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\begin{document}
\title{Summarized Results for 100,057\\ Second Run\\Zero Omitted}
\date{\today}
\maketitle

\section*{General}
\noindent{\bf All: } 100,055 Graphs\\
{\bf Relatively Prime: } 30,239 Graphs\\
{\bf Primitive Root: } 30,240 Graphs\\
{\bf Relatively Prime and Primitive Root: } 9,192 Graphs\\
{\bf Neither: } 48,768 Graphs\\
{\bf Not Primitive Root: } 69,815 Graphs\\
\vspace{11 pt}\\
{\bf Largest Cycle: } 100,052 ($g = 58,303$)\\
{\bf Longest Tail: }  1,589 ($g = 18,115$)\\
{\bf Shortest Maximum Cycle: } 1 (24 values of $g$)\\

\section*{Number of Components}
\noindent{\bf Observed All: }
\begin{equation*}760,744/100,055 \approx 7.603\end{equation*}


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} \frac{1}{2}\log{n} =
\frac{1}{2}\log{100,057} \approx 5.76\end{equation*}\\

\noindent{\bf Observed Not PR: }
\begin{equation*}396,228/69,815 \approx 5.68\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}
\begin{equation*}\sum_{i=1}^{100,057} \frac{1}{i} \approx 12.09
\end{equation*}\\
\noindent{\bf Observed PR: }
\begin{equation*}364,516/30,240 \approx 12.05\end{equation*}





\section*{Number of Cyclic Nodes} \noindent{\bf Observed All:}
\begin{equation*}3,041,638,223/100,055\approx 30,399.66
\end{equation*}


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} \sqrt{\pi n / 2} = \sqrt{\pi100,057/2}
\approx 396.457\end{equation*}
\noindent{\bf Observed Not PR: }
\begin{equation*}15,944,783 / 69,815 \approx 228.39\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}100,056\end{equation*}
\noindent{\bf Observed PR:}
\begin{equation*}3,025,693,440/30,240 = 100,056\end{equation*}


\section*{Number of Tail Nodes} \noindent{\bf Observed All:
}\begin{equation*}6,969,564,912
/100,055 \approx 69,657.33\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} n-\sqrt{\pi n / 2} = 100,057-\sqrt{\pi100,057/2}
\approx 99,660.55\end{equation*}\\

\noindent{\bf Observed Not PR: }
\begin{equation*}6,969,534,672 / 69,815 \approx 99,828.61\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical: }
\begin{equation*}1\end{equation*}
\noindent{\bf Observed PR:}\begin{equation*}30,240/30,240 =
1\end{equation*}\\
\noindent{\bf Observed RPPR: }\begin{equation*}9,192 / 9,192 = 1\end{equation*}\\



\section*{Number of Terminal Nodes} \noindent{\bf Observed
All:}\begin{equation*} 5,224,643,043/100,055 \approx 52,217.71
\end{equation*}\\
\noindent{\bf Observed RP:}\begin{equation*} 1,572,707,711 /
30,239 \approx 52,009.25
\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} e^{-1}n = e^{-1} * 100,057 \approx
36,808.9\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*} 5,224,612,803
 / 69,815 \approx 74,835.10
\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*} 1 \end{equation*}
\noindent{\bf Observed PR:}\begin{equation*}
30,240 / 30,240 = 1\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
9,192 / 9,192 = 1\end{equation*} \\


\section*{Number of Image Nodes} \noindent{\bf Observed
All:}\begin{equation*}
100,057-52,217.71 \approx 47,839.29\end{equation*}\\
\noindent{\bf Observed RP:}\begin{equation*}
100,057 - 52,009.25 \approx 48,047.75\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}(1-e^{-1})n = (1-e^{-1})*100,057 \approx
63,248.09\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*} 100,057 -
74,835.10 \approx 25,221.90\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}n-1=100,057 - 1 =
100,056\end{equation*}
\noindent{\bf Observed
PR:}\begin{equation*}
100,057 - 1 = 100,056\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
100,057 - 1 = 100,056\end{equation*} \\


\section*{Average Tail Length} \noindent{\bf Observed
All:}\begin{equation*}
114.216\end{equation*}\\
\noindent{\bf Observed RP:}\begin{equation*}
114.86\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
\sqrt{\pi n/8} = \sqrt{\pi100,057/8}\approx 198
\end{equation*}
\noindent{\bf Observed None:}\begin{equation*}
113.94\end{equation*}\\
\noindent{\bf Observed Not PR:}\begin{equation*}
114.22\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}1
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}
1\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
1\end{equation*} \\




\section*{Average Cycle Length} \noindent{\bf Observed
All:}\begin{equation*}
115.32\end{equation*}\\
\noindent{\bf Observed RP:}\begin{equation*}
115.41\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
\sqrt{\pi n/8} = \sqrt{\pi100,057/8}\approx 198
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*} 114.67
\end{equation*}


\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}\frac{n+1}{2} =
\frac{100,057+1}{2} = 50,029
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}
50,191.4\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
50,454.6\end{equation*} \\

\section*{Maximum Cycle Length} \noindent{\bf Observed:}
\noindent{\bf Observed All:}\begin{equation*}
19,028.01\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
c_1\sqrt{n}\approx 0.78248\sqrt{100,057} \approx 247.5
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*}
143.02\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*} 0.62432965n =
0.62432965*100057 \approx 62,468.55
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}
62,627.74\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
62,896.84\end{equation*} \\



\section*{Maximum Tail Length} \noindent{\bf Observed
All:}\begin{equation*}
218.15\end{equation*}\\
\noindent{\bf Observed RP:}\begin{equation*}
218.69\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
c_2\sqrt{n}\approx 1.73746\sqrt{100,057} \approx 549.59
\end{equation*}
\noindent{\bf Observed None:}\begin{equation*}
311.53\end{equation*}\\
\noindent{\bf Observed Not PR:}\begin{equation*}
312.20\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}1
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}
1\end{equation*} \\
\noindent{\bf Observed RPPR:}\begin{equation*}
1\end{equation*} \\



\end{document}
