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\begin{center}
\vskip 1cm{\LARGE\bf
Abundancy ``Outlaws" of the Form $\frac{\sigma(N) + t}{N}$}
%\vskip .1in of the Form $\frac{\sigma(N) + t}{N}$}

\vskip 1cm \large
William G. Stanton  and Judy A. Holdener\\
Department of Mathematics\\
Kenyon College\\
Gambier, Ohio 43022 \\
USA \\
\href{mailto:stantonw@kenyon.edu}{\tt stantonw@kenyon.edu}\\
\href{mailto:holdenerj@kenyon.edu}{\tt holdenerj@kenyon.edu} \\
\end{center}

\vskip .2 in


\begin{abstract}
The abundancy index of a positive integer $n$ is defined to be the
rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of
divisors function $\sigma(n)=\sum_{d|n}d$.  An abundancy outlaw is
a rational number greater than 1 that fails to be in the image of
of the map $I$. In this paper, we consider rational numbers of the
form $(\sigma(N)+t)/N$ and prove that under certain conditions
such rationals are abundancy outlaws.
\end{abstract}

%\newtheorem{theorem}{Theorem}
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\numberwithin{equation}{section}





\section{Introduction}
The abundancy index of a positive integer $n$ is defined to be the
rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of
divisors function, $\sigma(n)=\sum_{d|n}d$. Positive integers
having integer-valued abundancy indices are said to be
\textit{multiperfect} numbers, and if $I(n)=2$ in particular, then
$n$ is \emph{perfect}.  More generally, the abundancy index of a
number $n$ can be thought of as a measure of its perfection; if
$I(n)<2$ then $n$ is said to be \textit{deficient}, and if
$I(n)>2$ then $n$ is \textit{abundant}.  In this way, the
abundancy index is a useful tool in gaining a better understanding
of perfect numbers. In fact, the following theorem provides
conditions equivalent to the existence of an odd perfect number
\cite{HoldJ2006}.

\begin{theorem} \label{thm:oddperfequiv}
There exists an odd perfect number if and only if there exist
positive integers $p, n,$ and $\alpha$ such that $p\equiv \alpha
\equiv 1\mod 4$, where $p$ is a prime not dividing $n$, and
\begin{displaymath}
I(n)=\frac{2p^\alpha(p-1)}{p^{\alpha+1}-1}.
\end{displaymath}
\end{theorem}
So, for example, if one could find an integer $n$ having abundancy
index equal to $5/3$, then one would be able to produce an odd
perfect number. Hence, it is useful to try to characterize those
rational numbers in $(1, \infty)$ that do not appear as the
abundancy index of some positive integer.  We will call such
numbers \emph{abundancy outlaws}.


\begin{definition}
A rational number $r/s$ greater than 1 is said to be an
\textit{abundancy outlaw} if $I(x)=r/s$ has no solution among the
positive integers.
\end{definition}

In this paper, we consider the sequence of rational numbers in
$(1,\infty)$.  For each numerator $a > 1$, list the rationals
$a/b$, with gcd$(a,b)=1$, so that denominators $1\leq b < a$
appear in ascending order:
\begin{displaymath}
\frac{2}{1}, \frac{3}{1}, \frac{3}{2}, \frac{4}{1}, \frac{4}{3},
\frac{5}{1}, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}, \frac{6}{1},
\frac{6}{5}, \frac{7}{1}, \frac{7}{2}, \frac{7}{3}, \frac{7}{4},
\frac{7}{5}, \frac{7}{6}, ...
\end{displaymath}
While each term in the sequence is either an abundancy index or an
abundancy outlaw, it is generally difficult to determine the
status of a given rational. The sequence can be partitioned into
three sets: those rationals that are known to be abundancy
indices, those that are known to be outlaws, and those with
abundancy index/outlaw status unknown.  Our goal is to capture
outlaws from the third category, increasing the size of the second
category. Since rationals of the form $(\sigma(N)-t)/N$, with $t
\geq 1$,  are known to be outlaws (see Property 2.3 below), we
consider rational numbers of the form $(\sigma(N)+t)/N$. We prove
that under certain conditions $(\sigma(N)+t)/N$ is an abundancy
outlaw. It is worth noting that our original interest in such
rationals stemmed from our interest in the fraction
$5/3=(\sigma(3)+1)/3$. Unfortunately, our results do not allow us
to say anything about rationals of the form $(\sigma(p)+1)/p$
where $p$ is a prime. Such elusive rationals remain in category
three. Nonetheless, we do prove that $(\sigma(2p)+1)/(2p)$ is an
abundancy outlaw for all primes $p>3$. (Since $I(6)=
(\sigma(2^2)+1)/2^2$ and $I(18)= (\sigma(6)+1)/6$, this provides a
complete characterization of fractions of this form.)

\section{Preliminaries}
It is useful to think of the abundancy index $I$ as a function
mapping the natural numbers $n\geq 2$ into the set of rational
numbers in $(1,\infty)$.  Defining $D$ to be the image of $I$:
\begin{displaymath}
D=\{I(n):n\in \mathbb{N},n\geq 2\},
\end{displaymath}
we can ask many questions about $D$. For instance, how are the
abundancy indices distributed among the set $(1,\infty)$?
Certainly, we can find elements of $D$ arbitrarily close to 1
because $I(p)=(p+1)/p$ for all primes $p$.  Moreover, it is not
hard to show that $I(n!)\geq \sum_i^n 1/i$, and therefore $D$ is
unbounded. In fact, $D$ is dense in $(1,\infty)$
\cite{Laatsch1986}.  Even more interesting, P. Weiner proved that
the set of abundancy outlaws is \emph{also} dense in $(1,\infty)$
\cite{Weiner2000}!  Hence it seems that the situation is both
complex and interesting. For our purposes, the following
properties will be helpful.

\smallskip

\begin{description}
    \item[Property 2.1]  $I(kN) \geq I(N)$ for all natural numbers
$k$ and $N \geq 2$. (See \cite{Laatsch1986}, page 84.)
\medskip
    \item[Property 2.2] If $I(n) = k/m$ with
$\gcd(k,m)=1$, then $m|n$. This follows directly from setting
$\sigma(n)/n = k/m$. Clearly, $m|(nk)$ and since $k$ and $m$ are
relatively prime, it must be that $m|n$.
\medskip
    \item[Property 2.3] If $m  < k < \sigma(m)$ and $k$
is relatively prime to $m$, then $k/m$ is an abundancy outlaw.
Hence if $r/s$ is an abundancy index with $\gcd(r,s)=1$, then $r
\geq \sigma(s)$. (See \cite{Weiner2000}, page 309. The property
also appears in \cite{Anderson1974}.)
\end{description}

\medskip

Property 2.3 reveals a class of abundancy outlaws. Indeed, it was
using this property that Weiner was able to prove that the set of
outlaws is dense in $(1,\infty)$.  It also worth noting that
Property 2.3 implies that $(k+1)/k$ is an abundancy index if and
only if $k$ is prime.  Similarly, $(k+2)/k$ is an abundancy outlaw
whenever $k$ is an odd composite number. If $p$ is a prime greater
than 2, then it is unknown whether $(p+2)/p$ is an outlaw. (See
\cite{Ryan2002}, pages 512-513 for more discussion about this.)

Finally, recent progress has been made by R. Ryan in finding
abundancy outlaws.  In 2002, Ryan produced an example of an
abundancy outlaw not captured by Property 2.3. (See Theorem B.6 in
\cite{Ryan2002}.) Because the conditions describing Ryan's outlaw
are quite technical, we will not restate them here. Nonetheless,
we want to mention that the search for outlaws employed in the
next section was inspired by Ryan's work.

\section{A search for abundancy outlaws}
As Property 2.1 implies, multiplying any number $N = \prod_{i=1}^n
p_i^{k_i}$ by one of its prime divisors, $p_j$, will serve to
increase its abundancy. The following lemma measures this
increase.
\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1,p_2,...,p_n$.
Then
\begin{displaymath}
\frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) - 1} =
\frac{\sigma(p_j N)}{p_j \sigma(N)}
\end{displaymath}
for all $1 \leq j \leq n$.
\end{lemma}

\begin{proof}
The result follows from the fact that
\begin{displaymath}
\begin{array}{lll}
p_j \sigma(N) &=& p_j \sigma(p_j^{k_j}) \sigma(N/p_j^{k_j})\\
              &=& p_j (\sum_{i=1}^{k_j} p_j^i) \sigma(N/p_j^{k_j}) \\
              &=& (\sigma(p_j^{k_j + 1}) - 1) \sigma(N/p_j^{k_j}).
\end{array}
\end{displaymath}
Therefore,
\begin{eqnarray}
\frac{\sigma\left(p_j N\right)}{p_j \sigma(N)} &=&
\frac{\sigma(p_j^{k_j + 1})
\sigma(N/p_j^{k_j})}{(\sigma(p_j^{k_j +
1}) - 1) \sigma(N/p_j^{k_j})}
\\
                                    &=& \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j +
                                    1})-1}.
\end{eqnarray}
\end{proof}
Next we present criteria that can be used in the search for
abundancy outlaws. As the assumptions given in Theorem 3.2 below
indicate, our search focuses on those fractions $r/s$ (in reduced
form) that satisfy $I(N)<r/s<I(p_iN)$ for some prime divisor $p_i$
of a positive integer $N$.  To be more specific, keeping
Properties 2.1 and 2.2 in mind, we look for those values of $s$
having divisors that lead to abundancy values exceeding $I(p_iN)$.

\begin{theorem}
Let $r/s>1$ be a fraction in lowest terms such that there exists a
divisor $N = \prod_{i=1}^n p_i^{k_i}$ of $s$ satisfying the
following two conditions:
\begin{enumerate}
   \item $r/s < I(p_i N)$ for all $i \leq n$
   \item The product \, $\sigma(N) (s/N)$ has a divisor $M$ such that $(M,r) =
   1$ and $I(M) \geq \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) -
              1}$ for some positive integer $j \leq n$.
\end{enumerate}
Then $r/s$ is an abundancy outlaw.
\end{theorem}

\begin{proof}

Let $r/s>1$ be a fraction in lowest terms satisfying the above
hypotheses.  Suppose that $I(x) = r/s$ for some natural number
$x$. Since $r/s$ is in lowest terms, Property 2.2 ensures that $s
| x$. Then, because $N | s$, $N | x$, and therefore, $x = d N$ for
some positive integer $d$. However, since $r/s < I(p_i N)$ for $1
\leq i \leq n$, the first assumption requires that $p_i^{k_i + 1}
\nmid x$ for all $1 \leq i \leq n$, and thus, $d$ is relatively
prime to $N$.
\par

Consequently, we can write
\begin{displaymath}
I(x) = I(d N) = I(d) I(N) = \frac{r}{s}.
\end{displaymath}
Hence,
\begin{displaymath}
\sigma(d)\sigma(N)(s/N) = r d.
\end{displaymath}
Now, by the second assumption, we know that there exists a
positive integer $M$ such that
\begin{displaymath}
M | \sigma(N) (s/N)
\end{displaymath}
and $M$ is relatively prime to $r$.  Therefore, $M | d$ and we can
say that
\begin{displaymath}
I(x) = I(d)I(N) \geq I(M) I(N)
\end{displaymath}
by Property 2.1. Then, by assumption 2) and Lemma 3.1,
\begin{displaymath}
I(M) \geq \frac{\sigma(p_j^{k_j + 1})}{\sigma(p_j^{k_j + 1}) -1} =
\frac{\sigma(p_j N)}{p_j \sigma(N)}
\end{displaymath}
for some $1 \leq j \leq n$, so we know that
\begin{displaymath}
I(x) \geq I(M) I(N) \geq \frac{\sigma(p_j N)}{p_j \sigma(N)}
\frac{\sigma(N)}{N} = \frac{\sigma(p_j N)}{p_j N} = I(p_j N)
\end{displaymath}
Therefore, $I(x) \geq I(p_j N)$, which contradicts our assumption
that $I(x) = r/s < I(p_i N)$ for all $1 \leq i \leq n$.  We
conclude, then, that $r/s$ is an abundancy outlaw.
\end{proof}

\begin{example}
Theorem 3.2 can be used to show that $37/22$ is an abundancy
outlaw. Certainly $37/22 < I(2^2) = 7/4$.  Thus, assumption 1) is
satisfied for $N = 2$.  Next, note that $M = 3$ divides
$\sigma(2)\cdot (22/2)$, and because $gcd(3,37) = 1$, with $I(3)
> \frac{\sigma(4)}{\sigma(4) - 1} = 7/6$, assumption 2) is
satisfied as well.  A computer search reveals many more examples.
(See Table 3.4)
\end{example}

\begin{center}

$
\begin{array}{ccc}

\begin{tabular}{|l|c|l|}
\hline
Abundancy   &   &       \\
outlaw $r/s$  &  $s$   &   $\sigma(s)$ \\
\hline

$   29/12   $   &   $   2^2\cdot 3  $   &   $   28  $   \\
$    37/22  $   &   $    2\cdot 11  $   &   $   36  $   \\
$    43/20  $   &   $    2^2\cdot 5 $   &   $   42  $   \\
$    43/26  $   &   $    2\cdot 13  $   &   $   42  $   \\
$    55/34  $   &   $    2\cdot 17  $   &   $   54  $   \\
$    59/34  $   &   $    2\cdot 17  $   &   $   54  $   \\
$    61/24  $   &   $    2^3\cdot 3 $   &   $   60  $   \\
$    61/38  $   &   $    2\cdot 19  $   &   $   60  $   \\
$    65/38  $   &   $    2\cdot 19  $   &   $   60  $   \\
$    73/30  $   &   $    2\cdot 3\cdot 5    $   &   $   72  $   \\
$    73/46  $   &   $    2\cdot 23  $   &   $   72  $   \\
$    73/51  $   &   $    3\cdot 17  $   &   $   72  $   \\
$    77/46  $   &   $    2\cdot 23  $   &   $   72  $   \\
$    79/45  $   &   $    3^2\cdot 5 $   &   $   78  $   \\
$    79/46  $   &   $    2\cdot 23  $   &   $   72  $   \\
$    91/40  $   &   $    2^3\cdot 5 $   &   $   90  $   \\
$    91/58  $   &   $    2\cdot 29  $   &   $   90  $   \\
$    95/58  $   &   $    2\cdot 29  $   &   $   90  $   \\
$    97/42  $   &   $    2\cdot 3\cdot 7    $   &   $   96  $   \\
$    97/58  $   &   $    2\cdot 29  $   &   $   90  $   \\
$    97/62  $   &   $    2\cdot 31  $   &   $   96  $   \\
$    97/69  $   &   $    3\cdot 23  $   &   $   96  $   \\
$    101/58 $   &   $    2\cdot 29  $   &   $   90  $   \\
$    101/62 $   &   $    2\cdot 31  $   &   $   96  $   \\
$    103/62 $   &   $    2\cdot 31  $   &   $   96  $   \\
$    107/62 $   &   $    2\cdot 31  $   &   $   96  $   \\
$    115/74 $   &   $    2\cdot 37  $   &   $   114 $   \\
$    119/74 $   &   $    2\cdot 37  $   &   $   114 $   \\
$    121/56 $   &   $    2^3\cdot 7 $   &   $   120 $   \\
$    121/74 $   &   $    2\cdot 37  $   &   $   114 $   \\
$    121/87 $   &   $    3\cdot 29  $   &   $   120 $   \\
$    125/48 $   &   $    2^4\cdot 3 $   &   $   124 $   \\


