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\begin{center}
\vskip 1cm{\LARGE\bf 
More Upper Bounds on Taxicab and  \\
\vskip .1in
Cabtaxi Numbers
}
\vskip 1cm
\large
Po-Chi Su    \\
Hsing-Hua Senior High School\\
Miaoli 35144 \\
Taiwan \\
\href{mailto:poky@webmail.mlc.edu.tw}{\tt poky@webmail.mlc.edu.tw} 
\end{center}

\vskip .2 in



\begin{abstract}
For positive integers $a,b$ and integers $x,y$ such that
$S=a^3+b^3=x^3+y^3$, we prove that $ x+y\equiv a+b$ (mod 6);
moreover, we give a parametric function $r_i\rightarrow
(x(r_i),y(r_i))$ with $(x(r_i))^3+(y(r_i))^3=a^3+b^3$ for chosen
parameters $r_i$, and we conjecture that most such $S$ are multiples
of 18 if $S$ is large enough. Accordingly, {\it floating sieving} is
introduced and upper bounds on the Cabtaxi numbers $\Ca(n)$ with
$43\leq n\leq 57$, and the Taxicab numbers $\Ta(n)$ with $n=23,24$
are given. Among them, $\Ta(n)$ with $n=23,24$, and $\Ca(n)$ with
$n=43,44$, are included in the \emph{On-Line Encyclopedia of Integer
Sequences}.
\end{abstract}

\section{Introduction}
\label{intro}
The $n$-th \emph{Taxicab number $\Ta(n)$} (respectively, the $n$-th \emph{Cabtaxi number $\Ca(n)$}) is the smallest that can be expressed as a sum of two cubes of positive integers (respectively, integers)
in $n$ ways, which are called $n$ {\it decompositions} \cite{Boy1,Boy2}. For any positive $n$, Fermat proved the existence of $\Ta(n)$, as shown in the book by Hardy and Wright \cite[Theorem\ 412]{Har}. Clearly, $\Ca(n)\le \Ta(n)$ by definition. Specifically, Dardis found $\Ta(5)$ in 1994, but $\Ta(6)$ was not determined until 14 year later. Indeed, Wilson \cite{Wil} found an upper bound on $\Ta(6)$ in 1997,
$$
\begin{array}{ll}
\Ta(6) & \le 8,230,545,258,248,091,551,205,888 \\
      & =2^9\cdot3^3\cdot7\cdot13\cdot19^3\cdot31\cdot67^3\cdot79\cdot109^3. \\
\end{array}
$$
Rathbun improved this upper bound in 2002 to
$$
\begin{array}{ll}
\Ta(6) & \le 24,153,319,581,254,312,065,344         \\
      & = 2^6\cdot3^3\cdot7^4\cdot13\cdot19\cdot43\cdot73\cdot79^3\cdot97\cdot157. \\
\end{array}
$$
Calude et al. \cite{Cal} showed that this upper bound is $\Ta(6)$ with a probability of greater than 0.99. Finally, in 2008, Hollerbach showed that it was exactly $\Ta(6)$ \cite{Boy1,Boy2}.

Clearly, the determinations of $\Ta(n)$ or $\Ca(n)$ are not trivial. Indeed, these are problems \href{http://oeis.org/A011541}{\underline{A011541}} and \seqnum{A047696} in the \emph{On-Line Encyclopedia of Integer Sequences}, \emph{OEIS} \cite{Slo}. Up to now, those known $\Ta(n)$ and $\Ca(n)$ are given in Table \ref{t1}. Straightforward relations, called {\it magnifications}, among these values; for example, $\Ca(3) = 2^3\cdot \Ca(2)$, $\Ca(9) = 5^3\cdot 67^3 \cdot \Ca(7) = 2^3\cdot 5^3\cdot 67^3\cdot \Ca(6)$ are given in Fig.~\ref{fig1}. Magnifications among the best known upper bounds on $\Ta(n)$ and $\Ca(n)$ are described similarly by the diagrams in Figs.~\ref{fig13} and \ref{fig14} in the Appendix.


\begin{table}[h]
\centering
\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{$\Ta(n)$}  & \multicolumn{1}{|c|}{$\Ca(n)$} \\
\hline
$\Ta(1)=2$ & $\Ca(1)=1$ \\
\hline
$\Ta(2)=7\cdot13\cdot19$ & $\Ca(2)=7\cdot13$ \\
\hline
$\Ta(3)=3^3\cdot7\cdot31\cdot67\cdot223$ & $\Ca(3)=2^3\cdot7\cdot13$ \\
\hline
$\Ta(4)=2^{10}\cdot3^3\cdot7\cdot13\cdot19\cdot31\cdot37\cdot127$ & $\Ca(4)=2^3\cdot3^3\cdot7^3\cdot37$ \\
\hline
$\Ta(5)=2^6\cdot3^3\cdot7^4\cdot13\cdot19\cdot43\cdot73\cdot97\cdot157$ & $\Ca(5)=3^3\cdot7\cdot13\cdot31\cdot79$ \\
\hline
$\Ta(6)=2^6\cdot3^3\cdot7^4\cdot13\cdot19\cdot43\cdot73\cdot79^3\cdot97\cdot157$ & $\Ca(6)=3^3\cdot7^4\cdot19\cdot31\cdot37$\\
\hline
 & $\Ca(7)=2^3\cdot3^3\cdot7^4\cdot19\cdot31\cdot37$ \\
\hline
 & $\Ca(8)=2^3\cdot3^3\cdot7^4\cdot19\cdot23^3\cdot31\cdot37$ \\
\hline
 & $\Ca(9)=2^3\cdot3^3\cdot5^3\cdot7^4\cdot19\cdot31\cdot37\cdot67^3$ \\
\hline
 & $\Ca(10)=2^3\cdot3^3\cdot5^3\cdot7^4\cdot13^3\cdot19\cdot31\cdot37\cdot67^3$ \\
\hline
\end{tabular}
\caption{Known $\Ta(n)$ and $\Ca(n)$}\label{t1}
\end{table}


\begin{figure}[h]
  \begin{center}
  \includegraphics[scale=0.5]{f1.eps}
  \caption{Magnifications among $\Ta(n)$ and $\Ca(n)$}\label{fig1}
  \end{center}
\end{figure}

Following the discovery of $\Ta(n)$ with $1\le n\le 6$ and $\Ca(n)$ with $1\le n\le 10$, in 2008, Boyer found upper bounds on $\Ta(7),\ldots, \Ta(19)$ and $\Ca(11), \ldots, \Ca(30)$ respectively \cite{Boy1,Boy2}. As mentioned by Boyer later on his webpage, Moore improved the upper bounds on $\Ca(11), \Ca(12), \Ca(14)$, and Boyer and Wroblewski improved the upper bounds on $\Ta(11)$, $\ldots, \Ta(19)$ and $\Ca(13), \Ca(15), \ldots, \Ca(30)$. Boyer also gave the upper bounds on $\Ta(20)$, $\Ta(21)$, $\Ta(22)$ and $\Ca(31), \ldots, \Ca(42)$\cite{Boy2}. In this paper, the known upper bounds on $\Ta(n)$ and $\Ca(n)$, given on Boyer's webpage, are denoted by $\BTa(n)$ with $7\le n\le 22$, and $\BCa(n)$ with $11\le n\le 42$, respectively. Complete decompositions of $\BTa(n)$ with $7\le n\le 12$ and $\BCa(n)$ with $11\le n\le 22$ can be found in \cite{Boy2}.

For given positive integers $a$ and $b$, a condition for sieving integers $x\ge y$ to satisfy $S=a^3+b^3=x^3+y^3$ is given in Lemma \ref{lem1}. The relation $x + y \equiv a+ b$ (mod 6) is a crucial condition in sieving the upper bounds on $\Ta(n)$ and $\Ca(n)$. We conjecture that $\Ta(n)$ with $n\ge 7$ and $\Ca(n)$ with $n\ge 11$ are multiples of 18 (as discussed in Section \ref{para}). If $S=a^3+b^3$ is a multiple of 18, then the introduced \emph{sieving process} can be utilized to find the upper bounds on $\Ta(n)$ and $\Ca(n)$. Based on the sieving conditions in Theorem \ref{thm1}, together with $\BTa(n)$ and $\BCa(n)$ that were provided by Boyer, the sieving process is modified herein by applying the concept of floating sieving to reduce the number of computations ({Section \ref{sieve}}). Applying the floating sieving process, the upper bounds on $\Ta(23),\Ta(24)$ and $\Ca(43),\ldots, \Ca(57)$ together with their corresponding parameters are given in Section \ref{up}. The upper bounds on $\Ta(23), \Ta(24), \Ca(43)$ and $\Ca(44)$ were collected in \emph{OEIS}, October 2014.


\begin{section}{Magnifications}
\label{mag}
{\it Magnification}, introduced by Wilson \cite{Wil} and Boyer \cite{Boy1}, is an efficient and frequently used technique for finding the upper bounds on $\Ta(n)$ and $\Ca(n)$, that is finding a number with $n+1$ decompositions starting from a number with $n$ decompositions.


\subsection{Magnifications among $\BTa(n)$ and among $\BCa(n)$}
\label{sec2-1}
If $S$ can be described in $n$ ways as a sum of two cubes, $S=x^3_i+y^3_i$ with $i=1,2,\ldots,n$, and $k$ is an integer, then $Sk^3$ can be described in at least $n$ ways as a sum of two cubes: $Sk^3=(kx_i)^3+(ky_i)^3$ with $i=1,\ldots,n$. If a value $k$ is found such that there exists another sum of two cubes, $Sk^3=x^3_{n+1}+y^3_{n+1}$ with $(x_{n+1},y_{n+1})\ne (x_i,y_i)$ for $i=1,2,\ldots,n$, then $Sk^3$ can be described as $n+1$ sums of two cubes, yielding an upper bound on $\Ta(n+1)$ or $\Ca(n+1)$. Such a number $k$ is called a \emph{splitting factor} by Boyer \cite{Boy1}.

