\documentclass{amsart} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \usepackage{psfig,epsf} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} % Absolute value notation \newcommand{\abs}[1]{\lvert#1\rvert} % Blank box placeholder for figures (to avoid requiring any % particular graphics capabilities for printing this document). \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering % Set fboxsep to 0 so that the actual size of the box will match the % given measurements more closely. \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \begin{document} %For submitting manuscript add next 2 lines %\hoffset -.5 in %\hsize 6 in \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo27.eps} \vskip 1cm {\LARGE\bf Primes of the Form $\bf (b^n+1)/(b+1)$} \vskip 1.5cm \large Harvey Dubner \\ \smallskip 449 Beverly Road, Ridgewood, New Jersey 07450 \\ \medskip \large Torbj\"orn Granlund \\ \smallskip Notvarpsgr\"and 1, 1tr SE-116 66 Stockholm, Sweden \\ \medskip Email addresses: \href{mailto:hdubner1@compuserve.com}{hdubner1@compuserve.com} and \href{mailto:tege@swox.se}{tege@swox.se} \vskip2.5cm \bf {Abstract} \end{center} {\em Numbers of the form $(b^n+1)/(b+1)$ are tested for primality. A table of primes and probable primes is presented for $b$ up to 200 and large values of $n$. } \vspace*{+.1in} \noindent 1999 {\it Mathematics Subject Classification}: Primary 11A41 \noindent {\em Keywords}: prime numbers, generalized repunits \section{Introduction} A truly prodigious amount of computation has been devoted to investigating numbers of the form $b^n \pm 1$. The Cunningham project, to factor these numbers for b from 2 to 12, is perhaps the longest running computer project of all time \cite{Cunn}. The range of $b$ has been extended to 100 and even further in special cases \cite{RB92}\cite{RB94} . The Mersenne numbers, $2^n-1$ have been studied extensively for hundreds of years and the largest known prime is almost always a Mersenne prime. In \cite{HD93}, generalized repunit primes of the form $(b^n -1)/(b-1)$ were tabulated for bases up to 99 and large values of $n$. The purpose of this paper is to present the results of computer searches for primes of the form, $$Q(b,n)=\frac {b^n+1}{b+1}\leqno(1)$$ for bases up to 200 and large values of n. \section{Prime Search} For certain values of $n$ in (1) the denominator cannot divide the numerator and are thus excluded from this study, and $Q$ has algebraic factors for certain other values of $b,n$ so that it cannot be prime. The algebraic factors of $b^n+1$ can be determined using the theory of cyclotomic polynomials \cite{Cunn}, but virtually all the important results can be obtained by simple long division. Trying long division, it is easy to see that the denominator cannot divide the numerator when $n$ is even, and always divides it when $n$ is odd. Also, if $n$ is odd and composite then $b^k+1$ will divide $b^n+1$ when $k$ divides $n$ so that $Q$ cannot be prime. Thus $Q$ can be prime only if $n$ is an odd prime. For certain special forms of $b$, $Q$ has algebraic factors for all $n$. If $b=c^t$ is a perfect power where $t$ is greater than 2 and not a power of 2 then $Q$ has algebraic factors and is almost always composite. There are rare cases when $Q$ may be prime for small $n$ but again $Q(b,n)$ can only be prime when $n$ is prime. It is well known that all factors of $b^n+1$ with $n$ an odd prime must be primes of the form $p=2kn+1$. We divided each $Q(b,n)$ by all primes of this form with $k<100,000$, finding a small factor about half the time. Each remaining $Q$ was subjected to a Fermat test $$a^{Q-1}=1 \pmod Q$$ for some $a\ne b$. If the congruence failed, then $Q$ was composite. If it held then we tried the test again with a different $a$. If both tests succeeded, $Q$ was declared a probable prime (or {\em prp}). About a day was devoted to each value of $b$ using computers with a frequency of about 500 MegaHertz. Almost all the prp searching was done by the second author. \section{Prime Proving} Small prp's up to 12 digits were proved prime by simple division. For prp's up to about 800 digits the prime proving program, APRT-CLE of UBASIC was used \cite{APRT}. This program has an upper test limit of about 830 digits. For prp's greater