\hline
\end{tabular}
& \hspace{1cm} &
\begin{tabular}{|l|c|l|}
\hline
Abundancy   &   &       \\
outlaw $r/s$  &  $s$   &   $\sigma(s)$ \\
\hline

$    125/74 $   &   $    2\cdot 37  $   &   $   114 $   \\
$    125/87 $   &   $    3\cdot 29  $   &   $   120 $   \\
$    127/68 $   &   $    2^2\cdot 17    $   &   $   126 $   \\
$    127/74 $   &   $    2\cdot 37  $   &   $   114 $   \\
$    127/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    131/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    131/93 $   &   $    3\cdot 31  $   &   $   128 $   \\
$    133/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    133/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    133/93 $   &   $    3\cdot 31  $   &   $   128 $   \\
$    137/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    137/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    139/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    139/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    141/76 $   &   $    2^2\cdot 19    $   &   $   140 $   \\
$    143/82 $   &   $    2\cdot 41  $   &   $   126 $   \\
$    143/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    145/66 $   &   $    2\cdot 3\cdot 11   $   &   $   144 $   \\
$    145/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    145/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    149/86 $   &   $    2\cdot 43  $   &   $   132 $   \\
$    149/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    151/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    155/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    155/111    $   &   $    3\cdot 37  $   &   $   152 $   \\
$    157/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    157/99 $   &   $    3^2\cdot 11    $   &   $   156 $   \\
$    157/111    $   &   $    3\cdot 37  $   &   $   152 $   \\
$    161/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    163/94 $   &   $    2\cdot 47  $   &   $   144 $   \\
$    163/106    $   &   $    2\cdot 53  $   &   $   162 $   \\
$    167/106    $   &   $    2\cdot 53  $   &   $   162 $   \\




\hline
\end{tabular}
\end{array}
$

\vspace{0.3cm} \textbf{Table 3.4}:  A list of abundancy outlaws
found using Theorem 3.2.  These outlaws are captured by Theorem 3.2,
but not by Property 2.3.

\end{center}




\section{The main results}

Table 3.4 reveals some recognizable patterns.  Most obvious is the
indication that, for small odd values of $t$ (and $p$ prime),
reduced rational numbers of the form

\begin{displaymath}
\frac{r}{s}=\frac{\sigma(2^mp^n)+t}{2^mp^n}
\end{displaymath}
often lead to abundancy outlaws.  In fact, the results of this
section will show that $\frac{\sigma(2^mp^{2n+1})+1}{2^mp^{2n+1}}$
is an abundancy outlaw whenever gcd$(p, \sigma(2^m))=1$. In
particular, if $p>3$, then $(\sigma(2p)+1)/(2p)$ is
\textit{always} an abundancy outlaw. Our main result, however,
will address the more general situation where $r/s$ is a reduced
rational number of the form $(\sigma(N)+t)/N$. First we will need
a lemma.

\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$, where $p_i$ is a prime for all
$1 \leq i \leq n$.  Then, for a given $1 \leq j \leq n$ and a
positive integer $t$,
\begin{displaymath}
p_j < \frac{1}{t}\sigma\left(\frac{N}{p_j^{k_j}}\right)
\end{displaymath}
if and only if
\begin{displaymath}
\frac{\sigma(N) + t}{N} < I(p_j N).
\end{displaymath}
\end{lemma}

\begin{proof}
Assume that
\begin{displaymath}
\frac{\sigma(N) + t}{N} < I(p_j N)
\end{displaymath}
for a given natural number $j \leq n$.  This is equivalent to
\begin{displaymath}
p_j \sigma(N) + p_j t < \sigma(p_j N).
\end{displaymath}

Then, since $p_j \sigma(p_j^{k_j}) = \sigma(p_j^{k_j + 1}) - 1$,
we find that this inequality is equivalent to
\begin{displaymath}
\begin{array}{lll}
(\sigma(p_j^{k_j + 1}) - 1)\sigma\left(\frac{N}{p_j^{k_j}}\right)+
p_j t & < & \sigma(p_j N),
\end{array}
\end{displaymath}
or
\begin{displaymath}
\begin{array}{lll}
\sigma(p_j N) - \sigma\left(\frac{N}{p_j^{k_j}}\right) + p_j t & <
& \sigma(p_j N).
\end{array}
\end{displaymath}

 Therefore,
\begin{displaymath}
p_j < \frac{1}{t}\sigma\left(\frac{N}{p_j^{k_j}}\right).
\end{displaymath}

\end{proof}

We are now ready to present the main result.

\begin{theorem}
For a positive integer $t$, let $\frac{\sigma(N) + t}{N}$ be a
fraction in lowest terms, and let $N = \prod_{i=1}^n p_i^{k_i}$
for primes $p_1,p_2,...,p_n$. If there exists a positive integer
$j \leq n$ such that $p_j < \frac{1}{t} \sigma(N/p_j^{k_j})$ and
$\sigma(p_j^{k_j})$ has a divisor $D>1$ such that at least one of
the following is true:

\begin{enumerate}
  \item $I(p_j^{k_j}) I(D) > \frac{\sigma(N)
  + t}{N}$ and $gcd(D,t) = 1$
  \item $gcd(D,Nt) = 1$
\end{enumerate}

\noindent then $\frac{\sigma(N) + t}{N}$ is an abundancy outlaw.

\end{theorem}

\begin{proof}

Case 1: Let $N$ and $t$ be natural numbers such that
$\frac{\sigma(N) + t}{N}$ is a fraction in lowest terms, and let
$j \leq n$ be a natural number satisfying $p_j < \frac{1}{t}
\sigma(N/p_j^{k_j})$ and $D$ a divisor of $\sigma(p_j^{k_j})$
satisfying hypothesis (1). Suppose further that $I(x) =
\frac{\sigma(N)+t}{N}$ for some natural number $x$. Since
$(\sigma(N) + t,N) = 1$, $N | x$ by Property 2.2. Say $x = d N$,
where $d$ is a positive integer.  Since $p_j$ satisfies $p_j <
\frac{1}{t}\sigma(N/p_j^{k_j})$, Lemma 4.1 implies that $I(x) =
\frac{\sigma(N) +t}{N} < I(p_j N)$.  Hence, $gcd(p_j^{k_j}, d
N/p_j^{k_j}) = 1$ and
\begin{displaymath}
I(x) = I(p_j^{k_j}) I(d N/p_j^{k_j}) = \frac{\sigma(N) + t}{N}.
\end{displaymath}
Equivalently,
\begin{displaymath}
\sigma(p_j^{k_j}) \sigma(d N/p_j^{k_j}) = (\sigma(N) + t) d.
\end{displaymath}

Since $\sigma(p_j^{k_j}) | \sigma(N)$ and $gcd(D,t) = 1$, the
divisor $D$ of $\sigma(p_j^{k_j})$ satisfying hypothesis (1) also
divides $d$.  Hence $D$ divides $d N/p_j^{k_j}$, and by Property
2.1, $I(D) \leq I(d N/p_j^{k_j})$. Thus
\begin{displaymath}
I(p_j^{k_j}) I(D) \leq I(p_j^{k_j}) I(dN/p_j^{k_j}) = I(x).
\end{displaymath}

Given $I(x) = \frac{\sigma(N) + t}{N}$, we conclude then that
\begin{displaymath}
I(p_j^{k_j})I(D) \leq \frac{\sigma(N) + t}{N}.
\end{displaymath}
This contradicts the assumption that $I(p_j^{k_j}) I(D) >
\frac{\sigma(N) + t}{N}$.  Therefore, $\frac{\sigma(N) + t}{N}$ is
an
abundancy outlaw.\\

Case 2:  Now let $N=\prod_{i=1}^n p_i^{k_i}$ and $t$ be natural
numbers such that $\sigma(p_j)^{k_j}$ has a divisor $D$ satisfying
hypothesis (2) for some $1\leq j\leq n$, and assume that $I(x) =
\frac{\sigma(N) + t}{N}$ for some natural number $x$. Since
$\frac{\sigma(N) + t}{N}$ is in lowest terms, $N | x$, so that $x
= s N$ for some natural number $s$.  By assumption, there exists a
natural number $j \leq n$ so that $p_j < \frac{1}{t}
\sigma(N/p_j^{k_j})$. By Lemma 4.1, $I(x) = \frac{\sigma(N) +
t}{N} < I(p_j N)$.  Hence, $p_j \nmid s$, so that
\begin{displaymath}
\begin{array}{lll}
I(x) &=& I\left(p_j^{k_j} s \frac{N}{p_j^{k_j}}\right) \\
     &=& I(p_j^{k_j}) I\left(s \frac{N}{p_j^{k_j}}\right).
\end{array}
\end{displaymath}
Next, we factor out the part of $s$, $\prod_{i=1}^n
p_i^{\gamma_i}$, that has divisors in common with $N/p_j^{k_j}$,
so that
\begin{displaymath}
\overline{s} = \frac{s}{\prod_{i=1}^{n} p_i^{\gamma_i}},
\end{displaymath}
where $\gamma_i$ is a non-negative integer for all $i \leq n$.\par

Then, we can rewrite $I(x)$ once more in the following form:
\begin{displaymath}
I(x) = I(p_j^{k_j}) I(\overline{s}) I\left(\frac{N}{p_j^{k_j}}
\prod_{i=1}^n p_i^{\gamma_i}\right)
\end{displaymath}
Because $I(x) = \frac{\sigma(N) + t}{N}$, we can see, then, that
\begin{displaymath}
\begin{array}{lll}
I(p_r^{k_r})I(\overline{s}) I\left(\frac{N}{p_r^{k_r}}
\prod_{i=1}^n p_i^{\gamma_i} \right) &=& \frac{\sigma(N) + t}{N},
\end{array}
\end{displaymath}
or equivalently,
\begin{displaymath}
\begin{array}{lll} \sigma(p_r^{k_r})
\sigma(\overline{s}) \sigma\left(\frac{N}{p_r^{k_r}} \prod_{i=1}^n
p_i^{\gamma_i} \right) &=& (\sigma(N) + t) \overline{s}
\prod_{i=1}^n p_i^{\gamma_i}.
\end{array}
\end{displaymath}
Now consider the divisor $D$ of $\sigma(p_j^{k_j})$ satisfying
hypothesis (2). Since $\sigma(p_j^{k_j}) | \sigma(N)$ and
$gcd(D,Nt) = 1$, $D$ divides $\overline{s}$ (so $\overline{s}>1)$.
This, then, means that $I(\overline{s}) \geq I(D)$. Then, since
$p_j < \frac{1}{t} \sigma(N/p_j^{k_j})$,
\begin{equation*}
p_j \sigma(p_j^{k_j}) < \frac{1}{t} \sigma(N),
\end{equation*}
which implies that the following is true:
\begin{equation*}
\frac{1}{p_j \sigma(p_j^{k_j})} > \frac{t}{\sigma(N)}.
\end{equation*}
Thus, since $D | \sigma(p_j^{k_j})$, $D < p_j \sigma(p_j^{k_j})$,
so that $1/D > 1/p_j \sigma(p_j^{k_j})$.

Hence, the following are true:
\begin{eqnarray*}
I(\overline{s}) &\geq & I(D) \\
                &\geq & 1 + \frac{1}{D} \\
                &>& 1 + \frac{1}{p_j \sigma(p_j^{k_j})} \\
                &>&  1 + \frac{t}{\sigma(N)} \\
                &=&  \frac{\sigma(N) + t}{\sigma(N)} \\
                &=&  \frac{\sigma(N) + t}{I(p_j^{k_j}) I(N/p_j^{k_j})
                N}\\
                &\geq&  \frac{\sigma(N) + t}{I(p_j^{k_j}) I((N/p_j^{k_j})\prod_{i=1}^n p_i^{\gamma_i}) N}.
\end{eqnarray*}
Hence
\begin{displaymath}
I(p_j^{k_j})I(\overline{s}) I\left(\frac{N}{p_j^{k_j}}
\prod_{i=1}^n p_i^{\gamma_i} \right) = I(x) > \frac{\sigma(N) +
t}{N},
\end{displaymath}
which contradicts our original assumption that
$I(x)=\frac{\sigma(N)+t}{N}$. Thus, $\frac{\sigma(N) + t}{N}$ is
an abundancy outlaw.