For most values of $n$, the quotient ${\BCa(n)/\BCa(n-1)}$ or $\BTa(n) / \BTa(n-1)$ is $k^3$ or the product $k_1^3\cdot k_2^3$ (where $k,k_1$ and $k_2$ are primes). Starting from $\BTa(7)$ and $\BCa(11)$ on, the quotients $\BTa(n) / \BTa(n-1)$ and $\BCa(n) / \BCa(n-1)$ are given in Table \ref{t2} respectively. Some of the quotients are complicated (see below for examples). However, most of them are in a simple form.

\begin{enumerate}
\itemsep=-5pt
\item 	[1.] $\BTa(11) / \BTa(10)= (2^3\cdot 5^3\cdot 13^2\cdot 17^3\cdot 31\cdot 37\cdot 97^2\cdot 109^3)\ /\ ( 19\cdot 29^3\cdot 101^3\cdot 127^3)$
\item	[2.] $\BCa(11) / \BCa(10)= (2^4\cdot 3^3\cdot 37^2\cdot 43\cdot 61^3 )\ /\ ( 5^3\cdot 7\cdot 13^2\cdot 31\cdot 67^2)$
\item	[3.] $\BCa(13) / \BCa(12)= (2^2\cdot 3^3\cdot 7\cdot 13^3\cdot 109\cdot 193 )\ /\  (19\cdot 43\cdot 61^2\cdot 67)$
\item	[4.] $\BCa(14) / \BCa(13)= (2^4\cdot 19\cdot 31^3\cdot 43\cdot 61^2\cdot 67 )\ /\  (3^3\cdot 7\cdot 13^3 \cdot109\cdot 193)$
\item	[5.] $\BCa(15) / \BCa(14)= (3^3 \cdot7\cdot 13^3 \cdot73^3\cdot 109\cdot 193  )\ /\ ( 2^4\cdot 19\cdot 31^3\cdot 43\cdot 61^2\cdot 67)$
\item	[6.] $\BCa(19) / \BCa(18)= (5^3\cdot 11^3\cdot 37\cdot 43\cdot 61^2\cdot 67^3\cdot 109^2\cdot 157 )\ /\ ( 2^6\cdot 13\cdot 19^6\cdot 31^2\cdot 73^2\cdot 193)$
\item	[7.] $\BCa(20) / \BCa(19)= (2^6\cdot 5^3\cdot 13\cdot 19^3\cdot 31^2\cdot 73^2\cdot 103^3\cdot 193) \ /\ ( 11^3\cdot 37\cdot 43\cdot 61^2\cdot 67^3\cdot 109^2\cdot 157)$
\item	[8.] $\BCa(21) / \BCa(20)= (11^3\cdot 43\cdot 61^2\cdot 67^3\cdot 79^3\cdot 109^2\cdot 157) \ /\ (2^6\cdot 5^3\cdot 13\cdot 31^2\cdot 37^2\cdot 73^2\cdot 103^3\cdot 193)$
\end{enumerate}


\begin{table}[h]
\centering
\subtable[$\BTa(n)$]{
\label{t2-1}
         \begin{tabular}{|c|c|}
         \hline
          $n$ & \small $\BTa(n) / \BTa(n-1)$ \\
         \hline
          7	  & $101^3$ \\
         \hline
          8	  & $127^3$ \\
         \hline
          9   &	$139^3$ \\
         \hline
         10   &	$13^3\cdot 29^3$ \\
         \hline
         11	  & See 1.   \\
         \hline
         12	  & $3^3\cdot 19^3$ \\
         \hline
         13   &	$3^3\cdot 61^3$ \\
         \hline
         14   &	$397^3$ \\
         \hline
         15	  & $503^3$ \\
         \hline
         16   &	$2^3\cdot 607^3$ \\
         \hline
         17   &	$4261^3$ \\
         \hline
         18	  & $37^3\cdot 181^3$ \\
         \hline
         19   &	\small $5^6\cdot 457^3\cdot 521^3 / 4261^3$ \\
         \hline
         20   &	$4261^3$ \\
         \hline
         21   &	$127^3\cdot 197^3$ \\
         \hline
         22	  & $11^3\cdot 31^3\cdot 103^3$ \\
         \hline
         \end{tabular}

}
\qquad
\subtable[$\BCa(n)$]{
\label{t2-2}
        \begin{tabular}{|c|c|c|c|}
        \hline
        $n$   & \small $\BCa(n) / \BCa(n-1)$ & $n$ & \small$\BCa(n) / \BCa(n-1)$ \\
        \hline
        11	  & See 2.              & 27  & $5^6$ \\
        \hline
        12	  & $19^3$	            & 28  &	$7^3\cdot 13^3\cdot 97^3 / 5^6\cdot 17^3$ \\
        \hline
        13	  & See 3.              & 29  &	$17^3$ \\
        \hline
        14	  & See 4.              & 30  &	$5^6$   \\
        \hline
        15	  & See 5.              & 31  &	$29^3$ \\
        \hline
        16	  & $19^3$                & 32  &	$43^3$ \\
        \hline
        17    &	$2^6\cdot31^3 / 19^3$ & 33  &	$181^3$ \\
        \hline
        18	  & $19^3$                & 34  & $193^3$ \\
        \hline
        19	  & See 6.              & 35  &	\small $397^3\cdot 457^3 / 181^3\cdot 193^3$ \\
        \hline
        20	  & See 7.              & 36  & $181^3$ \\
        \hline
        21	  & See 8.               & 37  &	$101^3\cdot 229^3 / 181^3$ \\
        \hline
        22	  & $37^3 / 3^3$	        & $38$  &	$181^3$ \\
        \hline
        23	  & $3^3$	                & $39$  &	$163^3$ \\
        \hline
        24	  & $17^3$	            & $40$  &	$193^3$ \\
        \hline
        25	  & $139^3 / 17^3$	    & $41$  &	$223^3$ \\
        \hline
        26	  & $17^3$	            & $42$  &	$307^3$ \\
        \hline
        \end{tabular}

}
\caption{Quotients of consecutive $\BTa(n)$ and $\BCa(n)$}\label{t2}
\end{table}

The magnifications among $\BTa(7),\ldots,BTs(22)$ and
$\BCa(11),\ldots,\BCa(42)$ can be found in Figs.~\ref{fig2},
\ref{fig3}, \ref{fig4}, \ref{fig5} and also Figs.~\ref{fig13} and
\ref{fig14} in the Appendix.


\subsection{Consecutive magnifications}
\label{sec2-2}
 If $k_i$ with $1\le i\le h$ are splitting factors of $S$ that can be described by $n$ sums of two cubes, such that $k_i^3S$ can be described by $n+1$ sums of two cubes, then $k_1^3\cdot k_2^3\cdots k_h^3\cdot S$ may be described by $n+h$ sums of two cubes, as in Example \ref{ex1}. The $k$-th known upper bounds, from small to large, are denoted by $\BTa(n,k)$ and $\BCa(n,k)$ respectively. Some upper bounds on $\Ca(n+h)$ may be improved by applying the magnification technique over some $\BCa(n, k)$, although these are not the best known upper bounds when $k >1$, as in Examples \ref{ex2} and \ref{ex3}.


\begin{example}
\label{ex1}
$\Ta(6$) can be described as 6 sums of two cubes; $101^3\cdot \Ta(6)$ and $127^3\cdot \Ta(6)$ can be described as 7 sums of two cubes, and $101^3\cdot 127^3\cdot \Ta(6)$ can be described as 8 sums of two cubes. This gives
$$\Ta(8)\le \BTa(8) = 127^3\cdot \BTa(7) = 101^3\cdot 127^3\cdot  \Ta(6).$$
When $k=23, 29, 38, 43$, each $k^3\cdot \Ca(10)$ can be described as 11 sums of two cubes, and $23^3\cdot29^3\cdot38^3\cdot43^3\cdot \Ca(10)$ can be described by 14 sums of two cubes. (see also Examples \ref{ex4} and \ref{ex5} in Section \ref{sec4-1}).
\end{example}


\begin{figure}[h]
\centering
\includegraphics[scale=0.52]{f2.eps}
\caption{Magnifications among $\Ta(6)$, $\BTa(7)$ and $\BTa(8)$}
\label{fig2}
\end{figure}


\begin{example}
\label{ex2}
The second and third smallest of the numbers with 9 decompositions are denoted
by $\BCa(9, 2)$ and $\BCa(9, 3)$, respectively:
$$
\begin{array}{ll}
\BCa(9, 2) & = 2^6\cdot 3^3\cdot 7^7\cdot 19\cdot 31\cdot 73\cdot 97\cdot 139 ,\\
\BCa(9, 3) & = 2^7\cdot 3^6\cdot 7^3\cdot 13\cdot 19\cdot 37^3\cdot 43\cdot 67 .\\
\end{array}
$$
The upper bounds on $\Ca(11)$ that were derived by Boyer and Moore in 2008 are
$$	
\begin{array}{cl}
\Ca(11)  & \le 13^3\cdot 17^3\cdot \BCa(9, 2), \text{ and} \\
\BCa(11) & \le 61^3\cdot \BCa(9, 3) \\
\end{array}
$$
respectively. Although $\BCa(9, 3)$ is larger than $\BCa(9, 2)$, we have
$$\Ca(11) \le \BCa(11) = 61^3\cdot \BCa(9, 3) < 13^3\cdot 17^3\cdot \BCa(9, 2).$$
Fig.~\ref{fig3} presents the magnifications among the best known bounds.
\end{example}


\begin{figure}[h]
\centering
\includegraphics[scale=0.47]{f3.eps}
\caption{$\BCa(11),\BCa(12),\BCa(14)$ associated with $\BCa(9, 3)$}
\label{fig3}
\end{figure}


\begin{example}
\label{ex3}
The fifth smallest known with 10 decompositions is
\begin{enumerate}
\item[] $\BCa(10, 5) = 2^9\cdot 3^3\cdot 7^4\cdot 13^4\cdot 19^3\cdot 61\cdot 109\cdot 193$.
\end{enumerate}
The upper bounds associated with $\BCa(10, 5)$ are given below:

\begin{enumerate}
\itemsep=-5pt
\item[]$\BCa(13)=3^6\cdot 37^3 \cdot \BCa(10, 5)$,
\item[]$\BCa(15)=3^6\cdot 37^3\cdot 73^3\cdot \BCa(10, 5)=73^3\cdot \BCa(13)$,
\item[]$\BCa(16)=3^6\cdot 19^3\cdot 37^3\cdot 73^3\cdot \BCa(10, 5) =19^3\cdot \BCa(15)$,
\item[]$\BCa(17)=2^6\cdot 3^6\cdot 31^3\cdot 37^3\cdot 73^3\cdot \BCa(10, 5) =2^6\cdot 31^3\cdot \BCa(15)$,
\item[]$\BCa(18)=2^6\cdot 3^6\cdot 19^3\cdot 31^3\cdot 37^3\cdot 73^3\cdot \BCa(10, 5)=19^3\cdot \BCa(17)$,
\item[]$\BCa(20)=2^6\cdot 3^6\cdot 5^6\cdot 31^3\cdot 37^3\cdot 73^3\cdot 103^3\cdot  \BCa(10, 5)=2^6\cdot 5^6\cdot 31^3\cdot 103^3\cdot \BCa(15)$,
\end{enumerate}
which are summarized in Fig.~\ref{fig4}.
\end{example}

