solutions to equation(\ref{eq1}) are $(a,c,d,e,f,h,k)= (4,1,0,1,2,1,3)$,$(1,2,0,4,0,0,3)$,$(3,1,0,4,1,0,3)$,$(2,2,1,4,0,0,3)$, $(8,1,4,1,0,1,3)$,$(5,1,2,4,1,0,3)$,$(4,2,0,4,2,0,3)$. \quad$\square$ \begin{center} \begin{tabular}{|l|l|l|}\hline $\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\,\hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in$\mu$\,\, \hskip .1in $\nu$\\\hline 7 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \, &10\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \, &12\,\hskip .1in 8 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\\ 7 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 2 &10\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2&12\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3\\ 7 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 2 &10\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 3&13\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 4\\ 7 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &10\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 3 &13\,\hskip .1in 8 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \\ 7 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &10\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 4 \,\,\hskip .1in 3&13\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2&14\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &10\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 3 &14\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \\ 8 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2&10\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2&14\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 8 \,\,\hskip .1in 2\\ 8 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 4 \,\,\hskip .1in 2&11\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 2&15\,\hskip .1in 10\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 2 &11\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &16\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2&11\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 3&16\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 5\\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 2&11\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 3&16\,\hskip .1in 6 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 5\\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &11\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 4 \,\,\hskip .1in 3 &16\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 5 \\ 8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2&11\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 3&16\,\hskip .1in 8 \,\,\hskip .1in 5 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2&11\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 3&16\,\hskip .1in 9 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 7 \,\,\hskip .1in 5\\ 9 \,\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2 &11\,\hskip .1in 10\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3 &16\,\hskip .1in 9 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \\ 10\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 3&12\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 3&16\,\hskip .1in 9 \,\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 3\\ 10\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 3&12\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 2&16\,\hskip .1in 11\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 5\\ 10\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3 &12\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2&16\,\hskip .1in 11\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 5\\ 10\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3&12\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 4 \,\,\hskip .1in 3&16\,\hskip .1in 11\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 5\\ 10\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 2&12\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 3&16\,\hskip .1in 13\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 9 \,\,\hskip .1in 3\\ 10\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 3 &12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 3&18\,\hskip .1in 10\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 8 \,\,\hskip .1in 2\\ 10\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 3&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2&18\,\hskip .1in 13\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 9 \,\,\hskip .1in 3\\ 10\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 2&12\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3& \,\hskip .1in \,\,\hskip .1in \,\,\hskip .1in \,\,\hskip .1in \,\,\hskip .1in \,\,\hskip .1in \\ \hline \end{tabular} \end{center} \centerline{Table 1} \begin{center} \begin{tabular}{|l|l|l|}\hline $\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\,\hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in$\mu$\,\, \hskip .1in $\nu$\\\hline 5 \,\,\hskip .1in 1 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 19&6 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 19&8 \,\,\hskip .1in 3 \,\,\hskip .1in 