\end{proof}


\vspace{1cm}

If $t = 1$ we get the following corollary.

\begin{corollary}
Let $\frac{\sigma(N) + 1}{N}$ be a fraction in lowest terms, and
let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1,p_2,...,p_n$. If
there exists a natural number $j \leq n$ such that $p_j <
\sigma(N/p_j^{k_j})$ and $\sigma(p_j^{k_j})$ has a divisor $D$
such that at least one of the following is true:

\begin{enumerate}
  \item $I(p_j^{k_j}) I(D) > \frac{\sigma(N)
  + 1}{N}$
  \item $gcd(D,N) = 1$
\end{enumerate}

then $\frac{\sigma(N) + 1}{N}$ is an abundancy outlaw.
\end{corollary}

\section{Constructing Sequences of Abundancy Outlaws}
Using the results in the previous section, we can find and
construct sequences of abundancy outlaws.  The following lemma
will be helpful.

\begin{lemma}
Let $N = \prod_{i=1}^n p_i^{k_i}$ for primes $p_1, p_2,...,p_n$.
Then $N$ is relatively prime to $\sigma(N) + 1$ if and only if
$p_i$ is relatively prime to $\sigma(N/p_i^{k_i}) + 1$ for all $1
\leq i \leq n$.
\end{lemma}

\begin{proof}
Since $\sigma(p_i^{k_i}) \equiv 1 \; (mod \; p_i)$,
\begin{displaymath}
\begin{array}{lll}
\sigma(N) + 1 &=& \sigma(p_i^{k_i})\sigma(N/p_i^{k_i}) + 1 \\
              &\equiv& \sigma(N/p_i^{k_i}) + 1 \; (mod \; p_i).
\end{array}
\end{displaymath}
Thus, any prime divisor $p_i$ of $N$ and $\sigma(N) + 1$ must also
be a prime divisor of $\sigma(N/p_i^{k_i}) + 1$, and conversely,
if $p_i | (\sigma(N/p_i^{k_i}) + 1)$, then $p_i$ divides both $N$
and $\sigma(N) + 1$.
\end{proof}

\subsection*{Outlaws with even denominators}

\begin{corollary}
For all natural numbers $m$ and nonnegative integers $n$, and for
all odd primes $p$ such that $gcd(p,\sigma(2^m))=1$, the rational
number
\begin{equation*}
\frac{\sigma(2^m p^{2n + 1}) + 1}{2^m p^{2n + 1}}
\end{equation*}
is an abundancy outlaw.
\end{corollary}

\begin{proof}
To see that $(\sigma(2^m p^{2n+1})+1)/2^m p^{2n+1}$ is in lowest
terms, apply Lemma 5.1.  Since $p$ and $2n+1$ are odd,
$\sigma(p^{2n+1}) + 1$ is odd, so $gcd(2^m, \sigma(p^{2n+1}) + 1)
= 1$.  Also, $gcd(p^{2n+1}, \sigma(2^m) + 1) = 1$, because
$\sigma(2^m) + 1 = 2^{m+1}$.  Next, we apply Corollary 4.3. Since
$2 < \sigma(p^{2n+1})$, we consider divisors $D$ of $\sigma(2^m)$.
Clearly, $gcd(\sigma(2^m), 2^m) = 1$, and because $p$ does not
divide $\sigma(2^m)$, $gcd(\sigma(2^m), 2^m p^{2n+1}) = 1$. Hence,
$D = \sigma(2^m)$ is the divisor required to apply Corollary 4.3.
  Thus, $(\sigma(2^m p^{2n+1}))/2^m p^{2n+1}$
is an abundancy outlaw.\\
\end{proof}

\begin{corollary}
For all primes $p > 3$,
\begin{displaymath}
\frac{\sigma(2p) + 1}{2p}
\end{displaymath}
is an abundancy outlaw.  If $p=2$ or $p=3$ then $\frac{\sigma(2p)
+ 1}{2p}$ is an abundancy index.
\end{corollary}

\begin{proof}
This result follows directly from Corollary 5.2.  To prove that
$\frac{\sigma(2p) + 1}{2p}$ is an abundancy index when $p=2$ and
$p=3$, note that $I(6)=\frac{\sigma(2^2) + 1}{2^2}$ and
$I(18)=\frac{\sigma(6) + 1}{6}$.
\end{proof}

\begin{remark}
Corollary 5.3 actually has a very simple proof (and in fact, the
results presented in section 4 represent our attempt to push this
simple proof as far as possible). Clearly, $(\sigma(2p) + 1)/(2p)
= (3p + 4)/(2p)$ is in lowest terms.  Thus, if $I(N) = (\sigma(2p)
+ 1)/2p$, $2p | N$. Because $p > 3$, it can be shown that $I(4p) >
(\sigma(2p) + 1)/2p$, so $4 \nmid N$.  Therefore, $\sigma(2) |
\sigma(N)$, and since $\sigma(2) = 3$ does not divide $\sigma(2p)
+ 1$, $3$ divides $N$.  Hence, $I(N) > I(6 p) > 2 > (\sigma(2p) +
1)/2p$. This is a contradiction, so $(\sigma(2p) + 1)/2p$ is an
abundancy outlaw, and we have captured the following sequence of
outlaws:
\begin{displaymath}
\frac{19}{10}, \frac{25}{14}, \frac{37}{22}, \frac{43}{26},
\frac{55}{34}, \frac{61}{38}, \frac{73}{46}, \frac{91}{58},
\frac{97}{62}, \frac{115}{74}, \frac{127}{82}, \frac{133}{86},
\frac{145}{94}, \frac{163}{106}, \frac{181}{118},
\frac{187}{122},...
\end{displaymath}
\end{remark}


Corollary 5.3 also captures another (potentially infinite) set of
outlaws having even denominators...

\begin{corollary}
If $N$ is an even perfect number,
\begin{equation*}
\frac{\sigma(2N) + 1}{2N}
\end{equation*}
is an abundancy outlaw.
\end{corollary}

\begin{proof}
Since $N$ is an even perfect number, $N = 2^{p-1} (2^p - 1)$,
where $p$ and $2^p - 1$ are both prime.  Hence,
\begin{equation*}
2N = 2^p (2^p - 1).
\end{equation*}
Applying Corollary 5.2, we need only show that $gcd(2^p - 1,
\sigma(2^p)) = 1$.  This follows from the fact that
\begin{equation*}
2(2^p - 1) = (2^{p+1} - 1) - 1,
\end{equation*}
which is clearly relatively prime to $\sigma(2^p) = 2^{p+1} - 1.$
Therefore, $(\sigma(2N) + 1)/2N$ is an abundancy outlaw.
\end{proof}

\subsection*{Outlaws with odd denominators}


\begin{corollary}
Let $M$ be an odd natural number, and let $p$, $\alpha$, and $t$
be odd natural numbers such that $p \nmid M$ and  $p < \frac{1}{t}
\sigma(M)$.  Then, if $(\sigma(p^\alpha M) + t)/p^\alpha M$ is in
lowest terms,
\begin{equation*}
\frac{\sigma(p^\alpha M) + t}{p^\alpha M}
\end{equation*}
is an abundancy outlaw.
\end{corollary}

\begin{proof}
Since $(\sigma(p^\alpha M) + t)/p^\alpha M$ is in lowest terms, we
apply Theorem 4.2 to show that it is an abundancy outlaw.  Since
$p$ and $\alpha$ are both odd, $\sigma(p^{\alpha})$ is even.
Hence, $D = 2$ divides $\sigma(p^{\alpha})$, and since $p$, $M$,
and $t$ are all odd, $gcd(D, p M t) = 1$.  Thus, $(\sigma(p^\alpha
M) + t)/p^\alpha M$ is an abundancy outlaw.
\end{proof}

\begin{corollary}
For primes $p$ and $q$, with $3 < q$, $p < q$, and $gcd(p,q+2) =
gcd(q,p+2) = 1$,
\begin{displaymath}
\frac{\sigma(pq) + 1}{pq}
\end{displaymath}
is an abundancy outlaw.
\end{corollary}

\begin{proof}
The case for $p = 2$ follows from Corollary 5.3.  Now suppose that
$q > p > 2$.  In this case, neither $p$ nor $q$ divides
$\sigma(pq) + 1 = pq + p + q + 2$, because $gcd(p,q+2) =
gcd(q,p+2) = 1$. Thus, $\frac{\sigma(pq) + 1}{pq}$ is in lowest
terms. The result now follows from Corollary 5.6.
\end{proof}

Corollary 5.7 produces outlaws with ease.  To illustrate, let $p$
and $q$ be odd primes with $3 < p < q$, and assume $q \equiv 1 \;
(mod \; p)$.  Then $p \nmid q + 2$ and $q \nmid p + 2$. Since
Dirichlet's theorem on arithmetic progressions of primes ensures
the existence of an infinite sequence of primes $q$ satisfying $q
\equiv 1 \; (mod  \; p)$, Corollary 5.7 reveals an infinite class
of outlaws corresponding to each odd prime $p>3$. The sequences
corresponding to the primes 5, 7, and 11 follow.

\begin{description}
    \item[$p=5$ : ] $\frac{73}{55}, \frac{193}{155}, \frac{253}{205}, \frac{373}{305}, \frac{433}{355}, \frac{613}{505}, \frac{793}{655},
    \frac{913}{755}, \frac{1093}{905}, \frac{1153}{955}, \frac{1273}{1055}, \frac{1513}{1255},\frac{1633}{1355},\frac{1693}{1405},
    \frac{1873}{1555},\frac{1993}{1655},\frac{2413}{2005}, \frac{2533}{2105},...$
    \item[$p=7$ : ] $\frac{241}{203}, \frac{353}{301}, \frac{577}{497}, \frac{913}{791}, \frac{1025}{889}, \frac{1585}{1379}, \frac{1697}{1477}, \frac{1921}{1673}, \frac{2257}{1967}, \frac{2705}{2359}, \frac{3041}{2653},
    \frac{3377}{2947}, \frac{3601}{3143}, \frac{3713}{3241}, \frac{3937}{3437},\frac{4385}{3829},\frac{4945}{4319},...$
    \item[$p=11$ :] $\frac{289}{253}, \frac{817}{737}, \frac{1081}{979}, \frac{2401}{2189}, \frac{3985}{3641}, \frac{4249}{3883}, \frac{4777}{4367},
    \frac{5041}{4609}, \frac{5569}{5093}, \frac{7417}{6787},\frac{7945}{7271},\frac{8209}{7513},\frac{8737}{7997},
    \frac{10321}{9449},\frac{10585}{9691},\frac{11377}{10417},...$
\end{description}




\begin{remark}
It is interesting to note that if $p$ and $q = p+2$ are twin
primes then Corollary 5.7 does not apply, and in this case
$$\frac{\sigma(p(p+2))+1}{p(p+2)}=\frac{\sigma(p)+1}{p}=\frac{p+2}{p}.$$
Once again, we are faced with that elusive ratio $\frac{p+2}{p}$!
\end{remark}


\section{Further explorations}

R. Ryan considered the equation
\begin{equation}
I(x) = \frac{p+2}{p} = \frac{\sigma(p) + 1}{p},
\end{equation}
where $p$ is an odd prime \cite{Ryan2002}.  Whether or not a
solution $x$ exists is still an open question. This problem is
interesting for two reasons. First, $\frac{p+2}{p}$ is just barely
out of reach of Weiner's 2000 result, since $p + 2 = \sigma(p) +
1$ for all primes $p$. Second, as we have already mentioned, if
$\frac{5}{3} = \frac{3+2}{3}$ is an abundancy index then there
exists an odd perfect number.

It seems to be just as difficult to find a solution to the above
equation as it is to show that no such $x$ exists.  R. Ryan
reports that $I(x) = \frac{p+2}{p}$ has no solution for $x <
10^{16}$ \cite{Ryan2002}. The investigations that led us to
Theorems 3.2 and 4.2 were motivated in part by a desire to show
that equation 5.1 has no solutions. However, $\frac{p+2}{p}$ has
proven to be an elusive fraction. The techniques we have employed
in this paper cannot be applied to it. The difficulty lies in the
fact that
\begin{displaymath}
\frac{p+2}{p} > \frac{p}{p-1} > I(p^{\alpha})
\end{displaymath}
for all primes $p$ and natural numbers $\alpha$.  Thus, one can
never ``trap" $\frac{p+2}{p}$ between two numbers:
\begin{displaymath}
I(p^{\alpha}N)<\frac{p+2}{p} < I(p^{\alpha+1}N).
\end{displaymath}
Hence, Theorems 3.2 and 4.2 fail to capture this potential outlaw.
In fact, proving that $\frac{p+2}{p}$ is an abundancy outlaw seems
to require the discovery of an entirely new category of abundancy outlaws.\\

There are also many interesting questions to consider relating to
the size of these sets of abundancy outlaws, in the sense of
asymptotic density. Based on some preliminary computer
experiments, the proportion of outlaws captured by Theorem 4.2
seems to approach $1.7$ percent.  See the appendix for empirical
data. Our future work will involve a deeper exploration of the
sizes of the sets of outlaws, indices, and rationals of unknown
status.

\color{black}{
\section{Appendix}
The following is a table containing empirical data relating to the
asymptotic density of the set of outlaws captured by Theorem 3.4.
The second and third columns give the number of outlaws with
numerators less than or equal to $n$ captured by Theorem 3.4 and
Proposition 2.3, respectively. The fourth column gives the number
of rationals in the image of the abundancy index of the first one
million natural numbers with numerators less than or equal to $n$.
The fifth column gives the total number of rationals with
numerators less than or equal to $n$, and the sixth column gives
the value of column 2 divided by column 5.  In other words, the
sixth column gives the proportion of outlaws captured by Theorem
3.4 in the set of rationals with numerator less than or equal to
$n$.

\begin{tabular}{|l|l|l|l|l|l|}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \hline
  $n$ & Thm. 3.4 & Prop. 2.3 & Abundancy Index & Total rationals & Proportion \\
  \hline
  10 & 0 & 4 & 20 & 32 & 0 \\
  100 & 44 & 720 & 553 & 3044 & 0.0145 \\
  1000 & 5170 & 74927 & 7803 & 304192 & 0.01700 \\
  10000 & 518193 & 750174 & 62064 & 30397486 & 0.01705 \\
  \hline
\end{tabular}

\vspace{0.3cm} \textbf{Table 7.1}:  A table of empirical data on
the asymptotic densities of the abundancy outlaws and the
abundancy indices.\\ \\


As a way to visualize the distribution of abundancy outlaws, we
include a list of the rationals with numerators less than or equal
to 100 (under the ordering described in the introduction). Each
rational number $q$ is colored according to its abundancy
index/outlaw status:}

\color{black}
\begin{itemize}
\color{black}
  \item  \color{blue} {Blue}:  \color{black}$q$ is in the image of the abundancy index of the first one million natural numbers, or a natural number from $2$ to $11$ (the abundancy index of a known multiperfect number)

  \item \color{green} {Green}: \color{black}  $q$ is an outlaw captured by Property 2.3.

  \item \color{red} {Red}:  \color{black} $q$ is an outlaw captured by Theorem 4.2.