In Fig.~\ref{fig5}, the magnifications among $\BCa(15), \BCa(16), \BCa(17)$ and $\BCa(18)$ are displayed in the shape of a parallelogram. Similar situations also hold in $\BCa(23)$, $\BCa(24)$, $\BCa(25)$ and $\BCa(26)$, as displayed in 
Fig.~\ref{fig7}, and $\BCa(35)$, $\BCa(36)$, $\BCa(37)$, $\BCa(38)$, as displayed in Fig.~\ref{fig9}, respectively.


\begin{figure}[h]
\centering
\includegraphics[scale=0.46]{f4.eps}
\caption{$\BCa(n)$ associated with $\BCa(10, 5)$}
\label{fig4}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[scale=0.47]{f5.eps}
\caption{Magnifications among $\BCa(15)$, $\BCa(16)$, $\BCa(17)$, $\BCa(18)$}\label{fig5}
\end{figure}
\end{section}


\begin{section}{Parameters of decompositions}
\label{para}
If $S=a^3+b^3$, then $a^3+b^3$ is called a decomposition of $S$ as a sum of two cubes, abbreviated as a {\it decomposition} of $S$. Hence $\Ta(n)$ (resp., $\Ca(n)$) is the smallest positive integer with $n$ decompositions with nonnegative integral summands (or respectively integral summands). We will show that each decomposition of $S$ can be expressed as a parametric function;
see Theorem \ref{thm1}.


\subsection{Decompositions as sums of two cubes}
\label{sec3-1}
Consider decompositions $S=x^3_1+y^3_1=x^3_2+y^3_2$ as sums of two cubes, and observe that $(x_1+y_1)-(x_2+y_2)$ is a multiple of 6 in Table \ref{t3}, and will be proved in Lemma \ref{lem1}.


\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$S$     & $x_1$& $y_1$& $x_2$& $y_2$& $(x_1+y_1)-(x_2+y_2)$ \\	
\hline
1,729   & 10 &  9 & 12 &  1 &  6 \\
\hline
4,104   & 15 &  9 & 16 &  2	&  6 \\
\hline
20,683  & 24 & 19 & 27 & 10	&  6 \\
\hline
39,312	& 33 & 15 & 34 &  2	& 12 \\
\hline
40,033	& 33 & 16 & 34 &  9	&  6 \\
\hline
65,728	& 33 & 31 & 40 & 12	& 12 \\
\hline
64,232	& 36 & 26 & 39 & 17	&  6 \\
\hline
134,379	& 43 & 38 & 51 & 12	& 18 \\
\hline
149,389	& 50 & 29 & 53 &  8	& 18 \\
\hline
171,288	& 54 & 24 & 55 & 17	&  6 \\
\hline
\end{tabular}
\caption{Relations between pairs of decompositions}\label{t3}
\end{table}


\begin{lemma}
\label{lem1}
For integers $a,b,x,y$ with $a < x$ and $a^3+b^3=x^3+y^3$, then
$$x+y\equiv a+b \text{ (mod 6)}.$$
\end{lemma}

\begin{proof}
We notice that $a^3-a=(a-1)a(a+1)$ is a multiple of 6, and then $a^3\equiv a$ (mod 6) holds for any integer $a$. Hence, $x+y\equiv a+b$ (mod 6).
\end{proof}

Based on observations on known $\Ta(n)$ with $n=4,5,6$, $\BTa(n)$ with $7\le n\le 12$, $\Ca(n)$ with $7\le n\le 10$, and $\BCa(n)$ with $11\le n\le 22$, note that $6|(x+y)$ in each decomposition $S=x^3+y^3$ and therefore $x^3+y^3$ is a multiple of 18 because of $x^3+y^3=(x+y)((x+y)^2-3xy)$. We conjecture that $\Ta(n)$ with $n\ge 7$ and $\Ca(n)$ with $n\ge 11$ are all multiples of 18. This is because when the magnification technique is used to find the upper bound on $\Ta(n+1)$ and $\Ca(n+1)$ from $\BTa(n)$ and $\BCa(n)$ of multiples of 18, the one derived will also be a multiple of 18. Section \ref{sieve} introduces the sieving process that can be used to find the upper bounds on $\Ta(n)$ and $\Ca(n)$.


\subsection{Parameters of decompositions}
\label{sec3-2}
All sums of two cubes can be expressed as a parametric function (see Theorem \ref{thm1}), and this fact leads to a sieving process in Section \ref{sieve}.

For a given $S=a^3+b^3$ with positive integers $a\ge b$, to determine integers $x\ge y$ such that $S=a^3+b^3=x^3+y^3$, $k=x+y$ is introduced and $x,y$ must be obtained for appropriate $k$ (or else may have no solution).
$$
\begin{cases}
   x+y=k, \\
   x^3+y^3=S.
\end{cases}
$$
Hence, $x=x(k,S)$ and $y=y(k,S)$ can be expressed as functions of $k$ and $S$. Substituting $y=k-x$ into $x^3+(k-x)^3=S$ yields $3x^2-3kx+(k^3-S)/k=0$ and therefore,
$$x=(3k+\sqrt{(3k)^2-4\cdot3\cdot(k^3-S)/k})/6,$$
$$y=(3k-\sqrt{(3k)^2-4\cdot3\cdot(k^3-S)/k})/6.$$
The necessary and sufficient condition for $x$ and $y$ to be positive integers is that the expression $-3k^2+12S/k\ge 0$ inside the square root is a perfect square. Since $-3k^2+12S/k\ge 0$ and $k^3=(x+y)^3>x^3+y^3=S$, we then have $\sqrt[3]{S}<k\le\sqrt[3]{4S}$.

Repeating the above for $k=x+y=6r$ yields $x^2-6rx+12r^2-S/18r=0$. Substituting $y=6r-x$ yields
$$x=3r+\sqrt{-3r^2+(S/18r)},$$
$$y=3r-\sqrt{-3r^2+(S/18r)}.$$
The necessary and sufficient condition for $x$ and $y$ to be positive integers is that $-3r^2+S/18r$ is a perfect square. Moreover, $\sqrt[3]{S/216}<r\le\sqrt[3]{S/54}$ because $-3r^2+S/18r\ge 0$ and $(6r)^3=(x+y)^3>x^3+y^3=S$. The above is summarized as follows:


\begin{theorem}
\label{thm1}
If $S=x^3+y^3,x+y=k$ and $x\ge y$, then
$$x=\frac{3k+\sqrt{-sk^2+12S/k}}{6},y=\frac{3k-\sqrt{-3k^2+12S/k}}{6}.$$
Moreover,
\begin{enumerate}
\item [(a)]  $x=\frac{3k+\sqrt{-3k^2+12S/k}}{6}$ and $y=\frac{3k-\sqrt{-3k^2+12S/k}}{6}$ are positive rationals if and only if $k|S$, $\sqrt[3]{S}<k\le\sqrt[3]{4S}$, and $-3r^2+12S/k$ are perfect squares.
\item [(b)]  When $k=6r$, $x=3r+\sqrt{-3r^2+(S/18r)}$ and $y=3r-\sqrt{-3r^2+(S/18r)}$ are positive integers if and only if $18|S$, $r$ is a divisor of $S/18$, $\sqrt[3]{S/216}<r\le\sqrt[3]{S/54}$ and $-3r^2+(S/18r)$ is a perfect square.
\end{enumerate}
\end{theorem}

Among the sieving conditions in Theorem \ref{thm1}, the condition of perfect square is most crucial, as shown in Examples \ref{ex4} and \ref{ex5}. The condition that $r$ is a factor of $S/18$ is required in the Sieving Process and floating sieving process. When $n$ parameters
$r=r_1,r_2,\ldots,r_n$ are sieved, then the parametric functions
$$r_i\rightarrow (x_i,y_i)=(x(r_i),y(r_i))$$
provides $n$ decompositions of $S=(x(r_i))^3+(y(r_i))^3$ with $1\le i\le n$.
\end{section}


\begin{section}{Sieving and floating sieving}
\label{sieve}
Based on the parametric expression given in Theorem \ref{thm1}, a sieving process for upper bounds on $\Ta(n)$ and $\Ca(n)$ is given in Section \ref{sec4-1}. Upper bounds on $\Ta(n$) with $n = 7, 8, 9$ and on $\Ca(n)$ with $n = 11, \ldots, 16$, derived by this process are given in Examples \ref{ex4}, \ref{ex5} and \ref{ex6} with illustrations. To reduce computational load, the floating sieving process is introduced in Section \ref{sec4-2} with illustrations by applying the concept of floating sieving.


\subsection{Sieving process}
\label{sec4-1}
For given integers $a$ and $b$, the conditions that govern the parameters given in Theorem \ref{thm1} can be used to sieve for possible integers $x,y$ that satisfy $S=a^3+b^3=x^3+y^3$, providing a means of exploring upper bounds on $\Ta(n)$ and $\Ca(n)$ respectively.

