Update Estimation from ckiseki

codebook/6_Math/Estimation.tex

@@ -1,12 +1,45 @@
-\begin{itemize}
-\item Estimation
-
-\begin{itemize}
-    %\itemsep-0.5em
-    \item The number of divisors of $n$ is at most around $100$ for $n<5e4$, $500$ for $n<1e7$, $2000$ for $n<1e10$, $200000$ for $n<1e19$.
-    \item The number of ways of writing $n$ as a sum of positive integers, disregarding the order of the summands. $1, 1, 2, 3, 5, 7, 11, 15, 22, 30$ for $n=0\sim 9$, $627$ for $n=20$, $\sim 2e5$ for $n=50$, $\sim 2e8$ for $n=100$.
-    \item Total number of partitions of $n$ distinct elements: $B(n)=1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597,$\\
-    $27644437, 190899322, \ldots$.
-\end{itemize}
-
-\end{itemize}
+{
+  \setlength{\tabcolsep}{1pt}
+  \setlength{\columnsep}{0pt}
+
+  \noindent
+  \begin{tabular}{@{}c|*{20}{c@{\ }}@{}}
+    $n$    & 2 & 3 & 4 & 5 & 6  & 7  & 8  & 9  & 20  & 30   & 40  & 50  & 100 \\
+    \hline
+    $p(n)$ & 2 & 3 & 5 & 7 & 11 & 15 & 22 & 30 & 627 & 5604 & 4e4 & 2e5 & 2e8 \\
+  \end{tabular}
+
+  \noindent
+  \begin{tabular}{@{}c|*{20}{c@{\ }}@{}}
+    $n$
+    & 100 & 1e3 & 1e6 & 1e9  & 1e12 & 1e15  & 1e18 \\
+    \hline
+    $d(i)$ % max _ { i <= n } d(i)
+    & 12  & 32  & 240 & 1344 & 6720 & 26880 & 103680 \\
+  \end{tabular}
+
+  % \vspace{-2.0em}
+  % \begin{center}
+  %   \begin{tabular}{c|*{20}c}
+  %     $n$  & 1 & 2 & 3 & 4  & 5   & 6   & 7    & 8     & 9     & 10    & 11 \\
+  %     \hline
+  %     $n!$ & 1 & 2 & 6 & 24 & 120 & 720 & 5040 & 40320 & 3.6e5 & 3.6e6 & 4e7 \\
+  %   \end{tabular}
+  % \end{center}
+
+  \noindent
+  \begin{tabular}{c|*{20}c}
+    $n$             & 1 & 2 & 3  & 4  & 5   & 6   & 7    & 8     & 9
+                    & 10     & 11  & 12  & 13  & 14  & 15 \\
+                    \hline
+    $\binom{2n}{n}$ & 2 & 6 & 20 & 70 & 252 & 924 & 3432 & 12870 & 48620
+                    & 184756 & 7e5 & 2e6 & 1e7 & 4e7 & 1.5e8 \\
+  \end{tabular}
+
+  \noindent
+  \begin{tabular}{c|*{20}c}
+    $n$             & 2 & 3  & 4  & 5   & 6   & 7    & 8     & 9 & 10     & 11  & 12  & 13  \\
+                    \hline
+    $B_n$           & 2 & 5 & 15 & 52 & 203 & 877 & 4140 & 21147 & 115975 & 7e5 & 4e6 & 3e7 \\
+  \end{tabular}
+}