  \item \color{black} {Black}: \color{black} $q$ is not in any of the other categories (the outlaw status of $q$ is ``unknown")
\end{itemize}

\noindent In the following list, $450$ rationals are blue, $720$ are green, $44$ are red, and $1961$ are black.\\ \\

\noindent \color{blue}{2}, \color{blue}{3}, \color{blue}{3/2},
\color{blue}{4}, \color{blue}{4/3}, \color{blue}{5},
\color{blue}{5/2}, \color{black}{5/3}, \color{green}{5/4},
\color{blue}{6}, \color{blue}{6/5}, \color{blue}{7},
\color{blue}{7/2}, \color{blue}{7/3}, \color{blue}{7/4},
\color{black}{7/5}, \color{green}{7/6}, \color{blue}{8},
\color{blue}{8/3}, \color{blue}{8/5}, \color{blue}{8/7},
\linebreak
 \color{blue}{9}, \color{black}{9/2}, \color{blue}{9/4}, \color{blue}{9/5}, \color{black}{9/7}, \color{green}{9/8}, \color{blue}{10}, \color{blue}{10/3}, \color{black}{10/7}, \color{green}{10/9}, \color{blue}{11}, \color{black}{11/2}, \color{blue}{11/3}, \color{blue}{11/4}, \color{black}{11/5}, \color{green}{11/6}, \color{black}{11/7},  \linebreak
 \color{green}{11/8}, \color{green}{11/9}, \color{green}{11/10}, \color{black}{12}, \color{blue}{12/5}, \color{blue}{12/7}, \color{blue}{12/11}, \color{black}{13}, \color{black}{13/2}, \color{black}{13/3}, \color{blue}{13/4}, \color{blue}{13/5}, \color{blue}{13/6}, \color{black}{13/7}, \color{green}{13/8},  \linebreak
  \color{blue}{13/9}, \color{green}{13/10}, \color{black}{13/11}, \color{green}{13/12}, \color{black}{14}, \color{black}{14/3}, \color{blue}{14/5}, \color{blue}{14/9}, \color{black}{14/11}, \color{blue}{14/13}, \color{black}{15}, \color{black}{15/2}, \color{blue}{15/4}, \color{blue}{15/7}, \color{blue}{15/8},  \linebreak
\color{black}{15/11}, \color{black}{15/13}, \color{green}{15/14},
\color{black}{16}, \color{black}{16/3}, \color{blue}{16/5},
\color{blue}{16/7}, \color{blue}{16/9}, \color{blue}{16/11},
\color{blue}{16/13}, \color{green}{16/15}, \color{black}{17},
\color{black}{17/2}, \color{black}{17/3}, \color{blue}{17/4},
\linebreak
 \color{blue}{17/5}, \color{black}{17/6}, \color{black}{17/7}, \color{black}{17/8}, \color{black}{17/9}, \color{green}{17/10}, \color{black}{17/11}, \color{green}{17/12}, \color{black}{17/13}, \color{green}{17/14}, \color{green}{17/15}, \color{green}{17/16}, \color{black}{18}, \color{blue}{18/5},  \linebreak
\color{blue}{18/7}, \color{blue}{18/11}, \color{black}{18/13},
\color{blue}{18/17}, \color{black}{19}, \color{black}{19/2},
\color{black}{19/3}, \color{black}{19/4}, \color{blue}{19/5},
\color{blue}{19/6}, \color{blue}{19/7}, \color{black}{19/8},
\color{black}{19/9}, \color{red}{19/10}, \color{black}{19/11},
\linebreak \color{green}{19/12}, \color{black}{19/13},
\color{green}{19/14}, \color{green}{19/15}, \color{green}{19/16},
\color{black}{19/17}, \color{green}{19/18}, \color{black}{20},
\color{black}{20/3}, \color{blue}{20/7}, \color{blue}{20/9},
\color{black}{20/11}, \color{black}{20/13}, \color{black}{20/17},
\linebreak \color{blue}{20/19}, \color{black}{21},
\color{black}{21/2}, \color{black}{21/4}, \color{blue}{21/5},
\color{blue}{21/8}, \color{blue}{21/10}, \color{blue}{21/11},
\color{blue}{21/13}, \color{green}{21/16}, \color{black}{21/17},
\color{black}{21/19}, \color{green}{21/20}, \color{black}{22},
\linebreak
 \color{black}{22/3}, \color{blue}{22/5}, \color{blue}{22/7}, \color{blue}{22/9}, \color{black}{22/13}, \color{green}{22/15}, \color{black}{22/17}, \color{black}{22/19}, \color{green}{22/21}, \color{black}{23}, \color{black}{23/2}, \color{black}{23/3}, \color{black}{23/4}, \color{black}{23/5},  \linebreak
 \color{black}{23/6}, \color{black}{23/7}, \color{black}{23/8}, \color{black}{23/9}, \color{black}{23/10}, \color{black}{23/11}, \color{green}{23/12}, \color{black}{23/13}, \color{green}{23/14}, \color{green}{23/15}, \color{green}{23/16}, \color{black}{23/17}, \color{green}{23/18},  \linebreak
\color{black}{23/19}, \color{green}{23/20}, \color{green}{23/21},
\color{green}{23/22}, \color{black}{24}, \color{black}{24/5},
\color{blue}{24/7}, \color{blue}{24/11}, \color{blue}{24/13},
\color{blue}{24/17}, \color{blue}{24/19}, \color{blue}{24/23},
\color{black}{25}, \color{black}{25/2},  \linebreak
 \color{black}{25/3}, \color{black}{25/4}, \color{black}{25/6}, \color{blue}{25/7}, \color{blue}{25/8}, \color{blue}{25/9}, \color{black}{25/11}, \color{green}{25/12}, \color{black}{25/13}, \color{red}{25/14}, \color{green}{25/16}, \color{black}{25/17}, \color{green}{25/18}, \color{black}{25/19},  \linebreak
 \color{green}{25/21}, \color{green}{25/22}, \color{black}{25/23}, \color{green}{25/24}, \color{black}{26}, \color{black}{26/3}, \color{black}{26/5}, \color{blue}{26/7}, \color{blue}{26/9}, \color{blue}{26/11}, \color{blue}{26/15}, \color{blue}{26/17}, \color{black}{26/19}, \color{green}{26/21},  \linebreak
 \color{black}{26/23}, \color{green}{26/25}, \color{black}{27}, \color{black}{27/2}, \color{black}{27/4}, \color{black}{27/5}, \color{blue}{27/7}, \color{blue}{27/8}, \color{blue}{27/10}, \color{blue}{27/11}, \color{black}{27/13}, \color{black}{27/14}, \color{green}{27/16}, \color{blue}{27/17},  \linebreak
 \color{black}{27/19}, \color{green}{27/20}, \color{green}{27/22}, \color{black}{27/23}, \color{green}{27/25}, \color{green}{27/26}, \color{black}{28}, \color{black}{28/3}, \color{black}{28/5}, \color{blue}{28/9}, \color{blue}{28/11}, \color{blue}{28/13}, \color{blue}{28/15}, \color{blue}{28/17},  \linebreak
\color{black}{28/19}, \color{black}{28/23}, \color{green}{28/25},
\color{green}{28/27}, \color{black}{29}, \color{black}{29/2},
\color{black}{29/3}, \color{black}{29/4}, \color{black}{29/5},
\color{black}{29/6}, \color{black}{29/7}, \color{black}{29/8},
\color{black}{29/9}, \color{black}{29/10},  \linebreak
 \color{black}{29/11}, \color{red}{29/12}, \color{black}{29/13}, \color{black}{29/14}, \color{black}{29/15}, \color{green}{29/16}, \color{black}{29/17}, \color{green}{29/18}, \color{black}{29/19}, \color{green}{29/20}, \color{green}{29/21}, \color{green}{29/22}, \color{black}{29/23},  \linebreak
 \color{green}{29/24}, \color{green}{29/25}, \color{green}{29/26}, \color{green}{29/27}, \color{green}{29/28}, \color{black}{30}, \color{black}{30/7}, \color{blue}{30/11}, \color{blue}{30/13}, \color{black}{30/17}, \color{blue}{30/19}, \color{black}{30/23}, \color{blue}{30/29}, \color{black}{31},  \linebreak
\color{black}{31/2}, \color{black}{31/3}, \color{black}{31/4},
\color{black}{31/5}, \color{black}{31/6}, \color{black}{31/7},
\color{blue}{31/8}, \color{blue}{31/9}, \color{blue}{31/10},
\color{blue}{31/11}, \color{blue}{31/12}, \color{blue}{31/13},
\color{blue}{31/14}, \color{black}{31/15},  \linebreak
 \color{blue}{31/16}, \color{black}{31/17}, \color{green}{31/18}, \color{black}{31/19}, \color{green}{31/20}, \color{green}{31/21}, \color{green}{31/22}, \color{black}{31/23}, \color{green}{31/24}, \color{blue}{31/25}, \color{green}{31/26}, \color{green}{31/27}, \color{green}{31/28},  \linebreak
 \color{black}{31/29}, \color{green}{31/30}, \color{black}{32}, \color{black}{32/3}, \color{black}{32/5}, \color{black}{32/7}, \color{blue}{32/9}, \color{blue}{32/11}, \color{blue}{32/13}, \color{blue}{32/15}, \color{blue}{32/17}, \color{blue}{32/19}, \color{blue}{32/21}, \color{blue}{32/23},  \linebreak
 \color{blue}{32/25}, \color{green}{32/27}, \color{black}{32/29}, \color{blue}{32/31}, \color{black}{33}, \color{black}{33/2}, \color{black}{33/4}, \color{black}{33/5}, \color{black}{33/7}, \color{black}{33/8}, \color{blue}{33/10}, \color{black}{33/13}, \color{blue}{33/14}, \color{blue}{33/16}, \linebreak
\color{black}{33/17}, \color{black}{33/19}, \color{green}{33/20},
\color{black}{33/23}, \color{black}{33/25}, \color{green}{33/26},
\color{green}{33/28}, \color{black}{33/29}, \color{black}{33/31},
\color{green}{33/32}, \color{black}{34}, \color{black}{34/3},
\color{black}{34/5}, \color{black}{34/7},  \linebreak
\color{black}{34/9}, \color{black}{34/11}, \color{black}{34/13},
\color{black}{34/15}, \color{black}{34/19}, \color{black}{34/21},
\color{black}{34/23}, \color{black}{34/25}, \color{green}{34/27},
\color{black}{34/29}, \color{black}{34/31}, \color{green}{34/33},
\color{black}{35}, \color{black}{35/2},  \linebreak
\color{black}{35/3}, \color{black}{35/4}, \color{black}{35/6},
\color{black}{35/8}, \color{black}{35/9}, \color{blue}{35/11},
\color{blue}{35/12}, \color{blue}{35/13}, \color{black}{35/16},
\color{black}{35/17}, \color{green}{35/18}, \color{blue}{35/19},
\color{green}{35/22}, \color{black}{35/23},  \linebreak
\color{green}{35/24}, \color{green}{35/26}, \color{green}{35/27},
\color{black}{35/29}, \color{black}{35/31}, \color{green}{35/32},
\color{green}{35/33}, \color{green}{35/34}, \color{black}{36},
\color{black}{36/5}, \color{black}{36/7}, \color{blue}{36/11},
\color{blue}{36/13}, \color{blue}{36/17},  \linebreak
\color{blue}{36/19}, \color{blue}{36/23}, \color{black}{36/25},
\color{blue}{36/29}, \color{black}{36/31}, \color{green}{36/35},
\color{black}{37}, \color{black}{37/2}, \color{black}{37/3},
\color{black}{37/4}, \color{black}{37/5}, \color{black}{37/6},
\color{black}{37/7}, \color{black}{37/8}, \color{black}{37/9},
\linebreak \color{black}{37/10}, \color{black}{37/11},
\color{blue}{37/12}, \color{black}{37/13}, \color{black}{37/14},
\color{black}{37/15}, \color{blue}{37/16}, \color{black}{37/17},
\color{green}{37/18}, \color{black}{37/19}, \color{green}{37/20},
\color{black}{37/21}, \color{red}{37/22},  \linebreak
\color{black}{37/23}, \color{green}{37/24}, \color{black}{37/25},
\color{green}{37/26}, \color{green}{37/27}, \color{green}{37/28},
\color{black}{37/29}, \color{green}{37/30}, \color{black}{37/31},
\color{green}{37/32}, \color{green}{37/33}, \color{green}{37/34},
\color{green}{37/35},  \linebreak \color{green}{37/36},
\color{black}{38}, \color{black}{38/3}, \color{black}{38/5},
\color{black}{38/7}, \color{black}{38/9}, \color{blue}{38/11},
\color{blue}{38/13}, \color{black}{38/15}, \color{black}{38/17},
\color{blue}{38/21}, \color{black}{38/23}, \color{black}{38/25},
\color{green}{38/27}, \linebreak
 \color{black}{38/29}, \color{black}{38/31}, \color{green}{38/33}, \color{green}{38/35}, \color{blue}{38/37}, \color{black}{39}, \color{black}{39/2}, \color{black}{39/4}, \color{black}{39/5}, \color{black}{39/7}, \color{black}{39/8}, \color{blue}{39/10}, \color{blue}{39/11}, \color{black}{39/14}, \linebreak
 \color{black}{39/16}, \color{blue}{39/17}, \color{black}{39/19}, \color{green}{39/20}, \color{black}{39/22}, \color{black}{39/23}, \color{black}{39/25}, \color{green}{39/28}, \color{black}{39/29}, \color{black}{39/31}, \color{green}{39/32}, \color{green}{39/34}, \color{green}{39/35}, \linebreak
 \color{black}{39/37}, \color{green}{39/38}, \color{black}{40}, \color{black}{40/3}, \color{black}{40/7}, \color{black}{40/9}, \color{blue}{40/11}, \color{blue}{40/13}, \color{blue}{40/17}, \color{blue}{40/19}, \color{blue}{40/21}, \color{black}{40/23}, \color{blue}{40/27}, \color{blue}{40/29}, \linebreak
 \color{black}{40/31}, \color{green}{40/33}, \color{blue}{40/37}, \color{green}{40/39}, \color{black}{41}, \color{black}{41/2}, \color{black}{41/3}, \color{black}{41/4}, \color{black}{41/5}, \color{black}{41/6}, \color{black}{41/7}, \color{black}{41/8}, \color{black}{41/9}, \color{black}{41/10}, \color{black}{41/11}, \linebreak
 \color{black}{41/12}, \color{black}{41/13}, \color{black}{41/14}, \color{black}{41/15}, \color{black}{41/16}, \color{black}{41/17}, \color{black}{41/18}, \color{black}{41/19}, \color{green}{41/20}, \color{black}{41/21}, \color{red}{41/22}, \color{black}{41/23}, \color{green}{41/24}, \linebreak
 \color{black}{41/25}, \color{green}{41/26}, \color{black}{41/27}, \color{green}{41/28}, \color{black}{41/29}, \color{green}{41/30}, \color{black}{41/31}, \color{green}{41/32}, \color{green}{41/33}, \color{green}{41/34}, \color{green}{41/35}, \color{green}{41/36}, \color{black}{41/37}, \linebreak
 \color{green}{41/38}, \color{green}{41/39}, \color{green}{41/40}, \color{black}{42}, \color{black}{42/5}, \color{blue}{42/11}, \color{blue}{42/13}, \color{blue}{42/17}, \color{blue}{42/19}, \color{blue}{42/23}, \color{black}{42/25}, \color{black}{42/29}, \color{black}{42/31}, \linebreak