$\newline$
\textbf{Sieving process}
\begin{enumerate}
\itemsep=-5pt
\item Input $S=a^3+b^3$ and $k$, let $counter = 0$.
\item List all positive factors $r_1<r_2<r_3\cdots<r_t$ of $Sk^3/18$.
\item For $i =1,\cdots, t$, \\
      If $\sqrt[3]{Sk^3/216}<r_i \le \sqrt[3]{Sk^3/54}$, \\
      \indent\hspace{4mm} If $-3r_i^2+Sk^3/18r_i$ is a perfect square, \\
      \indent\hspace{8mm} output $r_i$, $x(r_i)=3r_i+\sqrt{-3r^2_i+Sk^3/18r_i}$ and $y(r_i)=3r_i-\sqrt{-3r^2_i+Sk^3/18r_i}$.\\
      \indent\hspace{8mm} $counter \leftarrow counter+1, i\leftarrow i+1$, return to Step 3. \\
      \indent\hspace{4mm} otherwise $i\leftarrow i+1$, return to Step 3. \\
      Otherwise, $i\leftarrow i+1$, return to Step 3.
\item Output $counter$.
\end{enumerate}

Let $S = a^3+b^3$. To find $x, y$ with $Sk^3=x^3+y^3$, the function \emph{counter} gives the number of decompositions, it is more likely to increase for prime $k$. If the condition $\sqrt[3]{Sk^3/216}<r\le\sqrt[3]{Sk^3/54}$ above is replaced by $0<r\le\sqrt[3]{Sk^3/54}$, then all integral solutions of $Sk^3=k^3(a^3+b^3)=x^3+y^3$ can be derived, yielding a sieving process for Cabtaxi numbers. In the following examples, the sieving process is illustrated in terms of $\Ta(6)$ with $k=101,127$ and $\Ca(10)$ with $k=23,29,38,43,127$.


\begin{example}
\label{ex4}
For $S=\Ta(6)=2^6\cdot 3^3\cdot 7^4\cdot 13\cdot 19\cdot 43\cdot 73\cdot 79^3\cdot 97\cdot 157$ and $k=101$, $Sk^3$ has 143,360 positive factors, of the 61,440 positive factors of $Sk^3/18$, 629 lie between $\sqrt[3]{Sk^3/216}$ and $\sqrt[3]{Sk^3/54}$, and finally $-3r^2+Sk^3/18r$ is a perfect square for only 7 of them. Therefore, 7 decompositions are derived by using Theorem \ref{thm1}, yielding $\Ta(7)\le 101^3\cdot \Ta(6)$.
\end{example}


\begin{example}
\label{ex5}
For $S=\Ta(6)\cdot 101^3= 2^6\cdot 3^3\cdot 7^4\cdot 13\cdot 19\cdot 43\cdot 73\cdot 79^3\cdot 97\cdot 101^3\cdot 157$ and $k=127$, $Sk^3$ has $573,440$ factors, of the 245,760 positive factors of $Sk^3/18$, 2,004 lie between $\sqrt[3]{Sk^3/216}$ and $\sqrt[3]{Sk^3/54}$, and finally $-3r^2+Sk^3/18r$ is a perfect square for only 8 of them. Therefore, 8 decompositions are derived by using Theorem \ref{thm1}, yielding
$$\Ta(8)\le 101^3\cdot 127^3\cdot \Ta(6).$$
Notably, the numbers 101 and 127 above are primes, and $counter = 9$ is derived for $101^3\cdot 127^3\cdot 139^3\cdot \Ta(6)$, which therefore has 9 decompositions, so
$$\Ta(9)\le 101^3\cdot 127^3\cdot 139^3\cdot \Ta(6).$$
\end{example}


\begin{example}
\label{ex6}
For $S = \Ca(10)$, then $counter = 11$ is derived when $k=23, 29, 38, 43$ and 46. Moreover, each of $23^3\cdot \Ca(10)$, $23^3\cdot 29^3\cdot \Ca(10)$, $23^3\cdot 29^3\cdot 38^3\cdot \Ca(10)$, and $23^3\cdot 29^3\cdot 38^3\cdot 43^3\cdot \Ca(10)$ have 11, 12, 13 and 14 parameters respectively. The above bounds on $\Ca(12)$, $\Ca(13)$, $\Ca(14)$ can be improved further. Consider the prime $k=127\le 23\cdot 29$, for which $counter = 12$, meaning that $127^3\cdot \Ca(10)$ also has 12 parameters, a better bound, so
$$\Ca(12)\le 127^3\cdot \Ca(10).$$
Similar arguments show that
\begin{enumerate}
\itemsep=-5pt
\item[]$29^3\cdot 127^3\cdot \Ca(10)$ has 13 parameters, an upper bound of $\Ca(13$),
\item[]$29^3\cdot 43^3\cdot 127^3\cdot \Ca(10)$ has 14 parameters, an upper bound of $\Ca(14)$,
\item[]$23^3\cdot 29^3\cdot 38^3\cdot 127^3\cdot \Ca(10)$ has 15 parameters, an upper bound of $\Ca(15)$,
\item[]$23^3\cdot 29^3\cdot 38^3\cdot 43^3\cdot 127^3\cdot \Ca(10)$ has 16 parameters, an upper bound of $\Ca(16)$.
\end{enumerate}

We let $\SCa(11),\ldots,\SCa(16)$ denote the upper bounds on
$\Ca(11),\ldots,\Ca(16)$ obtained by sieving process based on $Ca(10)$.
These bounds are summarized below for reference, though they are not as
good as bounds given by Boyer. The magnifications among them are given
in Fig.~\ref{fig6}. The same technique can also be used to derive upper
bounds on $\Ca(43),\ldots, \Ca(57)$ based on $\BCa(42)$; see Theorem
\ref{thm2}, and upper bounds on $\Ta(23)$, $\Ta(24)$ based on
$\BTa(22)$ as well; see Theorem \ref{thm3}.

\bigskip
$
\begin{array}{ccl}
\Ca(11) & \le & \BCa(11)\le \SCa(11) = 23^3\cdot \Ca(10) \\
       & =   & 2^3\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19\cdot 23^3\cdot 31\cdot 37\cdot 67^3, \\
\Ca(12) & \le & \BCa(12)\le \SCa(12) = 127^3\cdot \Ca(10)                  \\
       & =   & 2^3\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19\cdot 31\cdot 37\cdot 67^3\cdot 127^3,  \\
\Ca(13) & \le & \BCa(13)\le \SCa(13) = 29^3\cdot 127^3\cdot \Ca(10)  \\
       & =   & 2^3\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19\cdot 29^3\cdot 31\cdot 37\cdot 67^3\cdot 127^3, \\
\Ca(14) & \le & \BCa(14)\le \SCa(14) = 29^3\cdot 43^3\cdot 127^3\cdot \Ca(10) \\
       & =   & 2^3\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19\cdot 29^3\cdot 31\cdot 37\cdot 43^3\cdot 67^3\cdot 127^3, \\
\Ca(15) & \le & \BCa(15)\le \SCa(15) = 2^3\cdot 19^3\cdot 23^3\cdot 29^3\cdot 127^3\cdot \Ca(10) \\
       & =   & 2^6\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19^4\cdot 23^3\cdot 29^3\cdot 31\cdot 37\cdot 67^3\cdot 127^3, \\
\Ca(16) & \le & \BCa(16)\le \SCa(16) = 2^3\cdot 19^3\cdot 23^3\cdot 29^3\cdot 43^3\cdot 127^3\cdot \Ca(10) \\
       & =   & 2^6\cdot 3^3\cdot 5^3\cdot 7^4\cdot 13^3\cdot 19^4\cdot 23^3\cdot 29^3\cdot 31\cdot 37\cdot 43^3\cdot 67^3\cdot 127^3. \\
\end{array}
$


\begin{figure}[h]
\centering
\includegraphics[scale=0.49]{f6.eps}
\caption{Magnifications among $\Ca(10), \SCa(11), \ldots, \SCa(16)$}\label{fig6}
\end{figure}
\end{example}


\subsection{Floating sieving process}
\label{sec4-2}
Upper bounds for $\Ta(7)$, $\Ta(8)$, $\Ta(9)$, and for $\Ca(11),\ldots,\Ca(16)$ were derived by the above sieving process. However, the computing time that is needed for sieving increases as an exponential function of $n$ in both $\Ca(n)$ and $\Ta(n)$. The process is therefore modified by considering the exponent sum of its standard factorization, rather than a value itself, and this modified process is {the} called \emph{floating sieving process}.

Recall that each parameter $r$ is a divisor of $S/18$ as shown in Theorem \ref{thm1}. Let $r_i=\displaystyle \prod^m_{j=1}p^{\beta_{i,j}}_j$ for primes $p_1<p_2<\cdots <p_m$ with $1\le i\le n$ be the standard product of prime powers of the parameter of $\BCa(n)$, abbreviated as $r_i=(\beta_{i,1},\beta_{i,2},\ldots,\beta_{i,m})$ with base $(p_1,p_2,\ldots,p_m)$. When a number $S$ that can be described as $n+1$ sums of two cubes, we may try $S=k^3\cdot \BCa(n)$ first for a splitting factor $k$. Since $S=k^3\cdot \BCa(n)$ itself already has $n$ decompositions with parameters $kr_1,kr_2,\ldots,kr_n$, the key lies in finding an \emph{additional parameter} $r_{n+1}$. Let $a_i=\displaystyle \sum^m_{j=1}\beta_{i,j}$ with $1\le i\le n$ be the exponent sum of $r_i$, and let $[L,U]$ be an interval that contains all $a_i$. Based on the assumption that $k$ is a prime, the exponent sums of $kr_1,kr_2,\ldots,kr_n$ lie in the interval $[L+1,U+1]$. The exponent-sums of additional parameters are likely in the interval $[L+1,U+1]$ as well, rather than in the range of the parameter $[\sqrt[3]{S/216},\sqrt[3]{S/54}]$ so many and huge numbers can be avoided.

More specifically, an additional parameter $r_{n+1}=k^{\beta_{m+1}}\cdot\displaystyle \prod^{m}_{i=1}p^{\beta_i}_i$, associated with $\newline k^3\cdot \BCa(n)=k^3\cdot\displaystyle\prod^m_{i=1}p^{\alpha_i}_i$, may satisfy the conditions $0\le\beta_1\le\alpha_1-1$, $0\le\beta_2\le\alpha_2-2$ (since $18=2\cdot3^2$ is a divisor of $S$), $0\le \beta_i\le \alpha_i/2$ with $3\le i\le m$, and $\beta_{m+1}= 0$ or 3. Based on a comparison with original sieving over all $(\beta_1,\beta_2,\ldots,\beta_{m+1})$ for $0\le\beta_i\le\alpha_i$ with $i\le m$, and $0\le \beta_{m+1}\le 3$, only restricted values of $r$ are sieved for, efficiently reducing the time needed for sieving. Therefore, the number of searches will be reduced to about $1 / 2^{m+1}$ of the original number, where $m$ is the number of prime factors in the standard factorization of $\BCa(n)$.