14\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 16\,\hskip .1in 13\\ 5 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 19&6 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 36\,\hskip .1in 18&8 \,\,\hskip .1in 3 \,\,\hskip .1in 15\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 14\,\hskip .1in 13\\ 5 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 18&6 \,\,\hskip .1in 3 \,\,\hskip .1in 12\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 14\,\hskip .1in 12&8 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 19\\ 5 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 36\,\hskip .1in 18&6 \,\,\hskip .1in 3 \,\,\hskip .1in 13\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 12\,\hskip .1in 12&10\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 19\\ 5 \,\,\hskip .1in 3 \,\,\hskip .1in 12\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 14\,\hskip .1in 12&6 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 36\,\hskip .1in 19&10\,\hskip .1in 9 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 18\\ 5 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 20\,\hskip .1in 0 \,\,\hskip .1in 36\,\hskip .1in 19&6 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 19&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 19\\ 5 \,\,\hskip .1in 3 \,\,\hskip .1in 13\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 12\,\hskip .1in 12&7 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 19&18\,\hskip .1in 34\,\hskip .1in 22\,\hskip .1in 21\,\hskip .1in 16\,\hskip .1in 33\,\hskip .1in 11\\ 6 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 19&8 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 19&30\,\hskip .1in 23\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 13\,\hskip .1in 1 \,\,\hskip .1in 15\\ 6 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 18&8 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 19& \\\hline \end{tabular} \end{center} \centerline{Table 1\'} \begin{center} \begin{tabular}{|l|l|l|}\hline $\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\,\hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in$\mu$\,\, \hskip .1in $\nu$\\\hline 6 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip.1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2\\ 6 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2&12\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\ 6 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 2&12\,\hskip .1in 8 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2\\ 7 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2&12\,\hskip .1in 8 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1\\ 7 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 1 &12\,\hskip .1in 9 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \\ 7 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1&12\,\hskip .1in 9 \,\,\hskip .1in 1 \,\,\hskip .1in 8 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2\\ 7 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2&12\,\hskip .1in 10\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\ 8 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &10\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1&12\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1\\ 8 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 &10\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &12\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 8 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1&10\,\hskip .1in 9 \,\,\hskip .1in 0 \,\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2&13\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 1&11\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 1&13\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 2\\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &11\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &13\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \\ 8 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1&11\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2&13\,\hskip .1in 6 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\ 8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1&11\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1&13\,\hskip .1in 8 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 2\\ 8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &11\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 2 &13\,\hskip .1in 10\,\hskip .1in 0 \,\,\hskip .1in 8 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 8 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1&11\,\hskip .1in 9 \,\,\hskip .1in 1 \,\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2&13\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 3\\ 8 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1&11\,\hskip .1in 10\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1&13\,\hskip .1in 12\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\ 9 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 2 &11\,\hskip .1in 10\,\hskip .1in 0 \,\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &14\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 9 