 \color{black}{42/37}, \color{blue}{42/41}, \color{black}{43}, \color{black}{43/2}, \color{black}{43/3}, \color{black}{43/4}, \color{black}{43/5}, \color{black}{43/6}, \color{black}{43/7}, \color{black}{43/8}, \color{black}{43/9}, \color{black}{43/10}, \color{black}{43/11}, \color{black}{43/12}, \linebreak
 \color{black}{43/13}, \color{black}{43/14}, \color{black}{43/15}, \color{black}{43/16}, \color{black}{43/17}, \color{black}{43/18}, \color{black}{43/19}, \color{red}{43/20}, \color{black}{43/21}, \color{black}{43/22}, \color{black}{43/23}, \color{green}{43/24}, \color{black}{43/25}, \linebreak
 \color{red}{43/26}, \color{black}{43/27}, \color{green}{43/28}, \color{black}{43/29}, \color{green}{43/30}, \color{black}{43/31}, \color{green}{43/32}, \color{green}{43/33}, \color{green}{43/34}, \color{green}{43/35}, \color{green}{43/36}, \color{black}{43/37}, \color{green}{43/38}, \linebreak
 \color{green}{43/39}, \color{green}{43/40}, \color{black}{43/41}, \color{green}{43/42}, \color{black}{44}, \color{black}{44/3}, \color{black}{44/5}, \color{black}{44/7}, \color{black}{44/9}, \color{black}{44/13}, \color{blue}{44/15}, \color{blue}{44/17}, \color{black}{44/19}, \color{black}{44/21}, \linebreak
 \color{black}{44/23}, \color{black}{44/25}, \color{blue}{44/27}, \color{black}{44/29}, \color{black}{44/31}, \color{green}{44/35}, \color{black}{44/37}, \color{green}{44/39}, \color{black}{44/41}, \color{blue}{44/43}, \color{black}{45}, \color{black}{45/2}, \color{black}{45/4}, \color{black}{45/7}, \linebreak
 \color{black}{45/8}, \color{black}{45/11}, \color{blue}{45/13}, \color{blue}{45/14}, \color{blue}{45/16}, \color{blue}{45/17}, \color{blue}{45/19}, \color{blue}{45/22}, \color{blue}{45/23}, \color{black}{45/26}, \color{green}{45/28}, \color{blue}{45/29}, \color{black}{45/31}, \linebreak
 \color{green}{45/32}, \color{green}{45/34}, \color{black}{45/37}, \color{green}{45/38}, \color{black}{45/41}, \color{black}{45/43}, \color{green}{45/44}, \color{black}{46}, \color{black}{46/3}, \color{black}{46/5}, \color{black}{46/7}, \color{black}{46/9}, \color{black}{46/11}, \color{black}{46/13}, \linebreak
 \color{black}{46/15}, \color{black}{46/17}, \color{black}{46/19}, \color{black}{46/21}, \color{black}{46/25}, \color{black}{46/27}, \color{black}{46/29}, \color{black}{46/31}, \color{green}{46/33}, \color{green}{46/35}, \color{black}{46/37}, \color{green}{46/39}, \color{black}{46/41}, \linebreak
 \color{black}{46/43}, \color{green}{46/45}, \color{black}{47}, \color{black}{47/2}, \color{black}{47/3}, \color{black}{47/4}, \color{black}{47/5}, \color{black}{47/6}, \color{black}{47/7}, \color{black}{47/8}, \color{black}{47/9}, \color{black}{47/10}, \color{black}{47/11}, \color{black}{47/12}, \color{black}{47/13}, \linebreak
 \color{black}{47/14}, \color{black}{47/15}, \color{black}{47/16}, \color{black}{47/17}, \color{black}{47/18}, \color{black}{47/19}, \color{black}{47/20}, \color{black}{47/21}, \color{black}{47/22}, \color{black}{47/23}, \color{green}{47/24}, \color{black}{47/25}, \color{red}{47/26}, \linebreak
 \color{black}{47/27}, \color{green}{47/28}, \color{black}{47/29}, \color{green}{47/30}, \color{black}{47/31}, \color{green}{47/32}, \color{green}{47/33}, \color{green}{47/34}, \color{green}{47/35}, \color{green}{47/36}, \color{black}{47/37}, \color{green}{47/38}, \color{green}{47/39}, \linebreak
 \color{green}{47/40}, \color{black}{47/41}, \color{green}{47/42}, \color{black}{47/43}, \color{green}{47/44}, \color{green}{47/45}, \color{green}{47/46}, \color{black}{48}, \color{black}{48/5}, \color{black}{48/7}, \color{blue}{48/11}, \color{blue}{48/13}, \color{blue}{48/17}, \color{blue}{48/19},  \linebreak
\color{blue}{48/23}, \color{blue}{48/25}, \color{blue}{48/29},
\color{blue}{48/31}, \color{blue}{48/35}, \color{blue}{48/37},
\color{blue}{48/41}, \color{blue}{48/43}, \color{blue}{48/47},
\color{black}{49}, \color{black}{49/2}, \color{black}{49/3},
\color{black}{49/4}, \color{black}{49/5}, \linebreak
 \color{black}{49/6}, \color{black}{49/8}, \color{black}{49/9}, \color{black}{49/10}, \color{black}{49/11}, \color{black}{49/12}, \color{blue}{49/13}, \color{blue}{49/15}, \color{blue}{49/16}, \color{blue}{49/17}, \color{blue}{49/18}, \color{black}{49/19}, \color{black}{49/20}, \linebreak
 \color{black}{49/22}, \color{black}{49/23}, \color{green}{49/24}, \color{black}{49/25}, \color{blue}{49/26}, \color{black}{49/27}, \color{black}{49/29}, \color{green}{49/30}, \color{black}{49/31}, \color{green}{49/32}, \color{red}{49/33}, \color{green}{49/34}, \color{green}{49/36}, \linebreak
 \color{black}{49/37}, \color{green}{49/38}, \color{green}{49/39}, \color{green}{49/40}, \color{black}{49/41}, \color{black}{49/43}, \color{green}{49/44}, \color{green}{49/45}, \color{green}{49/46}, \color{black}{49/47}, \color{green}{49/48}, \color{black}{50}, \color{black}{50/3}, \color{black}{50/7}, \linebreak
 \color{black}{50/9}, \color{black}{50/11}, \color{black}{50/13}, \color{blue}{50/17}, \color{blue}{50/19}, \color{black}{50/21}, \color{black}{50/23}, \color{black}{50/27}, \color{black}{50/29}, \color{black}{50/31}, \color{black}{50/33}, \color{black}{50/37}, \color{green}{50/39}, \linebreak
 \color{black}{50/41}, \color{black}{50/43}, \color{black}{50/47}, \color{green}{50/49}, \color{black}{51}, \color{black}{51/2}, \color{black}{51/4}, \color{black}{51/5}, \color{black}{51/7}, \color{black}{51/8}, \color{black}{51/10}, \color{black}{51/11}, \color{blue}{51/13}, \color{blue}{51/14}, \linebreak
 \color{blue}{51/16}, \color{black}{51/19}, \color{blue}{51/20}, \color{black}{51/22}, \color{black}{51/23}, \color{black}{51/25}, \color{black}{51/26}, \color{green}{51/28}, \color{black}{51/29}, \color{black}{51/31}, \color{green}{51/32}, \color{black}{51/35}, \color{black}{51/37}, \linebreak
 \color{green}{51/38}, \color{green}{51/40}, \color{black}{51/41}, \color{black}{51/43}, \color{green}{51/44}, \color{green}{51/46}, \color{black}{51/47}, \color{green}{51/49}, \color{green}{51/50}, \color{black}{52}, \color{black}{52/3}, \color{black}{52/5}, \color{black}{52/7}, \color{black}{52/9},  \linebreak
\color{black}{52/11}, \color{blue}{52/15}, \color{blue}{52/17},
\color{blue}{52/19}, \color{blue}{52/21}, \color{blue}{52/23},
\color{black}{52/25}, \color{black}{52/27}, \color{blue}{52/29},
\color{black}{52/31}, \color{blue}{52/33}, \color{black}{52/35},
\color{black}{52/37},  \linebreak \color{black}{52/41},
\color{black}{52/43}, \color{green}{52/45}, \color{black}{52/47},
\color{green}{52/49}, \color{green}{52/51}, \color{black}{53},
\color{black}{53/2}, \color{black}{53/3}, \color{black}{53/4},
\color{black}{53/5}, \color{black}{53/6}, \color{black}{53/7},
\color{black}{53/8}, \linebreak
 \color{black}{53/9}, \color{black}{53/10}, \color{black}{53/11}, \color{black}{53/12}, \color{black}{53/13}, \color{black}{53/14}, \color{black}{53/15}, \color{black}{53/16}, \color{black}{53/17}, \color{black}{53/18}, \color{black}{53/19}, \color{black}{53/20}, \color{black}{53/21},  \linebreak
\color{black}{53/22}, \color{black}{53/23}, \color{green}{53/24},
\color{black}{53/25}, \color{black}{53/26}, \color{black}{53/27},
\color{green}{53/28}, \color{black}{53/29}, \color{green}{53/30},
\color{black}{53/31}, \color{green}{53/32}, \color{black}{53/33},
\color{green}{53/34}, \linebreak
 \color{black}{53/35}, \color{green}{53/36}, \color{black}{53/37}, \color{green}{53/38}, \color{green}{53/39}, \color{green}{53/40}, \color{black}{53/41}, \color{green}{53/42}, \color{black}{53/43}, \color{green}{53/44}, \color{green}{53/45}, \color{green}{53/46}, \color{black}{53/47}, \linebreak
 \color{green}{53/48}, \color{green}{53/49}, \color{green}{53/50}, \color{green}{53/51}, \color{green}{53/52}, \color{black}{54}, \color{black}{54/5}, \color{black}{54/7}, \color{black}{54/11}, \color{black}{54/13}, \color{blue}{54/17}, \color{blue}{54/19}, \color{blue}{54/23}, \color{black}{54/25}, \linebreak
 \color{blue}{54/29}, \color{black}{54/31}, \color{black}{54/35}, \color{black}{54/37}, \color{black}{54/41}, \color{black}{54/43}, \color{black}{54/47}, \color{green}{54/49}, \color{blue}{54/53}, \color{black}{55}, \color{black}{55/2}, \color{black}{55/3}, \color{black}{55/4}, \color{black}{55/6}, \linebreak
 \color{black}{55/7}, \color{black}{55/8}, \color{black}{55/9}, \color{black}{55/12}, \color{black}{55/13}, \color{black}{55/14}, \color{black}{55/16}, \color{blue}{55/17}, \color{blue}{55/18}, \color{blue}{55/19}, \color{black}{55/21}, \color{black}{55/23}, \color{green}{55/24},  \linebreak
\color{black}{55/26}, \color{black}{55/27}, \color{green}{55/28},
\color{black}{55/29}, \color{black}{55/31}, \color{green}{55/32},
\color{red}{55/34}, \color{green}{55/36}, \color{black}{55/37},
\color{green}{55/38}, \color{green}{55/39}, \color{black}{55/41},
\color{green}{55/42},  \linebreak \color{black}{55/43},
\color{green}{55/46}, \color{black}{55/47}, \color{green}{55/48},
\color{green}{55/49}, \color{green}{55/51}, \color{green}{55/52},
\color{black}{55/53}, \color{green}{55/54}, \color{black}{56},
\color{black}{56/3}, \color{black}{56/5}, \color{black}{56/9},
\color{black}{56/11}, \linebreak
 \color{blue}{56/13}, \color{blue}{56/15}, \color{blue}{56/17}, \color{blue}{56/19}, \color{blue}{56/23}, \color{blue}{56/25}, \color{blue}{56/27}, \color{blue}{56/29}, \color{blue}{56/31}, \color{blue}{56/33}, \color{black}{56/37}, \color{blue}{56/39}, \color{blue}{56/41}, \linebreak
 \color{black}{56/43}, \color{green}{56/45}, \color{black}{56/47}, \color{green}{56/51}, \color{black}{56/53}, \color{green}{56/55}, \color{black}{57}, \color{black}{57/2}, \color{black}{57/4}, \color{black}{57/5}, \color{black}{57/7}, \color{black}{57/8}, \color{black}{57/10}, \color{black}{57/11}, \linebreak
 \color{black}{57/13}, \color{blue}{57/14}, \color{blue}{57/16}, \color{blue}{57/17}, \color{black}{57/20}, \color{black}{57/22}, \color{black}{57/23}, \color{black}{57/25}, \color{blue}{57/26}, \color{blue}{57/28}, \color{black}{57/29}, \color{black}{57/31}, \color{green}{57/32}, \linebreak
 \color{black}{57/34}, \color{black}{57/35}, \color{blue}{57/37}, \color{green}{57/40}, \color{black}{57/41}, \color{black}{57/43}, \color{green}{57/44}, \color{green}{57/46}, \color{black}{57/47}, \color{blue}{57/49}, \color{green}{57/50}, \color{green}{57/52}, \color{black}{57/53}, \linebreak
 \color{green}{57/55}, \color{green}{57/56}, \color{black}{58}, \color{black}{58/3}, \color{black}{58/5}, \color{black}{58/7}, \color{black}{58/9}, \color{black}{58/11}, \color{black}{58/13}, \color{black}{58/15}, \color{black}{58/17}, \color{black}{58/19}, \color{black}{58/21}, \color{black}{58/23}, \linebreak
 \color{black}{58/25}, \color{black}{58/27}, \color{black}{58/31}, \color{black}{58/33}, \color{black}{58/35}, \color{black}{58/37}, \color{black}{58/39}, \color{black}{58/41}, \color{black}{58/43}, \color{green}{58/45}, \color{black}{58/47}, \color{black}{58/49}, \color{green}{58/51}, \linebreak
 \color{black}{58/53}, \color{green}{58/55}, \color{green}{58/57}, \color{black}{59}, \color{black}{59/2}, \color{black}{59/3}, \color{black}{59/4}, \color{black}{59/5}, \color{black}{59/6}, \color{black}{59/7}, \color{black}{59/8}, \color{black}{59/9}, \color{black}{59/10}, \color{black}{59/11}, \color{black}{59/12}, \linebreak
 \color{black}{59/13}, \color{black}{59/14}, \color{black}{59/15}, \color{black}{59/16}, \color{black}{59/17}, \color{black}{59/18}, \color{black}{59/19}, \color{black}{59/20}, \color{black}{59/21}, \color{black}{59/22}, \color{black}{59/23}, \color{green}{59/24}, \color{black}{59/25}, \linebreak
 \color{black}{59/26}, \color{black}{59/27}, \color{black}{59/28}, \color{black}{59/29}, \color{green}{59/30}, \color{black}{59/31}, \color{green}{59/32}, \color{black}{59/33}, \color{red}{59/34}, \color{black}{59/35}, \color{green}{59/36}, \color{black}{59/37}, \color{green}{59/38}, \linebreak
 \color{red}{59/39}, \color{green}{59/40}, \color{black}{59/41}, \color{green}{59/42}, \color{black}{59/43}, \color{green}{59/44}, \color{green}{59/45}, \color{green}{59/46}, \color{black}{59/47}, \color{green}{59/48}, \color{black}{59/49}, \color{green}{59/50}, \color{green}{59/51},  \linebreak
\color{green}{59/52}, \color{black}{59/53}, \color{green}{59/54},
\color{green}{59/55}, \color{green}{59/56}, \color{green}{59/57},
\color{green}{59/58}, \color{black}{60}, \color{black}{60/7},