$\newline$
\textbf{Floating sieving process}
\begin{enumerate}
\itemsep=-5pt
\item	Input primes $\{p_i\}$ and nonnegative integers $\{\alpha_i\}$ with $1\le i\le m$ with $(p_1,p_2)=(2,3)$ (for $\BCa(n)=\displaystyle\prod^m_{i=1}p_i^{\alpha_i}$), $k$ (for magnification $S=k^3\cdot\displaystyle\prod^m_{i=1}p_i^{\alpha_i}$), and $L, U$ (range for scanning).
\item	Input $\beta_i$ with $i=1,\ldots, m+1$, where $0\le \beta_1\le\alpha_1-1$, $0\le\beta_2\le\alpha_2-2$, $0\le\beta_i\le\alpha_i/2$, $i=3,\ldots, m$, and $\beta_{m+1}=0,3$. (candidates for scanning)

\item	For each sequence $(\beta_1,\beta_2,\ldots,\beta_{m+1})$, let $r=k^{\beta_{m+1}}\cdot\displaystyle\prod^m_{i=1}p_i^{\beta_i}$, and $a=\displaystyle\sum^{m+1}_{i=1}\beta_i$.
\item	If $L\le a\le U$,  \\
        \indent\hspace{4mm} if $-3r^2+S/18r$ is a perfect square, \\
        \indent\hspace{8mm}     output $r$, $x(r)=3r+\sqrt{-3r^2+S/18r}$, and $y=3r-\sqrt{-3r^2+S/18r}$, \\
        \indent\hspace{8mm}     return to Step 3.\\
        \indent\hspace{4mm} otherwise, return to Step 3. \\
        Otherwise, return to Step 3.
\end{enumerate}

The choices of $L$, $U$ and $\beta_i$ are crucial in floating sieving for upper bounds, as properly chosen values greatly reduce the number of computations. A risk of missing searching targets is taken; however, it is still worthwhile if performance efficiency is taken into consideration.


\begin{example}
\label{ex7}
Based on the floating sieving process, sets of parameters for $\BCa(22)$, $\BCa(30)$, $\BCa(42)$  and $\BTa(22)$ are given in Tables \ref{t8}--\ref{t12} in the Appendix. The first rows of these tables present the bases $(p_1, p_2, \ldots, p_m)$. The magnifications among $\BCa(23), \ldots, \BCa(26)$ are summarized in Table \ref{t4} with the base $(p_1, p_2, \ldots, p_{18})$ = (2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 43, 61, 67, 73, 79, 109, 139, 157), and the parameters for $\BCa(23), \ldots, \BCa(26)$ are thus provided.

\end{example}


\begin{table}[h]
\centering
\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{$\BCa(n)$} & \multicolumn{1}{|c|}{\emph{additional parameters}} \\
\hline
\begin{tabular}{cl}
  & $\BCa(23)$ \\
= & $3^3\cdot \BCa(22)$ \\
\end{tabular}
&
\begin{tabular}{l}
$r_{23}=(6,7,1,1,1,1,0,1,1,1,0,0,1,0,1,1,0,0)$ \\
\end{tabular} \\
\hline
\begin{tabular}{cl}
  & $\BCa(24)$ \\
= & $17^3\cdot \BCa(23)$ \\
\end{tabular}
&
\begin{tabular}{l}
$r_{24}=(2,2,1,2,1,1,3,1,0,1,0,1,1,0,1,1,0,0)$ \\
\end{tabular} \\
\hline
\begin{tabular}{cl}
  & $\BCa(25)$ \\
= & $139^3\cdot \BCa(23)$ \\
\end{tabular}
&
\begin{tabular}{l}
$r'_{24}=(2,2,3,2,1,1,0,1,1,1,1,1,1,0,1,1,0,0)$ \\
$r_{25}=(4,2,3,1,1,1,0,1,0,1,1,1,1,1,1,1,0,0)$ \\
\end{tabular}  \\
\hline
\begin{tabular}{cl}
  & $\BCa(26)$ \\
= & $139^3\cdot \BCa(24)$ \\
= & $17^3\cdot \BCa(25)$ \\
\end{tabular}
&
\begin{tabular}{l}
$139\cdot r_{24}$ \\
$17\cdot r'_{24}$ \\
$17\cdot r'_{25}$ \\
\end{tabular} \\
\hline
\end{tabular}
\caption{Parameters of $\BCa(23), \BCa(24), \BCa(25), \BCa(26)$}\label{t4}
\end{table}


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{f7.eps}
\caption{Magnifications among $\BCa(23)$, $\BCa(24)$, $\BCa(25)$, $\BCa(26)$}\label{fig7}
\end{figure}

\end{section}


\begin{section}{Upper bounds on $\Ca(43), \ldots, \Ca(57)$, and $\Ta(23)$, $\Ta(24)$}
\label{up}
To find an additional parameter of an upper bound on $\Ca(43)$ starting from $\BCa(42)$ by using sieving process, too much computations are required because $\BCa(42)$ has 29 prime factors. On the other hand, a value $k$ can be a splitting factor of $\BCa(n)$ as well as of $\BCa(n+h)$ simultaneously, splitting factors of $\BCa(30)$, rather than of $\BCa(42)$, tend to be sought because $\BCa(30)$ has 19 prime factors.

First, the parameters for $\BCa(31), \BCa(32), \BCa(35), \BCa(37), \BCa(38), \ldots$, and finally $\BCa(42)$ are obtained by a sequence of consecutive magnifications (Table \ref{t5}). Then 15 splitting factors of $\BCa(30)$, either primes or products of two primes, are provided along with their additional parameters, as in Table \ref{t6}. Combining these 15 splitting factors of $\BCa(30)$ and the 42 parameters for $\BCa(42)$ enables an upper bound on $\Ca(43)$ to be derived. In addition to an upper bound on $\Ca(43)$, upper bounds on $\Ca(44), \ldots, \Ca(57)$ can be derived in terms of the magnifications
$$\BCa(42)=Q^3\cdot \BCa(30) \text{ and } k^3\cdot \BCa(42)=k^3\cdot Q^3\cdot \BCa(30)$$
with respect to specific splitting factor $k$ of $\BCa(30)$, as in Figs.~\ref{t10} and \ref{t11}. A similar technique can be used to the derive of upper bounds on $\Ta(23)$ and $\Ta(24)$ from $\BTa(12)$ as in Section \ref{sec5-3}.


\subsection{The set of 42 parameters of $\BCa(42)$}
\label{sec5-1}
The set of 30 parameters of
$$\BCa(30)=2^9\cdot 3^9\cdot 5^9\cdot 7^7\cdot 11^3\cdot 13^6\cdot 17^3\cdot 19^3\cdot 31^1\cdot 37^4\cdot 43^1\cdot 61^3\cdot 67^3\cdot 73^1\cdot 79^3\cdot 97^3\cdot 109^3\cdot 139^3\cdot 157^1$$
is given in Table \ref{t9}, which yields 30 decompositions. A sequence of magnifications of $\BCa(31), \BCa(32), \BCa(35), \BCa(37), \ldots, \BCa(42)$, as shown in Fig.~\ref{t8}, follows, and Table \ref{t5} presents their corresponding additional parameters. Notably the base
$$
\begin{array}{ll}
 & {(p_1, p_2, \ldots, p_{29})} \\
= & ( 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 43, 61, 67, 73, 79, 97, \\
  & 101$, $109, 139, 157, 163, 181, 193, 223, 229, 307, 397, 457). \\
\end{array}
$$
The set of 42 parameters of $\BCa(42)$ is given in Tables \ref{t10} and \ref{t11} with bases $(p_1, p_2, \ldots, p_{29})$ in the first rows in the Appendix.


\begin{figure}[h]
\centering
\includegraphics[scale=0.472]{f8.eps}
\caption{Magnifications among $\BCa(30), \BCa(31),\BCa(32), \BCa(35), \BCa(37), \ldots, \BCa(42)$}\label{fig8}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[scale=0.46]{f9.eps}
\caption{Magnifications among $\BCa(35)$, $\BCa(36)$, $\BCa(37)$, $\BCa(38)$}\label{fig9}
\end{figure}


\begin{table}[h]
\centering
\begin{footnotesize}
\begin{tabular}{|l|c|}
\hline
\multicolumn{1}{|c|}{$\BCa(n)$} & \multicolumn{1}{|c|}{\emph{additional parameters}}  \\
\hline
\begin{tabular}{ll}
  & $\BCa(31)$ \\
= & $29^3\cdot \BCa(30)$ \\
\end{tabular} &	$r_{31} =(2,2,3,2,1,2,1,1,3,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(32)$ \\
= & $43^3\cdot \BCa(31)$ \\
\end{tabular} &	$r_{32} =(2,3,5,2,1,2,1,1,3,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(35)$ \\
= & $397^3\cdot 457^3\cdot \BCa(32)$ \\
\end{tabular} &
\begin{tabular}{l}
$r_{33} =(2,2,9,2,1,2,1,1,1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0)$ \\
$r_{34} =(2,5,3,1,1,2,1,1,1,1,1,2,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0,1)$ \\
$r_{35} =(4,2,7,3,1,2,1,1,1,1,2,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0)$ \\
\end{tabular}\\
\hline
\begin{tabular}{ll}
  & $\BCa(37)$ \\
= & $101^3\cdot 229^3\cdot \BCa(35)$ \\
\end{tabular} &
\begin{tabular}{l}
$r_{36} =(4,2,3,0,1,2,1,1,1,0,1,1,1,1,0,1,1,3,1,1,1,0,0,0,0,0,0,1,1)$ \\
$r_{37} =(6,3,3,5,1,2,1,1,1,0,2,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1)$ \\
\end{tabular}\\
\hline
\begin{tabular}{ll}
  & $\BCa(38)$ \\
= & $181^3\cdot \BCa(37)$ \\
\end{tabular} &	$r_{38} =(2,5,5,3,1,2,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,1)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(39)$ \\
= & $163^3\cdot \BCa(38)$ \\
\end{tabular} &	$r_{39} =(2,2,3,5,1,2,1,0,1,0,1,1,1,1,0,1,1,1,1,1,1,3,0,0,0,1,0,0,1)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(40)$ \\
= & $193^3\cdot \BCa(39)$ \\
\end{tabular} &	$r_{40} =(2,2,3,2,1,2,1,1,1,0,1,2,1,1,0,1,1,1,1,1,1,1,1,0,0,1,0,1,1)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(41)$ \\
= & $223^3\cdot \BCa(40)$ \\
\end{tabular} &	$r_{41} =(6,3,3,2,1,2,1,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,0,1,1)$ \\
\hline
\begin{tabular}{ll}
  & $\BCa(42)$ \\
= & $307^3\cdot \BCa(41)$ \\
\end{tabular} &	$r_{42} =(2,3,3,2,1,2,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,0,3,1,1)$ \\
\hline
\end{tabular}
\end{footnotesize}
\caption{Magnifications of $\BCa(30)$ and their additional parameters}\label{t5}
\end{table}


