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\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.


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\bibitem{Luca}
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\bibitem{ms}
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\end{thebibliography}


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\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.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.

\end{document}

                                                                                



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