\color{black}{60/11}, \color{black}{60/13}, \color{blue}{60/17},
\color{blue}{60/19},  \linebreak
 \color{blue}{60/23}, \color{blue}{60/29}, \color{blue}{60/31}, \color{blue}{60/37}, \color{black}{60/41}, \color{black}{60/43}, \color{black}{60/47}, \color{blue}{60/49}, \color{black}{60/53}, \color{blue}{60/59}, \color{black}{61}, \color{black}{61/2}, \color{black}{61/3}, \color{black}{61/4},  \linebreak
 \color{black}{61/5}, \color{black}{61/6}, \color{black}{61/7}, \color{black}{61/8}, \color{black}{61/9}, \color{black}{61/10}, \color{black}{61/11}, \color{black}{61/12}, \color{black}{61/13}, \color{black}{61/14}, \color{black}{61/15}, \color{black}{61/16}, \color{black}{61/17}, \color{black}{61/18},  \linebreak
 \color{black}{61/19}, \color{black}{61/20}, \color{black}{61/21}, \color{black}{61/22}, \color{black}{61/23}, \color{red}{61/24}, \color{black}{61/25}, \color{blue}{61/26}, \color{black}{61/27}, \color{black}{61/28}, \color{black}{61/29}, \color{green}{61/30}, \color{black}{61/31},  \linebreak
 \color{green}{61/32}, \color{black}{61/33}, \color{red}{61/34}, \color{black}{61/35}, \color{green}{61/36}, \color{black}{61/37}, \color{red}{61/38}, \color{blue}{61/39}, \color{green}{61/40}, \color{black}{61/41}, \color{green}{61/42}, \color{black}{61/43}, \color{green}{61/44}, \linebreak
 \color{green}{61/45}, \color{green}{61/46}, \color{black}{61/47}, \color{green}{61/48}, \color{black}{61/49}, \color{green}{61/50}, \color{green}{61/51}, \color{green}{61/52}, \color{black}{61/53}, \color{green}{61/54}, \color{green}{61/55}, \color{green}{61/56}, \color{green}{61/57},  \linebreak
\color{green}{61/58}, \color{black}{61/59}, \color{green}{61/60},
\color{black}{62}, \color{black}{62/3}, \color{black}{62/5},
\color{black}{62/7}, \color{black}{62/9}, \color{black}{62/11},
\color{black}{62/13}, \color{blue}{62/15}, \color{blue}{62/17},
\color{blue}{62/19}, \color{blue}{62/21}, \linebreak
 \color{blue}{62/23}, \color{blue}{62/25}, \color{black}{62/27}, \color{black}{62/29}, \color{black}{62/33}, \color{black}{62/35}, \color{black}{62/37}, \color{blue}{62/39}, \color{black}{62/41}, \color{black}{62/43}, \color{green}{62/45}, \color{black}{62/47}, \color{black}{62/49}, \linebreak
 \color{green}{62/51}, \color{black}{62/53}, \color{green}{62/55}, \color{green}{62/57}, \color{black}{62/59}, \color{blue}{62/61}, \color{black}{63}, \color{black}{63/2}, \color{black}{63/4}, \color{black}{63/5}, \color{black}{63/8}, \color{black}{63/10}, \color{black}{63/11}, \color{black}{63/13}, \linebreak
 \color{black}{63/16}, \color{blue}{63/17}, \color{blue}{63/19}, \color{blue}{63/20}, \color{blue}{63/22}, \color{blue}{63/23}, \color{blue}{63/25}, \color{blue}{63/26}, \color{blue}{63/29}, \color{blue}{63/31}, \color{blue}{63/32}, \color{blue}{63/34}, \color{black}{63/37}, \linebreak
 \color{black}{63/38}, \color{green}{63/40}, \color{blue}{63/41}, \color{black}{63/43}, \color{green}{63/44}, \color{green}{63/46}, \color{black}{63/47}, \color{green}{63/50}, \color{green}{63/52}, \color{black}{63/53}, \color{green}{63/55}, \color{green}{63/58}, \color{black}{63/59},  \linebreak
\color{black}{63/61}, \color{green}{63/62}, \color{black}{64},
\color{black}{64/3}, \color{black}{64/5}, \color{black}{64/7},
\color{black}{64/9}, \color{black}{64/11}, \color{black}{64/13},
\color{black}{64/15}, \color{blue}{64/17}, \color{blue}{64/19},
\color{blue}{64/21}, \color{blue}{64/23}, \linebreak
 \color{blue}{64/25}, \color{black}{64/27}, \color{blue}{64/29}, \color{blue}{64/31}, \color{blue}{64/33}, \color{blue}{64/35}, \color{blue}{64/37}, \color{blue}{64/39}, \color{blue}{64/41}, \color{blue}{64/43}, \color{green}{64/45}, \color{blue}{64/47}, \color{black}{64/49}, \linebreak
 \color{green}{64/51}, \color{black}{64/53}, \color{green}{64/55}, \color{green}{64/57}, \color{black}{64/59}, \color{blue}{64/61}, \color{green}{64/63}, \color{black}{65}, \color{black}{65/2}, \color{black}{65/3}, \color{black}{65/4}, \color{black}{65/6}, \color{black}{65/7}, \color{black}{65/8},  \linebreak
\color{black}{65/9}, \color{black}{65/11}, \color{black}{65/12},
\color{black}{65/14}, \color{black}{65/16}, \color{black}{65/17},
\color{black}{65/18}, \color{blue}{65/19}, \color{blue}{65/21},
\color{blue}{65/22}, \color{blue}{65/23}, \color{blue}{65/24},
\color{black}{65/27}, \linebreak
 \color{black}{65/28}, \color{blue}{65/29}, \color{black}{65/31}, \color{black}{65/32}, \color{black}{65/33}, \color{black}{65/34}, \color{green}{65/36}, \color{black}{65/37}, \color{red}{65/38}, \color{black}{65/41}, \color{green}{65/42}, \color{black}{65/43}, \color{green}{65/44},  \linebreak
\color{green}{65/46}, \color{black}{65/47}, \color{green}{65/48},
\color{black}{65/49}, \color{green}{65/51}, \color{black}{65/53},
\color{green}{65/54}, \color{green}{65/56}, \color{green}{65/57},
\color{green}{65/58}, \color{black}{65/59}, \color{black}{65/61},
\color{green}{65/62}, \linebreak
 \color{green}{65/63}, \color{green}{65/64}, \color{black}{66}, \color{black}{66/5}, \color{black}{66/7}, \color{black}{66/13}, \color{blue}{66/17}, \color{black}{66/19}, \color{black}{66/23}, \color{black}{66/25}, \color{black}{66/29}, \color{black}{66/31}, \color{black}{66/35}, \color{black}{66/37}, \linebreak
 \color{black}{66/41}, \color{blue}{66/43}, \color{black}{66/47}, \color{black}{66/49}, \color{black}{66/53}, \color{black}{66/59}, \color{black}{66/61}, \color{green}{66/65}, \color{black}{67}, \color{black}{67/2}, \color{black}{67/3}, \color{black}{67/4}, \color{black}{67/5}, \color{black}{67/6},  \linebreak
\color{black}{67/7}, \color{black}{67/8}, \color{black}{67/9},
\color{black}{67/10}, \color{black}{67/11}, \color{black}{67/12},
\color{black}{67/13}, \color{black}{67/14}, \color{black}{67/15},
\color{black}{67/16}, \color{black}{67/17}, \color{black}{67/18},
\color{black}{67/19}, \linebreak
 \color{black}{67/20}, \color{black}{67/21}, \color{black}{67/22}, \color{black}{67/23}, \color{black}{67/24}, \color{black}{67/25}, \color{black}{67/26}, \color{black}{67/27}, \color{black}{67/28}, \color{black}{67/29}, \color{green}{67/30}, \color{black}{67/31}, \color{black}{67/32}, \linebreak
 \color{black}{67/33}, \color{black}{67/34}, \color{black}{67/35}, \color{green}{67/36}, \color{black}{67/37}, \color{red}{67/38}, \color{black}{67/39}, \color{green}{67/40}, \color{black}{67/41}, \color{green}{67/42}, \color{black}{67/43}, \color{green}{67/44}, \color{green}{67/45}, \linebreak
 \color{green}{67/46}, \color{black}{67/47}, \color{green}{67/48}, \color{black}{67/49}, \color{green}{67/50}, \color{green}{67/51}, \color{green}{67/52}, \color{black}{67/53}, \color{green}{67/54}, \color{green}{67/55}, \color{green}{67/56}, \color{green}{67/57}, \color{green}{67/58}, \linebreak
 \color{black}{67/59}, \color{green}{67/60}, \color{black}{67/61}, \color{green}{67/62}, \color{green}{67/63}, \color{green}{67/64}, \color{green}{67/65}, \color{green}{67/66}, \color{black}{68}, \color{black}{68/3}, \color{black}{68/5}, \color{black}{68/7}, \color{black}{68/9}, \color{black}{68/11}, \linebreak
 \color{black}{68/13}, \color{black}{68/15}, \color{black}{68/19}, \color{black}{68/21}, \color{black}{68/23}, \color{black}{68/25}, \color{black}{68/27}, \color{black}{68/29}, \color{black}{68/31}, \color{black}{68/33}, \color{black}{68/35}, \color{black}{68/37}, \color{black}{68/39}, \linebreak
 \color{black}{68/41}, \color{black}{68/43}, \color{green}{68/45}, \color{black}{68/47}, \color{black}{68/49}, \color{black}{68/53}, \color{green}{68/55}, \color{green}{68/57}, \color{black}{68/59}, \color{black}{68/61}, \color{green}{68/63}, \color{green}{68/65}, \color{blue}{68/67},  \linebreak
 \color{black}{69}, \color{black}{69/2}, \color{black}{69/4}, \color{black}{69/5}, \color{black}{69/7}, \color{black}{69/8}, \color{black}{69/10}, \color{black}{69/11}, \color{black}{69/13}, \color{black}{69/14}, \color{black}{69/16}, \color{black}{69/17}, \color{black}{69/19}, \color{black}{69/20}, \linebreak
 \color{black}{69/22}, \color{black}{69/25}, \color{black}{69/26}, \color{black}{69/28}, \color{black}{69/29}, \color{black}{69/31}, \color{black}{69/32}, \color{black}{69/34}, \color{black}{69/35}, \color{black}{69/37}, \color{black}{69/38}, \color{green}{69/40}, \color{black}{69/41}, \linebreak
 \color{black}{69/43}, \color{green}{69/44}, \color{black}{69/47}, \color{black}{69/49}, \color{green}{69/50}, \color{green}{69/52}, \color{black}{69/53}, \color{green}{69/55}, \color{green}{69/56}, \color{green}{69/58}, \color{black}{69/59}, \color{black}{69/61}, \color{green}{69/62},  \linebreak
 \color{green}{69/64}, \color{green}{69/65}, \color{black}{69/67}, \color{green}{69/68}, \color{black}{70}, \color{black}{70/3}, \color{black}{70/9}, \color{black}{70/11}, \color{black}{70/13}, \color{black}{70/17}, \color{blue}{70/19}, \color{blue}{70/23}, \color{blue}{70/27}, \color{blue}{70/29}, \linebreak
 \color{black}{70/31}, \color{black}{70/33}, \color{blue}{70/37}, \color{black}{70/39}, \color{black}{70/41}, \color{black}{70/43}, \color{black}{70/47}, \color{green}{70/51}, \color{black}{70/53}, \color{green}{70/57}, \color{black}{70/59}, \color{black}{70/61}, \color{black}{70/67}, \linebreak
 \color{green}{70/69}, \color{black}{71}, \color{black}{71/2}, \color{black}{71/3}, \color{black}{71/4}, \color{black}{71/5}, \color{black}{71/6}, \color{black}{71/7}, \color{black}{71/8}, \color{black}{71/9}, \color{black}{71/10}, \color{black}{71/11}, \color{black}{71/12}, \color{black}{71/13}, \color{black}{71/14}, \linebreak
 \color{black}{71/15}, \color{black}{71/16}, \color{black}{71/17}, \color{black}{71/18}, \color{black}{71/19}, \color{black}{71/20}, \color{black}{71/21}, \color{black}{71/22}, \color{black}{71/23}, \color{black}{71/24}, \color{black}{71/25}, \color{black}{71/26}, \color{black}{71/27}, \linebreak
 \color{black}{71/28}, \color{black}{71/29}, \color{green}{71/30}, \color{black}{71/31}, \color{black}{71/32}, \color{black}{71/33}, \color{black}{71/34}, \color{black}{71/35}, \color{green}{71/36}, \color{black}{71/37}, \color{black}{71/38}, \color{black}{71/39}, \color{green}{71/40}, \linebreak
 \color{black}{71/41}, \color{green}{71/42}, \color{black}{71/43}, \color{green}{71/44}, \color{green}{71/45}, \color{green}{71/46}, \color{black}{71/47}, \color{green}{71/48}, \color{black}{71/49}, \color{green}{71/50}, \color{green}{71/51}, \color{green}{71/52}, \color{black}{71/53},  \linebreak
\color{green}{71/54}, \color{green}{71/55}, \color{green}{71/56},
\color{green}{71/57}, \color{green}{71/58}, \color{black}{71/59},
\color{green}{71/60}, \color{black}{71/61}, \color{green}{71/62},
\color{green}{71/63}, \color{green}{71/64}, \color{green}{71/65},
\color{green}{71/66}, \linebreak
 \color{black}{71/67}, \color{green}{71/68}, \color{green}{71/69}, \color{green}{71/70}, \color{black}{72}, \color{black}{72/5}, \color{black}{72/7}, \color{black}{72/11}, \color{black}{72/13}, \color{blue}{72/17}, \color{blue}{72/19}, \color{blue}{72/23}, \color{blue}{72/25}, \color{blue}{72/29}, \linebreak
 \color{blue}{72/31}, \color{blue}{72/35}, \color{blue}{72/37}, \color{blue}{72/41}, \color{blue}{72/43}, \color{blue}{72/47}, \color{blue}{72/49}, \color{blue}{72/53}, \color{blue}{72/55}, \color{blue}{72/59}, \color{black}{72/61}, \color{green}{72/65}, \color{blue}{72/67},  \linebreak
\color{blue}{72/71}, \color{black}{73}, \color{black}{73/2},
\color{black}{73/3}, \color{black}{73/4}, \color{black}{73/5},
\color{black}{73/6}, \color{black}{73/7}, \color{black}{73/8},
\color{black}{73/9}, \color{black}{73/10}, \color{black}{73/11},
\color{black}{73/12}, \color{black}{73/13}, \color{black}{73/14},
\linebreak
 \color{black}{73/15}, \color{black}{73/16}, \color{black}{73/17}, \color{blue}{73/18}, \color{blue}{73/19}, \color{blue}{73/20}, \color{black}{73/21}, \color{blue}{73/22}, \color{blue}{73/23}, \color{blue}{73/24}, \color{black}{73/25}, \color{black}{73/26}, \color{black}{73/27}, \linebreak
 \color{black}{73/28}, \color{black}{73/29}, \color{black}{73/30}, \color{blue}{73/31}, \color{blue}{73/32}, \color{black}{73/33}, \color{black}{73/34}, \color{black}{73/35}, \color{green}{73/36}, \color{black}{73/37}, \color{black}{73/38}, \color{black}{73/39}, \color{green}{73/40}, \linebreak