\subsection{Upper bounds on $\Ca(43), \ldots, \Ca(57)$}
\label{sec5-2}
Upper bounds on $\Ca(43)$ can be derived by a two-step strategy as follows:
\begin{enumerate}
\itemsep=-5pt
\item Find a sequence of magnifications of $\BCa(30), \ldots, \BCa(42)$ (Table \ref{t5}).
\item Find 15 splitting factors of $\BCa(30)$ by floating sieving (Table \ref{t6}),
\end{enumerate}
starting from $\BCa(30)$, and then upper bounds on $\Ca(44), \ldots, \Ca(55)$ are derived in a sequence of magnifications.

Floating sieving process is used to find splitting factors $k$ of $\BCa(30)$ with additional parameter $R$. Let $\BCa(30)=\displaystyle\prod^{19}_{i=1}p_i^{\alpha_i}$, and the corresponding 30 parameters of $r_i$ be $\displaystyle\prod^{19}_{j=1}p_j^{\beta_{i,j}}$. Additional parameter $R=\displaystyle k^{\beta_{20}}\cdot\prod^{19}_{i=1}p_i^{\beta_i}$, whose possible $\beta_i$, $i=1,\ldots,19$ are summarized in Table \ref{t7}, with $\beta_{20}=0$ or 3, can be found in the following.


\begin{table}[h]
\centering
\begin{tabular}{|c|c|r|}
\hline
$i$ & splitting factors $k_i$ & \multicolumn{1}{|c|}{additional parameters $R$} \\
\hline
1 &	487     & (4,5,3,2,1,2,1,0,0,1,1,0,1,1,0,1,1,1,1,0,0,0) \\
\hline
2 &	503     & (2,2,3,1,1,2,1,0,0,0,1,0,1,1,0,1,1,1,1,0,3,0) \\
\hline	
3 &$2\cdot607$& (11,2,3,2,1,2,1,1,0,1,1,0,0,1,1,1,1,1,1,1,0,0) \\
\hline	
4 &	1307    & (2,2,3,2,1,2,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,3) \\
\hline	
5 &$31\cdot103$& (2,3,3,3,3,2,1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0) \\
\hline	
6 &	3559    & (2,7,3,3,1,2,1,1,0,1,1,0,1,1,1,1,1,1,1,0,0,0) \\
\hline	
7 &	4057    & (2,2,5,2,1,2,1,0,0,0,2,0,1,1,1,1,1,1,1,1,0,0) \\
\hline	
8 &	4261    & (4,2,3,3,1,2,1,1,0,0,2,1,1,1,0,1,1,1,1,1,0,0) \\
\hline	
9 &	4339    & (4,2,3,2,1,2,1,1,0,1,1,0,1,1,1,1,1,1,1,1,0,0) \\
\hline	
10&	4957    & (2,5,3,2,1,2,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,0) \\
\hline	
11&$23\cdot283$& (2,3,3,1,1,2,1,1,3,0,2,1,1,1,0,1,1,1,1,0,0,0) \\
\hline
12&	6661    & (2,2,3,2,1,2,1,1,0,1,2,1,1,1,1,1,1,1,1,0,0,0) \\
\hline
13& 7489    & (2,2,3,2,1,1,1,1,0,0,2,0,1,1,1,1,1,1,1,0,0,0) \\
\hline
14&	8353    & (4,3,3,3,1,2,1,1,0,0,2,0,1,1,1,1,1,1,1,0,0,0) \\
\hline
15&	9043    & (2,2,3,2,1,2,1,1,0,1,2,0,1,1,0,1,1,1,1,1,0,0) \\
\hline
\end{tabular}
\caption{Splitting factors of $\BCa(30)$ and corresponding additional parameters}\label{t6}
\end{table}


The exponent-sums $\displaystyle\sum_{j=1}^{19}\beta_{i,j}, 1\le i\le 30$, of the set of 30 parameters $\newline r_i=\displaystyle\prod_{j=1}^{19}p_j^{\beta_{i,j}}$ of $\BCa(30)$ lie in the interval $[21,30]$. Set $L=22$ and $U=31$. Floating sieving is performed using possible values of $\beta_1, \ldots, \beta_{19}$ that are shown in Table \ref{t7}, and $\beta_{20}= 0$ or 3. Calculate $\displaystyle r=k^{\beta_{20}}\cdot\prod_{i=1}^{19}p_i^{\beta_i}$ whenever $\displaystyle 22\le \sum_{i=1}^{20}\beta_i\le 31$. Moreover, if $-3r^2+k^3\cdot \BCa(30)/18r$ is a perfect square, then this $r$ is an additional parameter, denoted by $R$, of $k^3\cdot \BCa(30)$. It follows that 15 splitting factors in the form of a prime or a product of two primes of $\BCa(30)$, together with their additional parameters are given in Table \ref{t6} with the base
$$
\begin{array}{ll}
  & (p_1, p_2, \ldots, p_{22}) \\
= & ( 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 61, 67, 73, 79, 97, 109, 139, 157, 503, 1307) \\
\end{array}
$$


\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$p_i$     & 2 &	3 &	5 &	7 &	11 & 13	& 17 & 19 &	31 & 37 & 43 & 61 &	67 & 73 & 79 & 97 &	109 & 139 &	157 \\
\hline
$\alpha_i$&	9 & 9 & 9 & 7 &	 3 &  6 &  3 &  3 &  1 &  4	&  1 &	3 &	 3 &  1 &  3 &	3 &	  3	&   3 &	  1 \\
\hline
          & 2 & 2 & 3 & 0 &  1 &  1 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &   0 &   0 \\
          & 4 & 3 & 5 & 1 &  3 &  2 &  3 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &   1 &   1 &   1 \\
$\beta_i$ &	6 & 5 & 9 & 2 &    &  4 &    &    &    &  2 &    &    &    &    &  3 &  3 &     &     &     \\
          & 8 & 7 &   & 3 &    &    &    &    &    &    &    &    &    &    &    &    &     &     &     \\
          &   &   &   & 5 &    &    &    &    &    &    &    &    &    &    &    &    &     &     &     \\
\hline
\end{tabular}
\caption{Possible $\beta_i$ for parameter $r$ of $\BCa(30)$}\label{t7}
\end{table}

We then show that upper bound on $\Ca(43)$ can be derived from a sequence of magnifications $\BCa(30), \BCa(31), \BCa(32), \BCa(35), \BCa(36), \BCa(38), \ldots, \BCa(42)$. Let $r_i$, $i=1, \ldots, 42$, be the set of parameters of $\BCa(42)$ and let
$$
\begin{array}{ccl}
Q & = & 29\cdot 43\cdot (397\cdot 457)\cdot  181\cdot  (101\cdot 229)\cdot  163\cdot 193\cdot 223\cdot 307 \\
  & = & 29\cdot 43\cdot 101\cdot 163\cdot 181\cdot 193\cdot 223\cdot 229\cdot 307\cdot 397\cdot 457, \\
\end{array}
$$
as presented in Fig.~\ref{fig10}.
Then the magnification $\BCa(42) = Q^3\cdot \BCa(30)$ holds. If $k_j$ is a splitting factor of $\BCa(30)$, relative prime with $Q$, and $k^3_j\cdot \BCa(30)$ has an additional parameter $R_j$, then a set of 43 parameters of $k_j^3\cdot \BCa(42) = Q^3\cdot k_j^3\cdot \BCa(30)$ is given below:
\begin{enumerate}
\item[] $k_j\cdot r_i, i=1,\ldots, 42$, because of $k_j^3\cdot \BCa(42)$ , and
\item[] $Q\cdot R_j$, because of $Q^3\cdot(k_j^3\cdot \BCa(30))$.
\end{enumerate}
Hence, the set of 43 parameters of $k_j^3\cdot \BCa(42)$ and an upper bound of $\Ca(43)$ are obtained.


\begin{example}
\label{ex8}
For $k_1=487$, all parameters $r_i$ with $i=1,\ldots, 42$ of $\BCa(42)$ are relative prime to 487. The magnification $\BCa(42) = Q^3\cdot \BCa(30)$ holds. Further let
\begin{enumerate}
\item[] $S = 487^3\cdot \BCa(42)$, and
\item[] $R = 2^4\cdot 3^5\cdot 5^3\cdot 7^2\cdot 11\cdot 13^2\cdot 17\cdot 31\cdot 37\cdot 61\cdot 67\cdot 79\cdot 97\cdot 109\cdot 139\cdot 487^0$,
\end{enumerate}
then $R$ is an additional parameter of $487^3\cdot \BCa(30)$. Clearly, the 42 parameters of $S$ are $487\cdot r_i, i=1,\ldots, 42$, because $S=487^3\cdot \BCa(42)$, and an additional parameter $r_{43}=Q\cdot R$, because $S = Q^3\cdot (487^3\cdot \BCa(30))$. Notice that $r_{43}$ differs from each of $487\cdot r_i , i=1, \ldots, 42$. Therefore, $487^3\cdot \BCa(42)$ has 43 parameters, so $487^3\cdot \BCa(42)$ is an upper bound on $\Ca(43)$, denoted by $\SCa(43)$.
\end{example}

Upper bounds on $\Ca(n)$ with $44\le n\le 57$ can be similarly derived, and are denoted by $\SCa(n)$ with $44\le n\le 57$, respectively.