than 800 digits and up to 1200 digits we used the VFYPR program of Tony Forbes, which is an extended version of the UBASIC program, that can test prp's up to 1600 digits and is about twice as fast as UBASIC \cite{TF}. For a Pentium/500 it takes about 40 hours to test a 1200-digit prp and the test time increases as about the 4th power of the number of digits. The test limit of 1200 digits was arbitrarily chosen because of computer time availability. One other prime-proving method was used in a few cases. The BLS method is based on being able to factor $Q-1$ so that the factored part exceeds $\root 3 \of Q$ \cite{BLS}. Since $${\frac {b^n+1}{b+1}-1}=\frac{b(b^{n-1}-1)}{b+1}$$ the BLS method in this case can sometimes use the extensive results of previous factorizations for the Cunningham project and other projects to reduce prime proving times from hours to seconds. The results are shown in the accompanying tables. An asterisk indicates a probable prime. [Numbers in square brackets give the appropriate sequence numbers in the \htmladdnormallink{On-Line Encyclopedia of Integer Sequences}{http://www.research.att.com/~njas/sequences/}.] %\newcommand{\noopsort}[1]{} \newcommand{\singleletter}[1]{#1} %\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \bibliographystyle{amsplain} \begin{thebibliography}{1} \bibitem{RB92} R.~P.~Brent, H.~J.~J.~te Riele, \emph{Factorizations of $a^n\pm~1,\ \ 13\le a<100$}, CWI Report NM-R9212, June 1992. \bibitem{RB94} R.~P.~Brent, P.~L.~Montgomery, H.~J.~J.~te Riele, \emph{Update 1 to: Factorizations of $a^n\pm~1,\ \ 13~\le~a~<~100$}, CWI Report NM-R9419, September 1994. \bibitem{BLS} J.~Brillhart, D.~H.~Lehmer, J.~I.~Selfridge, \emph{New primality criteria and factorizations of $2^m \pm 1$}, Math. Comp. \textbf{29} (1975), 620-647. \bibitem{Cunn} J.~Brillhart, D.~H.~Lehmer, J.~I.~Selfridge, B.~Tuckerman, and S.~S.~Wagstaff,~Jr., \emph{Factorization of $b^n \pm 1$, $b=2, 3, 5, 7, 10, 11, 12$ up to high powers}, Amer. Math. Soc., Providence, RI, 1988 \bibitem{APRT} H.~Cohen, A.~K.~Lenstra, \emph{Implementation of a new primality test}, Math. Comp. \textbf{48} (1987), 103-121. \bibitem{HD93} H.~Dubner, \emph{Generalized repunit primes}, Math. Comp. \textbf{61} (Oct 1993), 927-930. \bibitem{TF} T.~Forbes (tonyforbes@ltkz.demon.co.uk), personal communication concerning VFYPR prime proving program. \end{thebibliography} \begin{table} \caption{Primes of form $Q(b,n)=(b^n+1)/(b+1)$} {\fontsize 8 9 \selectfont \begin{tabular}{|r|l|r|} \hline & & max $n$\\ $b$ & \qquad $n$ for which $Q$ is prime or prp(*) & tested\\ \hline 2 & 3 5 7 11 13 17 19 23 31 43 61 79 101 127 167 191 & 32000\\ & 199 313 347 701 1709 2617 3539 5807* 10501* &\\ & 10691* 11279* 12391* 14479* [\htmladdnormallink{A978}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000978}] &\\ 3 & 3 5 7 13 23 43 281 359 487 577 1579 1663 1741 & 25000\\ & 3191 9209* 11257* 12743* 13093* 17027* [\htmladdnormallink{A7658}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007658}]&\\ 4 & 3 Algebraic &\\ 5 & 5 67 101 103 229 347 4013* [\htmladdnormallink{A57171}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057171}] & 20000\\ 6 & 3 11 31 43 47 59 107 811 2819* 4817* 9601* [\htmladdnormallink{A57172}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057172}] & 20000\\ 7 & 3 17 23 29 47 61 1619* 18251* [\htmladdnormallink{A57173}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057173}] & 20000\\ 8 & Algebraic &\\ 9 & 3 59 223 547 773 1009 1823* 3803* [\htmladdnormallink{A57175}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057175}] & 20000\\ 10 & 5 7 19 31 53 67 293 641 2137* 3011* [\htmladdnormallink{A57176}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057176}] & 20000\\ 11 & 5 7 179 229 439 557 6113* [\htmladdnormallink{A57177}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057177}] & 10000\\ 12 & 5 11 109 193 1483* [\htmladdnormallink{A57178}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057178}] & 10000\\ 13 & 3 11 17 19 919 1151 2791* 9323* [\htmladdnormallink{A57179}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057179}] & 10000\\ 14 & 7 53 503 1229 [\htmladdnormallink{A57180}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057180}] & 10000\\ 15 & 3 7 29 1091* 