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 2 &12\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 1 &14\,\hskip .1in 8 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \\ 9 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 4 \,\,\hskip .1in 1 &12\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &14\,\hskip .1in 9 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \\ 9 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &12\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 3 &14\,\hskip .1in 9 \,\,\hskip .1in 3 \,\,\hskip .1in 8 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 9 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2 &12\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 2 &14\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 7 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \\ 9 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 &12\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 3 &14\,\hskip .1in 10\,\hskip .1in 2 \,\,\hskip .1in 8 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 9 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &12\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 3 &14\,\hskip .1in 12\,\hskip .1in 1 \,\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \\ 9 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 2 &12\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 &15\,\hskip .1in 10\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \\ 9 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 0 \,\,\hskip .1in 4 \,\,\hskip .1in 1 &12\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 1 &16\,\hskip .1in 8 \,\,\hskip .1in 5 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \\ 9 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 7 \,\,\hskip .1in 0 \,\,\hskip .1in 1 \,\,\hskip .1in 2 &12\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2 &16\,\hskip .1in 10\,\hskip .1in 1 \,\,\hskip .1in 8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \\ 9 \,\,\hskip .1in 7 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &12\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 3 &18\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 9 \,\,\hskip .1in 5 \\ 9 \,\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &12\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 1 &18\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 10\,\hskip .1in 4 \\ 9 \,\,\hskip .1in 8 \,\,\hskip .1in 0 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 &12\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 3 &18\,\hskip .1in 5 \,\,\hskip .1in 6 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 10\,\hskip .1in 3 \\ 10\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1&12\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 3&18\,\hskip .1in 5 \,\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 10\,\hskip .1in 1\\ 10\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 1&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 3&18\,\hskip .1in 7 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 5 \,\,\hskip .1in 10\,\hskip .1in 1\\ 10\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 1 &12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 5 \,\,\hskip .1in 1&18\,\hskip .1in 8 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 4 \,\,\hskip .1in 10\,\hskip .1in 1\\ 10\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 3&18\,\hskip .1in 10\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 5 \,\,\hskip .1in 8 \,\,\hskip .1in 5\\ 10\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 2&18\,\hskip .1in 10\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 5\\ 10\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 5 \,\,\hskip .1in 1 &12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 4 \,\,\hskip .1in 4 \,\,\hskip .1in 1&18\,\hskip .1in 10\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 5\\ 10\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 6 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 2&18\,\hskip .1in 10\,\hskip .1in 5 \,\,\hskip .1in 7 \,\,\hskip .1in 4 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\ 10\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 2&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 2&18\,\hskip .1in 11\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 8 \,\,\hskip .1in 5\\ 10\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 2&18\,\hskip .1in 11\,\hskip .1in 4 \,\,\hskip .1in 3 \,\,\hskip .1in 7 \,\,\hskip .1in 9 \,\,\hskip .1in 1\\ 10\,\hskip .1in 5 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 5 \,\,\hskip .1in 2 \,\,\hskip .1in 4 \,\,\hskip .1in 1&18\,\hskip .1in 11\,\hskip .1in 4 \,\,\hskip .1in 5 \,\,\hskip .1in 7 \,\,\hskip .1in 8 \,\,\hskip .1in 1\\ 10\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 2&12\,\hskip .1in 7 \,\,\hskip .1in 3 \,\,\hskip .1in 6 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1&18\,\hskip .1in 11\,\hskip .1in 4 \,\,\hskip .1in 7 \,\,\hskip .1in 6 \,\,\hskip .1in 6 \,\,\hskip .1in 1\\\hline \end{tabular} \end{center} \centerline{Table 2} \begin{center} \begin{tabular}{|l|l|l|}\hline $\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\, \hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in $\mu$\,\, \hskip .1in $\nu$ &$\alpha$\,\, \hskip .1in $\beta$\,\,\hskip .1in $\gamma$\,\, \hskip .1in $\delta$\,\, \hskip .1in $\lambda$\,\, \hskip .1in$\mu$\,\, \hskip .1in $\nu$\\\hline 6 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 18&7 \,\,\hskip .1in 6 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 18 &18\,\hskip .1in 18\,\hskip .1in 18\,\hskip .1in 2 \,\,\hskip .1in 9 \,\,\hskip .1in 1 \,\,\hskip .1in 14\\ 6 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 18&8 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 18 &18\,\hskip .1in 18\,\hskip .1in 20\,\hskip .1in 1 \,\,\hskip .1in 14\,\hskip .1in 31\,\hskip .1in 18\\ 6 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 18&8 \,\,\hskip .1in 5 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 18 &18\,\hskip .1in 20\,\hskip .1in 33\,\hskip .1in 14\,\hskip .1in 20\,\hskip .1in 36\,\hskip .1in 1 \\ 6 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 18&8 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 18 &18\,\hskip .1in 23\,\hskip .1in 12\,\hskip .1in 17\,\hskip .1in 3 \,\,\hskip .1in 32\,\hskip .1in 1 \\ 6 \,\,\hskip .1in 3 \,\,\hskip .1in 35\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 2 \,\,\hskip .1in 17&8 \,\,\hskip .1in 6 \,\,\hskip .1in 1 \,\,\hskip .1in 4 \,\,\hskip .1in 1 \,\,\hskip .1in 1 \,\,\hskip .1in 18 &18\,\hskip .1in 27\,\hskip .1in 30\,\hskip .1in 3 \,\,\hskip .1in 34\,\hskip .1in 9 \,\,\hskip .1in 7 \\ 6 \,\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 18&8 \,\,\hskip .1in 9 \,\,\hskip .1in 35\,\hskip .1in 3 \,\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 17 &18\,\hskip .1in 28\,\hskip .1in 18\,\hskip .1in 7 \,\,\hskip .1in 20\,\hskip .1in 3 \,\,\hskip .1in 2 \\ 6 \,\,\hskip .1in 5 \,\,\hskip .1in 35\,\hskip .1in 4 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 17&10\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 5 \,\,\hskip .1in 18 &18\,\hskip .1in 28\,\hskip .1in 18\,\hskip .1in 7 \,\,\hskip .1in 20\,\hskip .1in 4 \,\,\hskip .1in 1 \\ 6 \,\,\hskip .1in 6 \,\,\hskip .1in 35\,\hskip .1in 3 \,\,\hskip .1in 35\,\hskip .1in 2 \,\,\hskip .1in 18&12\,\hskip .1in 7 \,\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 1 \,\,\hskip .1in 6 \,\,\hskip .1in 18 &18\,\hskip .1in 29\,\hskip .1in 2 \,\,\hskip .1in 3 \,\,\hskip .1in 29\,\hskip .1in 31\,\hskip .1in 6 \\ 6 \,\,\hskip .1in 6 \,\,\hskip .1in 35\,\hskip .1in 4 \,\,\hskip .1in 35\,\hskip .1in 2 \,\,\hskip .1in 17&18\,\hskip .1in 11\,\hskip .1in 0 \,\,\hskip .1in 10\,\hskip .1in 4 \,\,\hskip .1in 36\,\hskip .1in 5 &18\,\hskip .1in 29\,\hskip .1in 5 \,\,\hskip .1in 23\,\hskip .1in 21\,\hskip .1in 34\,\hskip .1in 3 \\ 6 \,\,\hskip .1in 9 \,\,\hskip .1in 33\,\hskip .1in 3 \,\,\hskip .1in 0 \,\,\hskip .1in 2 \,\,\hskip .1in 16&18\,\hskip .1in 18\,\hskip .1in 18\,\hskip .1in 2 \,\,\hskip .1in 1 \,\,\hskip .1in 9 \,\,\hskip .1in 14&18\,\hskip .1in 33\,\hskip .1in 35\,\hskip .1in 19\,\hskip .1in 24\,\hskip .1in 12\,\hskip .1in 15\\\hline \end{tabular} \end{center} \centerline{Table 2\'} \section{Acknowledgments} The author thanks Professor G. L. Cohen and Professor Huishi Li for useful advice and the referee for helpful suggestions which improved the quality of this paper. \begin{thebibliography}{9} \bibitem{deng1} Moujie Deng, On the Diophantine equation $2^x-2^y3^z-2\cdot3^u=9^k+1$, (in Chinese), \emph{Journal of Natural Science of Heilongjiang University}, \textbf{23} (2006), 87--91. \bibitem{deng2} Moujie Deng, On the Diophantine equation $2^x-2^y3^z-4\cdot3^w=3\cdot9^k+1$,(in Chinese), \emph{Journal of Inner Mongolia Normal University}, \textbf{37} (2008), 45--49. \bibitem{ian} D. E. Iannucci, D. Moujie and G. L. Cohen, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Cohen2/cohen50.html}{On perfect totient numbers}, \emph{J. Integer Sequences} \textbf {6} (2003), Article 03.4.5. \bibitem{Luca} F. Luca, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Luca/luca66.html}{On the distribution of perfect totients}, \emph{J. Integer Sequences} \textbf {9} (2006), Article 06.4.4. \bibitem{ms} A. L. Mohan and D. Suryanarayana, ``Perfect totient numbers'', in: \emph{Number Theory (Proc. Third Matscience Conf., Mysore, 1981)} Lect. Notes in Math. \textbf{938}, Springer-Verlag, New York, 1982, 101--105. \bibitem{cacho} L. Perez Cacho, Sobre la suma de indicadores de ordenes sucesivos, \emph{Revista Matematica Hispano-Americana} \textbf{5.3} (1939), 45--50. \bibitem{Shparlinski} I.~E.~Shparlinski, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Shparlinski/shpar43.html}{On the sum of iterations of the Euler function}, \emph{J. Integer Sequences} \textbf{9} (2006), Article 06.1.6. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11A25. \noindent \emph{Keywords:} totient, perfect totient number, diophantine equation. \bigskip \hrule \bigskip \noindent (Concerned with sequence \seqnum{A082897}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 14 2009; revised version received July 18 2009. Published in {\it Journal of Integer Sequences}, August 30 2009. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \end{document}