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{f10.eps}
\caption{Magnifications among $\BCa(30)$, $487^3\times \BCa(30)$, $\BCa(42)$, $\SCa(43)$}\label{fig10}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{f11.eps}
\caption{$\SCa(n)$ with $43\le n\le 57$ derived from $\BCa(30)$ by magnifications}\label{fig11}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{f12.eps}
\caption{Magnifications among $\BCa(42), \SCa(43), \ldots, \SCa(57)$}\label{fig12}
\end{figure}

The above mentioned upper bounds are summarized in Theorem \ref{thm2}. Among them, $\SCa(43)$ and $\SCa(44)$ were included in {\it OEIS}, August 2014.

\begin{theorem}
\label{thm2}
\indent
\begin{enumerate}
\itemsep=-5pt
\item[] $\Ca(43)\le \SCa(43)=487^3\cdot \BCa(42)$
\item[] $\Ca(44)\le \SCa(44)=503^3\cdot \SCa(43)$
\item[] $\Ca(45)\le \SCa(45)= (2\cdot 607)^3\cdot \SCa(44)$
\vspace{1.5mm}
\item[] $\Ca(46)\le \SCa(46)=1307^3\cdot \SCa(45)$
\item[] $\Ca(47)\le \SCa(47)= (31\cdot 103)^3\cdot \SCa(46)$
\item[] $\Ca(48)\le \SCa(48)=3559^3\cdot \SCa(47)$
\item[] $\Ca(49)\le \SCa(49)=4057^3\cdot \SCa(48)$
\item[] $\Ca(50)\le \SCa(50)=4261^3\cdot \SCa(49)$
\vspace{1.5mm}
\item[] $\Ca(51)\le \SCa(51)=4339^3\cdot \SCa(50)$
\item[] $\Ca(52)\le \SCa(52)=4957^3\cdot \SCa(51)$
\item[] $\Ca(53)\le \SCa(53)=(23\cdot 283)^3\cdot \SCa(52)$
\item[] $\Ca(54)\le \SCa(54)=6661^3\cdot \SCa(53)$
\item[] $\Ca(55)\le \SCa(55)=7489^3\cdot \SCa(54)$
\vspace{1.5mm}
\item[] $\Ca(56)\le \SCa(56)=8353^3\cdot \SCa(55)$
\item[] $\Ca(57)\le \SCa(57)=9043^3\cdot \SCa(56)$
\end{enumerate}
\end{theorem}

\noindent Remark: As pointed by Boyer that 673 is a splitting factor for $\\BCa(30)$ which is missing in Table \ref{t6}, we confirm that
$$r = 2^2\cdot 3^2\cdot 5^3\cdot 7^3\cdot 11\cdot 13^2\cdot 17\cdot 19\cdot 37\cdot 61\cdot 67^3\cdot 79\cdot 97\cdot 109\cdot 139\cdot 673^0$$
is its additional parameter. As a consequence, $\SCa(45), \ldots, \SCa(57)$ in Theorem \ref{thm2} can be improved easily by shifting the splitting factors properly.



\subsection{Upper bounds on $\Ta(23)$ and $\Ta(24)$ }
\label{sec5-3}
The strategy that was used in Section \ref{sec5-2} can be applied to search for upper bounds on $\Ta(23)$ and $\Ta(24)$ in terms of two splitting factors of $\BTa(12)$ and a sequence of magnifications $\BTa(12), \ldots, \BTa(16)$, $\BTa(19)$, $\BTa(21)$ and $\BTa(22)$ in the order.

By floating sieving, $47,627(=97\times491)$ and $91,037(=59\times1543)$ are two splitting factors of $\BTa(12)$ with additional parameters
$$
\begin{array}{ccl}
R_1 & = & 2^2\cdot 3\cdot 5\cdot 7\cdot 13^2\cdot 17\cdot 19\cdot 73\cdot 79\cdot 97^0\cdot 109\cdot 139\cdot 491^3, \\
R_2 & = & 2^2\cdot 3\cdot 5\cdot 7^2\cdot 13^2\cdot 17\cdot 19\cdot 59^3\cdot 79\cdot 97\cdot 109\cdot 139\cdot 157\cdot 1543^0 \\
\end{array}
$$
respectively. Let
$$
\begin{array}{ccl}
Q' & = & (3\cdot 61)\cdot  397\cdot 503\cdot (2\cdot 607)\cdot (5^2\cdot 37\cdot  181\cdot 457\cdot 521)\cdot  4261\cdot (127\cdot 197)\cdot   (11\cdot 31\cdot 103) \\
   & = & 2\cdot 3\cdot 5^2\cdot 11\cdot 31\cdot 37\cdot 61\cdot 103\cdot 127\cdot 181\cdot 197\cdot 397\cdot 457\cdot 503\cdot 521\cdot 607\cdot 4261. \\
\end{array}
$$
The magnification $\BTa(22) = Q'^3\cdot \BTa(12)$ holds. Then
$$(97\cdot 491)^3\cdot \BTa(22)= (97\cdot491)^3\cdot {Q'}^3\cdot \BTa(12)$$
has 22 parameters, which are $97\cdot 491\cdot r_i, i=1, \ldots, 22$, where $r_1, \ldots, r_{22}$ are the parameters of $\BTa(22)$ as shown in Table \ref{t12}, together with an additional parameter
$$
\begin{array}{ccl}
Q'\cdot R_1 & = & 2^3\cdot 3^2\cdot 5^3\cdot 7\cdot 11\cdot 13^2\cdot 17^1\cdot 19^1\cdot 31\cdot 37\cdot 61\cdot 73\cdot 79\cdot 103\cdot 109\cdot 127\cdot 139\cdot 181\cdot 197 \\
		   &   & \cdot 397\cdot 457\cdot 491^3\cdot 503\cdot 521\cdot 607\cdot 4261,
\end{array}
$$
which differs from the previous 22 parameters, so $(97\cdot 491)^3\cdot \BTa(22)$ has 23 parameters, and $(97\cdot491)^3\cdot \BTa(22)$ gives an upper bound on $\Ta(23)$, denoted by $\STa(23)$,
$$\STa(23) = (97\cdot491)^3\cdot \BTa(22).$$
Similarly,
$$\STa(24) = (59\cdot1543)^3\cdot \STa(23)$$
is an upper bound on $\Ta(24)$. Theorem \ref{thm3} summarizes the above results. Both upper bounds $\STa(23)$ and $\STa(24)$ were included in \emph{OEIS}, October 2014.


\begin{theorem}
\label{thm3}
\indent
\begin{enumerate}
\itemsep=-5pt
\item[] $\Ta(23)\le \STa(23)= 97^3\cdot 491^3\cdot \BTa(22)$,
\item[]$ \Ta(24)\le \STa(24)= 59^3\cdot 1543^3\cdot \STa(23)$.
\end{enumerate}
\end{theorem}
\end{section}

\section{Acknowledgement}
The author would like to thank Prof. C. Boyer for suggestive comments on the original version of this paper; in particular, for pointing out additional splitting factors for $\BCa(30)$.


\section{Appendix}
\bigskip


\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{f13.eps}
\caption{Magnifications among $\Ta(5)$, $\Ta(6)$, $\BTa(7), \ldots, \BTa(22)$, $\STa(23)$, $\STa(24)$}\label{fig13}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[scale=0.52]{f14.eps}
\caption{Magnifications among
$\BCa(19), \BCa(21) \ldots, \BCa(42)$, $\SCa(43), \ldots, \SCa(57)$}\label{fig14}
\end{figure}


\begin{landscape}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||c|}
\hline
  & 2 & 3 & 5 & 7 & 11 & 13 & 19 & 31 & 37 & 43 & 61 & 67 & 73 & 79 & 109 & 157 & $a_i$ \\
\hline
$r_1$ & 2 & 1 & 1 & 0 & 3 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 15  \\
\hline
$r_2$ & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 14  \\
\hline
$r_3$ & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 15  \\
\hline
$r_4$ & 2 & 1 & 1 & 1 & 1 & 1 & 3 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 15  \\
\hline
$r_5$ & 2 & 1 & 1 & 1 & 1 & 3 & 1 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 16  \\
\hline
$r_6$ & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 13  \\
\hline
$r_7$ & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 15  \\
\hline
$r_8$ & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 15  \\
\hline
$r_9$ & 2 & 1 & 1 & 4 & 1 & 3 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 17  \\
\hline
$r_{10}$ & 2 & 1 & 3 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 16  \\
\hline
$r_{11}$ & 2 & 2 & 1 & 0 & 1 & 1 & 1 & 0 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 15  \\
\hline
$r_{12}$ & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 16  \\
\hline
$r_{13}$ & 2 & 4 & 1 & 1 & 1 & 1 & 0 & 1 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 17  \\
\hline
$r_{14}$ & 4 & 1 & 1 & 1 & 3 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 17  \\
\hline
$r_{15}$ & 4 & 1 & 1 & 2 & 1 & 0 & 1 & 1 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 17  \\
\hline
$r_{16}$ & 4 & 2 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 3 & 0 & 0 & 17  \\
\hline
$r_{17}$ & 4 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 17  \\
\hline
$r_{18}$ & 4 & 2 & 1 & 2 & 1 & 0 & 1 & 0 & 2 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 17  \\
\hline
$r_{19}$ & 6 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 18  \\
\hline
$r_{20}$ & 6 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 18  \\
\hline
$r_{21}$ & 8 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 20  \\
\hline
$r_{22}$ & 8 & 1 & 3 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 21  \\
\hline
\end{tabular}
\caption{22 parameters for $\BCa(22)$}
\label{t8}
\end{table}
\end{landscape}