 \color{black}{73/41}, \color{green}{73/42}, \color{black}{73/43}, \color{green}{73/44}, \color{green}{73/45}, \color{red}{73/46}, \color{black}{73/47}, \color{green}{73/48}, \color{black}{73/49}, \color{green}{73/50}, \color{red}{73/51}, \color{green}{73/52}, \color{black}{73/53}, \linebreak
 \color{green}{73/54}, \color{red}{73/55}, \color{green}{73/56}, \color{green}{73/57}, \color{green}{73/58}, \color{black}{73/59}, \color{green}{73/60}, \color{black}{73/61}, \color{green}{73/62}, \color{green}{73/63}, \color{green}{73/64}, \color{green}{73/65}, \color{green}{73/66}, \linebreak
 \color{black}{73/67}, \color{green}{73/68}, \color{green}{73/69}, \color{green}{73/70}, \color{black}{73/71}, \color{green}{73/72}, \color{black}{74}, \color{black}{74/3}, \color{black}{74/5}, \color{black}{74/7}, \color{black}{74/9}, \color{black}{74/11}, \color{black}{74/13}, \color{black}{74/15}, \linebreak
 \color{black}{74/17}, \color{black}{74/19}, \color{black}{74/21}, \color{black}{74/23}, \color{black}{74/25}, \color{black}{74/27}, \color{black}{74/29}, \color{black}{74/31}, \color{black}{74/33}, \color{black}{74/35}, \color{black}{74/39}, \color{black}{74/41}, \color{black}{74/43}, \linebreak
 \color{green}{74/45}, \color{black}{74/47}, \color{black}{74/49}, \color{black}{74/51}, \color{black}{74/53}, \color{red}{74/55}, \color{green}{74/57}, \color{black}{74/59}, \color{black}{74/61}, \color{green}{74/63}, \color{green}{74/65}, \color{black}{74/67}, \color{green}{74/69}, \linebreak
 \color{black}{74/71}, \color{blue}{74/73}, \color{black}{75}, \color{black}{75/2}, \color{black}{75/4}, \color{black}{75/7}, \color{black}{75/8}, \color{black}{75/11}, \color{black}{75/13}, \color{black}{75/14}, \color{black}{75/16}, \color{black}{75/17}, \color{black}{75/19}, \color{blue}{75/22}, \linebreak
 \color{black}{75/23}, \color{black}{75/26}, \color{black}{75/28}, \color{blue}{75/29}, \color{black}{75/31}, \color{black}{75/32}, \color{black}{75/34}, \color{blue}{75/37}, \color{blue}{75/38}, \color{black}{75/41}, \color{black}{75/43}, \color{green}{75/44}, \color{black}{75/46}, \linebreak
 \color{black}{75/47}, \color{black}{75/49}, \color{green}{75/52}, \color{black}{75/53}, \color{green}{75/56}, \color{green}{75/58}, \color{black}{75/59}, \color{black}{75/61}, \color{green}{75/62}, \color{green}{75/64}, \color{black}{75/67}, \color{green}{75/68}, \color{black}{75/71}, \linebreak
 \color{black}{75/73}, \color{green}{75/74}, \color{black}{76}, \color{black}{76/3}, \color{black}{76/5}, \color{black}{76/7}, \color{black}{76/9}, \color{black}{76/11}, \color{black}{76/13}, \color{black}{76/15}, \color{black}{76/17}, \color{blue}{76/21}, \color{blue}{76/23}, \color{black}{76/25}, \linebreak
 \color{blue}{76/27}, \color{black}{76/29}, \color{black}{76/31}, \color{black}{76/33}, \color{blue}{76/35}, \color{blue}{76/37}, \color{black}{76/39}, \color{blue}{76/41}, \color{black}{76/43}, \color{green}{76/45}, \color{black}{76/47}, \color{blue}{76/49}, \color{black}{76/51}, \linebreak
 \color{black}{76/53}, \color{black}{76/55}, \color{black}{76/59}, \color{black}{76/61}, \color{green}{76/63}, \color{green}{76/65}, \color{black}{76/67}, \color{green}{76/69}, \color{black}{76/71}, \color{blue}{76/73}, \color{green}{76/75}, \color{black}{77}, \color{black}{77/2}, \linebreak
 \color{black}{77/3}, \color{black}{77/4}, \color{black}{77/5}, \color{black}{77/6}, \color{black}{77/8}, \color{black}{77/9}, \color{black}{77/10}, \color{black}{77/12}, \color{black}{77/13}, \color{black}{77/15}, \color{black}{77/16}, \color{black}{77/17}, \color{black}{77/18}, \color{black}{77/19}, \linebreak
 \color{blue}{77/20}, \color{blue}{77/23}, \color{blue}{77/24}, \color{black}{77/25}, \color{blue}{77/26}, \color{blue}{77/27}, \color{black}{77/29}, \color{black}{77/30}, \color{black}{77/31}, \color{black}{77/32}, \color{black}{77/34}, \color{green}{77/36}, \color{black}{77/37}, \linebreak
 \color{black}{77/38}, \color{black}{77/39}, \color{green}{77/40}, \color{black}{77/41}, \color{blue}{77/43}, \color{green}{77/45}, \color{red}{77/46}, \color{black}{77/47}, \color{green}{77/48}, \color{green}{77/50}, \color{red}{77/51}, \color{green}{77/52}, \color{black}{77/53}, \linebreak
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 \color{green}{77/72}, \color{black}{77/73}, \color{green}{77/74}, \color{green}{77/75}, \color{green}{77/76}, \color{black}{78}, \color{black}{78/5}, \color{black}{78/7}, \color{black}{78/11}, \color{black}{78/17}, \color{blue}{78/19}, \color{blue}{78/23}, \color{blue}{78/25}, \color{blue}{78/29}, \linebreak
 \color{black}{78/31}, \color{black}{78/35}, \color{black}{78/37}, \color{black}{78/41}, \color{black}{78/43}, \color{black}{78/47}, \color{black}{78/49}, \color{blue}{78/53}, \color{black}{78/55}, \color{black}{78/59}, \color{black}{78/61}, \color{black}{78/67}, \color{black}{78/71}, \linebreak
 \color{black}{78/73}, \color{green}{78/77}, \color{black}{79}, \color{black}{79/2}, \color{black}{79/3}, \color{black}{79/4}, \color{black}{79/5}, \color{black}{79/6}, \color{black}{79/7}, \color{black}{79/8}, \color{black}{79/9}, \color{black}{79/10}, \color{black}{79/11}, \color{black}{79/12}, \color{black}{79/13}, \linebreak
 \color{black}{79/14}, \color{black}{79/15}, \color{black}{79/16}, \color{black}{79/17}, \color{black}{79/18}, \color{black}{79/19}, \color{black}{79/20}, \color{black}{79/21}, \color{black}{79/22}, \color{black}{79/23}, \color{black}{79/24}, \color{black}{79/25}, \color{black}{79/26}, \linebreak
 \color{black}{79/27}, \color{black}{79/28}, \color{black}{79/29}, \color{black}{79/30}, \color{black}{79/31}, \color{black}{79/32}, \color{black}{79/33}, \color{black}{79/34}, \color{black}{79/35}, \color{green}{79/36}, \color{black}{79/37}, \color{black}{79/38}, \color{black}{79/39}, \linebreak
 \color{green}{79/40}, \color{black}{79/41}, \color{green}{79/42}, \color{black}{79/43}, \color{green}{79/44}, \color{red}{79/45}, \color{red}{79/46}, \color{black}{79/47}, \color{green}{79/48}, \color{black}{79/49}, \color{green}{79/50}, \color{black}{79/51}, \color{green}{79/52}, \linebreak
 \color{black}{79/53}, \color{green}{79/54}, \color{black}{79/55}, \color{green}{79/56}, \color{green}{79/57}, \color{green}{79/58}, \color{black}{79/59}, \color{green}{79/60}, \color{black}{79/61}, \color{green}{79/62}, \color{green}{79/63}, \color{green}{79/64}, \color{green}{79/65}, \linebreak
 \color{green}{79/66}, \color{black}{79/67}, \color{green}{79/68}, \color{green}{79/69}, \color{green}{79/70}, \color{black}{79/71}, \color{green}{79/72}, \color{black}{79/73}, \color{green}{79/74}, \color{green}{79/75}, \color{green}{79/76}, \color{green}{79/77}, \color{green}{79/78}, \linebreak
 \color{black}{80}, \color{black}{80/3}, \color{black}{80/7}, \color{black}{80/9}, \color{black}{80/11}, \color{black}{80/13}, \color{black}{80/17}, \color{blue}{80/19}, \color{blue}{80/21}, \color{blue}{80/23}, \color{blue}{80/27}, \color{blue}{80/29}, \color{blue}{80/31}, \color{blue}{80/33}, \linebreak
 \color{blue}{80/37}, \color{black}{80/39}, \color{black}{80/41}, \color{black}{80/43}, \color{black}{80/47}, \color{blue}{80/49}, \color{blue}{80/51}, \color{blue}{80/53}, \color{blue}{80/57}, \color{blue}{80/59}, \color{black}{80/61}, \color{green}{80/63}, \color{black}{80/67}, \linebreak
 \color{green}{80/69}, \color{black}{80/71}, \color{blue}{80/73}, \color{green}{80/77}, \color{blue}{80/79}, \color{black}{81}, \color{black}{81/2}, \color{black}{81/4}, \color{black}{81/5}, \color{black}{81/7}, \color{black}{81/8}, \color{black}{81/10}, \color{black}{81/11}, \color{black}{81/13},  \linebreak
\color{black}{81/14}, \color{black}{81/16}, \color{black}{81/17},
\color{black}{81/19}, \color{black}{81/20}, \color{blue}{81/22},
\color{blue}{81/23}, \color{black}{81/25}, \color{blue}{81/26},
\color{blue}{81/28}, \color{blue}{81/29}, \color{blue}{81/31},
\color{blue}{81/32}, \linebreak
 \color{blue}{81/34}, \color{black}{81/35}, \color{black}{81/37}, \color{black}{81/38}, \color{green}{81/40}, \color{black}{81/41}, \color{black}{81/43}, \color{green}{81/44}, \color{black}{81/46}, \color{black}{81/47}, \color{black}{81/49}, \color{green}{81/50}, \color{green}{81/52}, \linebreak
 \color{blue}{81/53}, \color{black}{81/55}, \color{green}{81/56}, \color{green}{81/58}, \color{black}{81/59}, \color{black}{81/61}, \color{green}{81/62}, \color{green}{81/64}, \color{green}{81/65}, \color{black}{81/67}, \color{green}{81/68}, \color{green}{81/70}, \color{black}{81/71}, \linebreak
 \color{black}{81/73}, \color{green}{81/74}, \color{green}{81/76}, \color{green}{81/77}, \color{black}{81/79}, \color{green}{81/80}, \color{black}{82}, \color{black}{82/3}, \color{black}{82/5}, \color{black}{82/7}, \color{black}{82/9}, \color{black}{82/11}, \color{black}{82/13}, \color{black}{82/15}, \linebreak
 \color{black}{82/17}, \color{black}{82/19}, \color{black}{82/21}, \color{black}{82/23}, \color{black}{82/25}, \color{black}{82/27}, \color{black}{82/29}, \color{black}{82/31}, \color{black}{82/33}, \color{black}{82/35}, \color{black}{82/37}, \color{black}{82/39}, \color{black}{82/43}, \linebreak
 \color{black}{82/45}, \color{black}{82/47}, \color{black}{82/49}, \color{black}{82/51}, \color{black}{82/53}, \color{black}{82/55}, \color{black}{82/57}, \color{black}{82/59}, \color{black}{82/61}, \color{green}{82/63}, \color{green}{82/65}, \color{black}{82/67}, \color{green}{82/69}, \linebreak
 \color{black}{82/71},  \color{black}{82/73}, \color{green}{82/75}, \color{green}{82/77}, \color{black}{82/79}, \color{green}{82/81}, \color{black}{83}, \color{black}{83/2}, \color{black}{83/3}, \color{black}{83/4}, \color{black}{83/5}, \color{black}{83/6}, \color{black}{83/7}, \color{black}{83/8}, \linebreak
 \color{black}{83/9}, \color{black}{83/10}, \color{black}{83/11}, \color{black}{83/12}, \color{black}{83/13}, \color{black}{83/14}, \color{black}{83/15}, \color{black}{83/16}, \color{black}{83/17}, \color{black}{83/18}, \color{black}{83/19}, \color{black}{83/20}, \color{black}{83/21}, \linebreak
 \color{black}{83/22}, \color{black}{83/23}, \color{black}{83/24}, \color{black}{83/25}, \color{black}{83/26}, \color{black}{83/27}, \color{black}{83/28}, \color{black}{83/29}, \color{black}{83/30}, \color{black}{83/31}, \color{black}{83/32}, \color{black}{83/33}, \color{black}{83/34}, \linebreak
 \color{black}{83/35}, \color{green}{83/36}, \color{black}{83/37}, \color{black}{83/38}, \color{black}{83/39}, \color{green}{83/40}, \color{black}{83/41}, \color{green}{83/42}, \color{black}{83/43}, \color{green}{83/44}, \color{black}{83/45}, \color{red}{83/46}, \color{black}{83/47}, \linebreak
 \color{green}{83/48}, \color{black}{83/49}, \color{green}{83/50}, \color{black}{83/51}, \color{green}{83/52}, \color{black}{83/53}, \color{green}{83/54}, \color{black}{83/55}, \color{green}{83/56}, \color{red}{83/57}, \color{green}{83/58}, \color{black}{83/59}, \color{green}{83/60}, \linebreak
 \color{black}{83/61}, \color{green}{83/62}, \color{green}{83/63}, \color{green}{83/64}, \color{green}{83/65}, \color{green}{83/66}, \color{black}{83/67}, \color{green}{83/68}, \color{green}{83/69}, \color{green}{83/70}, \color{black}{83/71}, \color{green}{83/72}, \color{black}{83/73}, \linebreak
 \color{green}{83/74}, \color{green}{83/75}, \color{green}{83/76}, \color{green}{83/77}, \color{green}{83/78}, \color{black}{83/79}, \color{green}{83/80}, \color{green}{83/81}, \color{green}{83/82}, \color{black}{84}, \color{black}{84/5}, \color{black}{84/11}, \color{black}{84/13}, \linebreak
 \color{black}{84/17}, \color{black}{84/19}, \color{blue}{84/23}, \color{blue}{84/25}, \color{blue}{84/29}, \color{blue}{84/31}, \color{blue}{84/37}, \color{blue}{84/41}, \color{blue}{84/43}, \color{blue}{84/47}, \color{blue}{84/53}, \color{black}{84/55}, \color{black}{84/59}, \linebreak
 \color{black}{84/61}, \color{blue}{84/65}, \color{black}{84/67}, \color{black}{84/71}, \color{black}{84/73}, \color{black}{84/79}, \color{blue}{84/83}, \color{black}{85}, \color{black}{85/2}, \color{black}{85/3}, \color{black}{85/4}, \color{black}{85/6}, \color{black}{85/7}, \color{black}{85/8}, \linebreak
 \color{black}{85/9}, \color{black}{85/11}, \color{black}{85/12}, \color{black}{85/13}, \color{black}{85/14}, \color{black}{85/16}, \color{black}{85/18}, \color{black}{85/19}, \color{blue}{85/21}, \color{blue}{85/22}, \color{blue}{85/23}, \color{blue}{85/24}, \color{blue}{85/26}, \linebreak