\begin{landscape}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 31 & 37 & 43 & 61 & 67 & 73 & 79 & 97 & 109 & 139 & 157  \\
\hline
$r_1$ & 2 & 2 & 3 & 1 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 \\
\hline
$r_2$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 0 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_3$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 2 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1  \\
\hline
$r_4$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 3 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0  \\
\hline
$r_5$ & 2 & 2 & 3 & 2 & 1 & 4 & 1 & 1 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0  \\
\hline
$r_6$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_7$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_8$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1  \\
\hline
$r_9$ & 2 & 2 & 3 & 3 & 1 & 2 & 3 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{10}$ & 2 & 2 & 3 & 5 & 1 & 4 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0\\
\hline
$r_{11}$ & 2 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1  \\
\hline
$r_{12}$ & 2 & 2 & 5 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0  \\
\hline
$r_{13}$ & 2 & 3 & 3 & 1 & 1 & 2 & 1 & 1 & 0 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1  \\
\hline
$r_{14}$ & 2 & 3 & 3 & 3 & 1 & 2 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1  \\
\hline
$r_{15}$ & 2 & 3 & 3 & 3 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0  \\
\hline
$r_{16}$ & 2 & 3 & 9 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0  \\
\hline
$r_{17}$ & 2 & 5 & 3 & 2 & 1 & 2 & 1 & 0 & 1 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{18}$ & 2 & 2 & 3 & 2 & 1 & 0 & 1 & 1 & 0 & 2 & 0 & 1 & 1 & 0 & 1 & 3 & 1 & 1 & 0  \\
\hline
$r_{19}$ & 4 & 2 & 3 & 2 & 3 & 2 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1  \\
\hline
$r_{20}$ & 4 & 2 & 3 & 3 & 1 & 1 & 1 & 1 & 1 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{21}$ & 4 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0  \\
\hline
$r_{22}$ & 4 & 3 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 3 & 1 & 0 & 1 & 0  \\
\hline
$r_{23}$ & 4 & 3 & 3 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{24}$ & 4 & 3 & 3 & 3 & 1 & 1 & 1 & 1 & 0 & 2 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{25}$ & 6 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{26}$ & 6 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{27}$ & 6 & 7 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{28}$ & 8 & 2 & 3 & 0 & 1 & 4 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0  \\
\hline
$r_{29}$ & 8 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
$r_{30}$ & 8 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0  \\
\hline
\end{tabular}
\caption{30 parameters for $\BCa(30)$}
\label{t9}
\end{table}
\end{landscape}


\begin{landscape}
\begin{table}[h]
\begin{center}
\resizebox{0.95\hsize}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 &\scriptsize 2 &\scriptsize 3 &\scriptsize 5 &\scriptsize 7 &\scriptsize 11 &\scriptsize 13 &\scriptsize 17 &\scriptsize 19 &\scriptsize 29 &\scriptsize 31 &\scriptsize 37 &\scriptsize 43 &\scriptsize 61 &\scriptsize 67 &\scriptsize 73 &\scriptsize 79 &\scriptsize 97 &\scriptsize 101 &\scriptsize 109 &\scriptsize
139 &\scriptsize 157 &\scriptsize 163 &\scriptsize 181 &\scriptsize 193 &\scriptsize 223 & \scriptsize 229 &\scriptsize 307 &\scriptsize 397 &\scriptsize 457  \\
\hline
$r_1$ & 2 & 2 & 3 & 1 & 3 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline
$r_2$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_3$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_4$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 3 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_5$ & 2 & 2 & 3 & 2 & 1 & 4 & 1 & 1 & 1 & 0 & 2 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_6$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_7$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_8$ & 2 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_9$ & 2 & 2 & 3 & 3 & 1 & 2 & 3 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{10}$ & 2 & 2 & 3 & 5 & 1 & 4 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{11}$ & 2 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{12}$ & 2 & 2 & 5 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{13}$ & 2 & 3 & 3 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{14}$ & 2 & 3 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{15}$ & 2 & 3 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{16}$ & 2 & 3 & 9 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 2 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{17}$ & 2 & 5 & 3 & 2 & 1 & 2 & 1 & 0 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{18}$ & 4 & 2 & 3 & 2 & 3 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{19}$ & 4 & 2 & 3 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{20}$ & 4 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{21}$ & 4 & 3 & 3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 0 & 3 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{22}$ & 4 & 3 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
\end{tabular}
}
\end{center}
\caption{42 parameters for $\BCa(42)$ (part a)}
\label{t10}
\end{table}
\end{landscape}


\begin{landscape}
\begin{table}[h]
\begin{center}
\resizebox{0.95\hsize}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 &\scriptsize 2 &\scriptsize 3 &\scriptsize 5 &\scriptsize 7 &\scriptsize 11 &\scriptsize 13 &\scriptsize 17 &\scriptsize 19 &\scriptsize 29 &\scriptsize 31 &\scriptsize 37 &\scriptsize 43 &\scriptsize 61 &\scriptsize 67 &\scriptsize 73 &\scriptsize 79 &\scriptsize 97 &\scriptsize 101 &\scriptsize 109 &\scriptsize
139 &\scriptsize 157 &\scriptsize 163 &\scriptsize 181 &\scriptsize 193 &\scriptsize 223 & \scriptsize 229 &\scriptsize 307 &\scriptsize 397 &\scriptsize 457  \\

\hline
$r_{23}$ & 4 & 3 & 3 & 3 & 1 & 1 & 1 & 1 & 1 & 0 & 2 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{24}$ & 6 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{25}$ & 6 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{26}$ & 6 & 7 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{27}$ & 8 & 2 & 3 & 0 & 1 & 4 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{28}$ & 8 & 2 & 3 & 3 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{29}$ & 8 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{30}$ & 2 & 2 & 3 & 2 & 1 & 0 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 1 & 0 & 1 & 3 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{31}$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 3 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{32}$ & 2 & 3 & 5 & 2 & 1 & 2 & 1 & 1 & 3 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{33}$ & 2 & 2 & 9 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
\hline
$r_{34}$ & 2 & 5 & 3 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\
\hline
$r_{35}$ & 4 & 2 & 7 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
\hline
$r_{36}$ & 4 & 2 & 3 & 0 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\
\hline
$r_{37}$ & 6 & 3 & 3 & 5 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\
\hline
$r_{38}$ & 2 & 5 & 5 & 3 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{39}$ & 2 & 2 & 3 & 5 & 1 & 2 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 0 & 1 & 1 & 1 & 1 & 0 & 1 \\
\hline
$r_{40}$ & 2 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{41}$ & 6 & 3 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
\hline
$r_{42}$ & 2 & 3 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 3 & 1 & 1 \\
\hline
\end{tabular}
}
\end{center}
\caption{42 parameters for $\BCa(42)$ (part b)}
\label{t11}
\end{table}
\end{landscape}


\begin{landscape}
\begin{table}[h]
\begin{center}
\resizebox{0.95\hsize}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 &\footnotesize 2 &\footnotesize 3 &\footnotesize 5 &\footnotesize 7 &\footnotesize 11 &\footnotesize 13 &\footnotesize 17 &\footnotesize 19 &\footnotesize 31 &\footnotesize 37 &\footnotesize 43 &\footnotesize 61 &\footnotesize 73 &\footnotesize 79 &\footnotesize 97 &\footnotesize 103 &\footnotesize 109 &\footnotesize 127 &\footnotesize
 139 &\footnotesize 157 &\footnotesize 181 &\footnotesize 197 & \footnotesize 397 &\footnotesize 457 &\footnotesize 503 &\footnotesize 521 &\footnotesize 607 &\footnotesize 4261 \\
\hline
$r_1$ & 3 & 2 & 9 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
\hline
$r_2$ & 3 & 2 & 3 & 1 & 1 & 4 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_3$ & 3 & 3 & 7 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 3 & 1 & 1 \\
\hline
$r_4$ & 3 & 3 & 3 & 0 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 3 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_5$ & 3 & 3 & 3 & 2 & 3 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_6$ & 3 & 2 & 3 & 1 & 1 & 0 & 1 & 1 & 1 & 2 & 0 & 1 & 0 & 1 & 3 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\	
\hline
$r_7$ & 3 & 2 & 3 & 0 & 1 & 2 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 \\
\hline
$r_8$ & 3 & 2 & 3 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_9$ & 3 & 2 & 5 & 2 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{10}$ & 3 & 3 & 3 & 0 & 1 & 2 & 1 & 1 & 1 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{11}$ & 3 & 2 & 3 & 2 & 1 & 2 & 3 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{12}$ & 3 & 3 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{13}$ & 3 & 5 & 3 & 0 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{14}$ & 3 & 5 & 3 & 1 & 1 & 2 & 1 & 0 & 2 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{15}$ & 3 & 5 & 5 & 2 & 1 & 2 & 1 & 1 & 2 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{16}$ & 5 & 3 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 3 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{17}$ & 5 & 2 & 5 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  	
\hline
$r_{18}$ & 5 & 2 & 3 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{19}$ & 5 & 2 & 3 & 2 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
\hline
$r_{20}$ & 7 & 7 & 3 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{21}$ & 9 & 2 & 5 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
$r_{22}$ &11 & 2 & 3 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\
\hline
\end{tabular}
}
\end{center}
\caption{22 parameters for $\BTa(22)$}
\label{t12}
\end{table}
\end{landscape}



\begin{thebibliography}{99}
\bibitem{Boy1}	C. Boyer, New upper bounds for Taxicab and Cabtaxi numbers.
{\it J. Integer Sequences}, {\bf 11} (2008), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL11/Boyer/boyer.html}{Article 08.1.6}.
\bibitem{Boy2}	C. Boyer, New upper bounds for Taxicab and Cabtaxi numbers. Available at\\
\href{http://www.christianboyer.com/taxicab/}{http://www.christianboyer.com/taxicab/}.
\bibitem{Cal} C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?,
{\it J. Univ. Comput. Sci.} {\bf 9} (2003), 1196--1203.
\bibitem{Har} G. H. Hardy and E. M. Wright,
{\it An Introduction to the Theory of Numbers}, Fifth edition,
Oxford University Press, 1980.

\bibitem{Wil} D. W. Wilson, The fifth Taxicab number is 48988659276962496,
{\it J. Integer Sequences}, {\bf 2} (1999), 
\href{https://cs.uwaterloo.ca/journals/JIS/wilson10.html}{Article 99.1.9}.

\bibitem{Slo}
 N. J. A. Sloane, {\it On-Line Encyclopedia of Integer Sequences},
\href{http://oeis.org}{http://oeis.org}.
\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary 11D25.\\

\noindent {\it Keywords}: Taxicab number, Cabtaxi number, magnification, splitting factor, sieving, floating sieving.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A011541} and
\seqnum{A047696}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received April 29 2015;
revised versions received November 17 2015; January 20 2016; March 28 2016.
Published in {\it Journal of Integer Sequences},
April 21 2016.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

                                                                                