 \color{black}{85/27}, \color{blue}{85/28}, \color{black}{85/29}, \color{blue}{85/31}, \color{blue}{85/32}, \color{black}{85/33}, \color{green}{85/36}, \color{black}{85/37}, \color{black}{85/38}, \color{black}{85/39}, \color{black}{85/41}, \color{green}{85/42}, \color{black}{85/43},  \linebreak
\color{red}{85/44}, \color{black}{85/46}, \color{black}{85/47},
\color{green}{85/48}, \color{black}{85/49}, \color{green}{85/52},
\color{black}{85/53}, \color{green}{85/54}, \color{green}{85/56},
\color{red}{85/57}, \color{green}{85/58}, \color{black}{85/59},
\color{black}{85/61}, \linebreak
 \color{green}{85/62}, \color{green}{85/63}, \color{green}{85/64}, \color{green}{85/66}, \color{black}{85/67}, \color{green}{85/69}, \color{black}{85/71}, \color{green}{85/72}, \color{black}{85/73}, \color{green}{85/74}, \color{green}{85/76}, \color{green}{85/77}, \color{green}{85/78}, \linebreak
 \color{black}{85/79}, \color{green}{85/81}, \color{green}{85/82}, \color{black}{85/83}, \color{green}{85/84}, \color{black}{86}, \color{black}{86/3}, \color{black}{86/5}, \color{black}{86/7}, \color{black}{86/9}, \color{black}{86/11}, \color{black}{86/13}, \color{black}{86/15}, \color{black}{86/17}, \linebreak
 \color{black}{86/19}, \color{black}{86/21}, \color{black}{86/23}, \color{black}{86/25}, \color{black}{86/27}, \color{black}{86/29}, \color{black}{86/31}, \color{black}{86/33}, \color{black}{86/35}, \color{black}{86/37}, \color{black}{86/39}, \color{black}{86/41}, \color{black}{86/45}, \linebreak
 \color{black}{86/47}, \color{black}{86/49}, \color{black}{86/51}, \color{black}{86/53}, \color{black}{86/55}, \color{black}{86/57}, \color{black}{86/59}, \color{black}{86/61}, \color{green}{86/63}, \color{red}{86/65}, \color{black}{86/67}, \color{green}{86/69}, \color{black}{86/71}, \linebreak
 \color{black}{86/73}, \color{green}{86/75}, \color{green}{86/77}, \color{black}{86/79}, \color{green}{86/81}, \color{black}{86/83}, \color{green}{86/85}, \color{black}{87}, \color{black}{87/2}, \color{black}{87/4}, \color{black}{87/5}, \color{black}{87/7}, \color{black}{87/8}, \color{black}{87/10},  \linebreak
\color{black}{87/11}, \color{black}{87/13}, \color{black}{87/14},
\color{black}{87/16}, \color{black}{87/17}, \color{black}{87/19},
\color{black}{87/20}, \color{black}{87/22}, \color{black}{87/23},
\color{black}{87/25}, \color{black}{87/26}, \color{black}{87/28},
\color{black}{87/31}, \linebreak
 \color{black}{87/32}, \color{black}{87/34}, \color{black}{87/35}, \color{black}{87/37}, \color{black}{87/38}, \color{green}{87/40}, \color{black}{87/41}, \color{black}{87/43}, \color{red}{87/44}, \color{black}{87/46}, \color{black}{87/47}, \color{black}{87/49}, \color{green}{87/50}, \linebreak
 \color{green}{87/52}, \color{black}{87/53}, \color{black}{87/55}, \color{green}{87/56}, \color{black}{87/59}, \color{black}{87/61}, \color{green}{87/62}, \color{green}{87/64}, \color{black}{87/65}, \color{black}{87/67}, \color{green}{87/68}, \color{green}{87/70}, \color{black}{87/71}, \linebreak
 \color{black}{87/73}, \color{green}{87/74}, \color{green}{87/76}, \color{green}{87/77}, \color{black}{87/79}, \color{green}{87/80}, \color{green}{87/82}, \color{black}{87/83}, \color{green}{87/85}, \color{green}{87/86}, \color{black}{88}, \color{black}{88/3}, \color{black}{88/5}, \color{black}{88/7}, \linebreak
 \color{black}{88/9}, \color{black}{88/13}, \color{black}{88/15}, \color{black}{88/17}, \color{black}{88/19}, \color{blue}{88/21}, \color{blue}{88/23}, \color{black}{88/25}, \color{blue}{88/27}, \color{blue}{88/29}, \color{black}{88/31}, \color{black}{88/35}, \color{black}{88/37}, \linebreak
 \color{black}{88/39}, \color{black}{88/41}, \color{blue}{88/43}, \color{blue}{88/45}, \color{black}{88/47}, \color{black}{88/49}, \color{blue}{88/51}, \color{blue}{88/53}, \color{black}{88/57}, \color{black}{88/59}, \color{black}{88/61}, \color{green}{88/63}, \color{black}{88/65}, \linebreak
 \color{black}{88/67}, \color{green}{88/69}, \color{black}{88/71}, \color{black}{88/73}, \color{green}{88/75}, \color{black}{88/79}, \color{green}{88/81}, \color{black}{88/83}, \color{green}{88/85}, \color{green}{88/87}, \color{black}{89}, \color{black}{89/2}, \color{black}{89/3}, \color{black}{89/4}, \linebreak
 \color{black}{89/5}, \color{black}{89/6}, \color{black}{89/7}, \color{black}{89/8}, \color{black}{89/9}, \color{black}{89/10}, \color{black}{89/11}, \color{black}{89/12}, \color{black}{89/13}, \color{black}{89/14}, \color{black}{89/15}, \color{black}{89/16}, \color{black}{89/17}, \color{black}{89/18}, \linebreak
 \color{black}{89/19}, \color{black}{89/20}, \color{black}{89/21}, \color{black}{89/22}, \color{black}{89/23}, \color{black}{89/24}, \color{black}{89/25}, \color{black}{89/26}, \color{black}{89/27}, \color{blue}{89/28}, \color{black}{89/29}, \color{black}{89/30}, \color{black}{89/31},  \linebreak
\color{blue}{89/32}, \color{black}{89/33}, \color{black}{89/34},
\color{black}{89/35}, \color{green}{89/36}, \color{black}{89/37},
\color{black}{89/38}, \color{black}{89/39}, \color{green}{89/40},
\color{black}{89/41}, \color{green}{89/42}, \color{black}{89/43},
\color{red}{89/44}, \linebreak
 \color{black}{89/45}, \color{black}{89/46}, \color{black}{89/47}, \color{green}{89/48}, \color{black}{89/49}, \color{green}{89/50}, \color{black}{89/51}, \color{green}{89/52}, \color{black}{89/53}, \color{green}{89/54}, \color{black}{89/55}, \color{green}{89/56}, \color{black}{89/57}, \linebreak
 \color{green}{89/58}, \color{black}{89/59}, \color{green}{89/60}, \color{black}{89/61}, \color{green}{89/62}, \color{green}{89/63}, \color{green}{89/64}, \color{black}{89/65}, \color{green}{89/66}, \color{black}{89/67}, \color{green}{89/68}, \color{green}{89/69}, \color{green}{89/70}, \linebreak
 \color{black}{89/71}, \color{green}{89/72}, \color{black}{89/73}, \color{green}{89/74}, \color{green}{89/75}, \color{green}{89/76}, \color{green}{89/77}, \color{green}{89/78}, \color{black}{89/79}, \color{green}{89/80}, \color{green}{89/81}, \color{green}{89/82}, \color{black}{89/83}, \linebreak
 \color{green}{89/84}, \color{green}{89/85}, \color{green}{89/86}, \color{green}{89/87}, \color{green}{89/88}, \color{black}{90}, \color{black}{90/7}, \color{black}{90/11}, \color{black}{90/13}, \color{black}{90/17}, \color{black}{90/19}, \color{black}{90/23}, \color{blue}{90/29}, \linebreak
 \color{blue}{90/31}, \color{blue}{90/37}, \color{blue}{90/41}, \color{blue}{90/43}, \color{blue}{90/47}, \color{blue}{90/49}, \color{black}{90/53}, \color{blue}{90/59}, \color{black}{90/61}, \color{black}{90/67}, \color{black}{90/71}, \color{black}{90/73}, \color{green}{90/77}, \linebreak
\color{black}{90/79}, \color{black}{90/83}, \color{blue}{90/89},
\color{black}{91}, \color{black}{91/2}, \color{black}{91/3},
\color{black}{91/4}, \color{black}{91/5}, \color{black}{91/6},
\color{black}{91/8}, \color{black}{91/9}, \color{black}{91/10},
\color{black}{91/11}, \color{black}{91/12}, \linebreak
 \color{black}{91/15}, \color{black}{91/16}, \color{black}{91/17}, \color{black}{91/18}, \color{black}{91/19}, \color{black}{91/20}, \color{black}{91/22}, \color{black}{91/23}, \color{black}{91/24}, \color{blue}{91/25}, \color{blue}{91/27}, \color{blue}{91/29}, \color{blue}{91/30}, \linebreak
 \color{blue}{91/31}, \color{blue}{91/32}, \color{blue}{91/33}, \color{blue}{91/34}, \color{blue}{91/36}, \color{black}{91/37}, \color{black}{91/38}, \color{red}{91/40}, \color{blue}{91/41}, \color{black}{91/43}, \color{black}{91/44}, \color{black}{91/45}, \color{black}{91/46}, \linebreak
 \color{black}{91/47}, \color{green}{91/48}, \color{green}{91/50}, \color{black}{91/51}, \color{black}{91/53}, \color{green}{91/54}, \color{black}{91/55}, \color{black}{91/57}, \color{red}{91/58}, \color{black}{91/59}, \color{green}{91/60}, \color{black}{91/61}, \color{green}{91/62}, \linebreak
 \color{green}{91/64}, \color{green}{91/66}, \color{black}{91/67}, \color{green}{91/68}, \color{green}{91/69}, \color{black}{91/71}, \color{green}{91/72}, \color{black}{91/73}, \color{green}{91/74}, \color{green}{91/75}, \color{green}{91/76}, \color{black}{91/79}, \color{green}{91/80}, \linebreak
 \color{green}{91/81}, \color{green}{91/82}, \color{black}{91/83}, \color{green}{91/85}, \color{green}{91/86}, \color{green}{91/87}, \color{green}{91/88}, \color{black}{91/89}, \color{green}{91/90}, \color{black}{92}, \color{black}{92/3}, \color{black}{92/5}, \color{black}{92/7}, \color{black}{92/9}, \linebreak
 \color{black}{92/11}, \color{black}{92/13}, \color{black}{92/15}, \color{black}{92/17}, \color{black}{92/19}, \color{black}{92/21}, \color{black}{92/25}, \color{black}{92/27}, \color{black}{92/29}, \color{black}{92/31}, \color{black}{92/33}, \color{black}{92/35}, \color{black}{92/37}, \linebreak
 \color{black}{92/39}, \color{black}{92/41}, \color{black}{92/43}, \color{black}{92/45}, \color{black}{92/47}, \color{black}{92/49}, \color{black}{92/51}, \color{black}{92/53}, \color{black}{92/55}, \color{black}{92/57}, \color{black}{92/59}, \color{black}{92/61}, \color{green}{92/63}, \linebreak
 \color{black}{92/65}, \color{black}{92/67}, \color{black}{92/71}, \color{black}{92/73}, \color{green}{92/75}, \color{green}{92/77}, \color{black}{92/79}, \color{green}{92/81}, \color{black}{92/83}, \color{green}{92/85}, \color{green}{92/87}, \color{black}{92/89}, \color{green}{92/91}, \linebreak
 \color{black}{93}, \color{black}{93/2}, \color{black}{93/4}, \color{black}{93/5}, \color{black}{93/7}, \color{black}{93/8}, \color{black}{93/10}, \color{black}{93/11}, \color{black}{93/13}, \color{black}{93/14}, \color{black}{93/16}, \color{black}{93/17}, \color{black}{93/19}, \color{black}{93/20}, \linebreak
 \color{black}{93/22}, \color{black}{93/23}, \color{blue}{93/25}, \color{black}{93/26}, \color{blue}{93/28}, \color{blue}{93/29}, \color{blue}{93/32}, \color{blue}{93/34}, \color{blue}{93/35}, \color{blue}{93/37}, \color{blue}{93/38}, \color{blue}{93/40}, \color{blue}{93/41}, \linebreak
 \color{blue}{93/43}, \color{blue}{93/44}, \color{blue}{93/46}, \color{blue}{93/47}, \color{black}{93/49}, \color{blue}{93/50}, \color{green}{93/52}, \color{black}{93/53}, \color{black}{93/55}, \color{green}{93/56}, \color{black}{93/58}, \color{black}{93/59}, \color{blue}{93/61}, \linebreak
 \color{green}{93/64}, \color{black}{93/65}, \color{black}{93/67}, \color{green}{93/68}, \color{green}{93/70}, \color{black}{93/71}, \color{black}{93/73}, \color{green}{93/74}, \color{green}{93/76}, \color{green}{93/77}, \color{black}{93/79}, \color{green}{93/80}, \color{green}{93/82}, \linebreak
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\section{Acknowledgment}The authors would like to thank Carl
Pomerance and an anonymous referee for their feedback, which
helped to improve both the content and the presentation of this
article.


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\begin{thebibliography}{10}

\bibitem {Anderson1974} C. W. Anderson,
The solution of $\Sigma(n)=\sigma(n)/n=a/b,$ $\Phi(n)=\phi(n)/n=a/b$
and some related considerations, unpublished manuscript, 1974.

\bibitem {HoldJ2006} J. Holdener, Conditions equivalent to the existence of odd perfect numbers,
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\bibitem {Laatsch1986} R. Laatsch, Measuring the abundancy of
integers, \textit{Math. Mag.} \textbf{59} (1986), 84--92.

\bibitem {Ryan2002} R. Ryan,
Results concerning uniqueness for $\sigma(x)/x= \sigma(p^nq^m)/(p^nq^m)$ and related topics,
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\bibitem {Weiner2000} P. A. Weiner,
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\end{thebibliography}

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\noindent 2000 {\it Mathematics Subject Classification}: Primary
11A25;
Secondary 11Y55, 11Y70.\\

\noindent {\it Keywords}: abundancy index, abundancy outlaw, sum
of divisors function, perfect numbers.

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\vspace*{+.1in}
\noindent
Received October 25 2006;
revised version received August 31 2007;  September 25 2007.
Published in {\it Journal of Integer Sequences}, September 25 2007.

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