2423* [\htmladdnormallink{A57181}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057181}] & 10000\\ 16 & 3 5 7 23 37 89 149 173 251 307 317 [\htmladdnormallink{A57182}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057182}] & 10000\\ 17 & 7 17 23 47 967 6653* 8297* [\htmladdnormallink{A57183}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057183}] & 10000\\ 18 & 3 7 23 73 733 941 1097 1933* 4651* [\htmladdnormallink{A57184}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057184}] & 10000\\ 19 & 17 37 157 163 631 7351* [\htmladdnormallink{A57185}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057185}] & 10000\\ 20 & 5 79 89 709 797 1163* 6971* [\htmladdnormallink{A57186}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057186}] & 10000\\ 21 & 3 5 7 13 37 347 [\htmladdnormallink{A57187}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057187}] & 10000\\ 22 & 3 5 13 43 79 101 107 227 353 7393* [\htmladdnormallink{A57188}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057188}] & 10000\\ 23 & 11 13 67 109 331 587 [\htmladdnormallink{A57189}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057189}] & 10000\\ 24 & 7 11 19 2207* 2477* 4951* [\htmladdnormallink{A57190}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057190}] & 10000\\ 25 & 3 7 23 29 59 1249* 1709* 1823* 1931* 3433* 8863* [\htmladdnormallink{A57191}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057191}] & 10000\\ 26 & 11 109 227 277 347 857 2297* 9043* & 10000\\ 27 & Algebraic &\\ 28 & 3 19 373 419 491 1031* & 10000\\ 29 & 7 & 10000\\ 30 & 139 173 547 829 2087* 2719* 3109* & 10000\\ 31 & 109 461 1061* & 10000\\ 32 & Algebraic &\\ 33 & 5 67 157 & 10000\\ 34 & 3 & 10000\\ 35 & 11 13 79 127 503 617 709 857 1499* 3823* & 10000\\ 36 & 31 191 257 367 3061* & 10000\\ 37 & 5 7 2707* & 10000\\ 38 & 5 167 1063* 1597* 2749* 3373* & 8000\\ 39 & 3 13 149 & 8000\\ 40 & 53 67 1217* 5867* 6143* & 8000\\ 41 & 17 691 & 8000\\ 42 & 3 709 1637* & 8000\\ 43 & 5 7 19 251 277 383 503 3019* 4517* & 8000\\ 44 & 7 & 8000\\ 45 & 103 157 & 8000\\ 46 & 7 23 59 71 107 223 331 2207* 6841* & 8000\\ 47 & 5 19 23 79 1783* 7681* & 8000\\ 48 & 5 17 131 & 8000\\ 49 & 7 19 37 83 1481* & 8000\\ 50 & 1153* & 8000\\ \hline \end{tabular} } \end{table} \begin{table} \caption{Primes of form $Q(b,n)=(b^n+1)/(b+1)$ - continued} {\fontsize 8 9 \selectfont \begin{tabular}{|r|l|r|} \hline & & max $n$\\ $b$ & \quad $n$ for which $Q$ is prime or prp(*) & tested\\ \hline 51 & 3 149 3253* & 6000\\ 52 & 7 163 197 223 467 5281* & 6000\\ 53 & & 6000\\ 54 & 7 19 67 197 991* & 6000\\ 55 & 3 5 179 229 1129* 1321* 2251* & 6000\\ 56 & 37 107 1063* 4019* & 6000\\ 57 & 53 227 & 6000\\ 58 & 3 17 1447* & 6000\\ 59 & 17 43 991* & 6000\\ 60 & 3 937* 1667* 3917* & 6000\\ 61 & 7 41 359 & 6000\\ 62 & 11 29 167 313 & 6000\\ 63 & 3 37 41 2131* 4027* & 6000\\ 64 & Algebraic &\\ 65 & 19 31 & 6000\\ 66 & 7 17 211 643 & 6000\\ 67 & 3 2347* 2909* 3203* & 6000\\ 68 & 757* 773* & 6000\\ 69 & 11 211 239 389 503 4649* & 6000\\ 70 & 3 61 97 & 6000\\ 71 & 5 37 5351* & 6000\\ 72 & 3 7 79 277 3119* & 6000\\ 73 & 7 & 6000\\ 74 & 13 31 37 109 & 6000\\ 75 & 5 83 & 6000\\ 76 & 3 5 191 269 & 6000\\ 77 & 37 317 & 6000\\ 78 & 3 7 31 661* 4217* & 6000\\ 79 & 3 107 457 491 2011* & 6000\\ 80 & 5 13 227 439 & 6000\\ 81 & 3 5 701* 829* 1031* 1033* & 6000\\ 82 & 293 1279* & 6000\\ 83 & 19 31 37 43 421 547 3037* & 6000\\ 84 & 7 13 139 359 971* 1087* 3527* & 6000\\ 85 & 167 3533* & 6000\\ 86 & 7 17 397 & 6000\\ 87 & 7 467 & 6000\\ 88 & 709* 1373* & 6000\\ 89 & 13 59 137 1103* 4423* & 6000\\ 90 & 3 47 & 6000\\ 91 & 3 11 43 397 & 6000\\ 92 & 37 59 113 & 6000\\ 93 & 89 571 601 3877* & 6000\\ 94 & 71 307 613 1787* 3793* & 6000\\ 95 & 43 & 6000\\ 96 & 37 103 131 263 & 6000\\ 97 & & 6000\\ 98 & 19 101 & 6000\\ 99 & 7 37 41 71 & 6000\\ 100 & 3 293 461 & 6000\\ \hline \end{tabular} } \end{table} \begin{table} \caption{Primes of form $Q(b,n)=(b^n+1)/(b+1)$ - continued} {\fontsize 8 9 \selectfont \begin{tabular}{|r|l|r|} \hline & & max $n$\\ $b$ & \quad $n$ for which $Q$ is prime or prp(*) & tested\\ \hline 101 & 7 229 & 6000\\ 102 & 3 & 6000\\ 103 & & 6000\\ 104 & 673* 839* 1031* & 6000\\ 105 & 11 149 1187* 1627* & 6000\\ 106 & 3 7 19 23 31 3989* & 6000\\ 107 & 103 983* & 6000\\ 108 & 13 223 & 6000\\ 109 & 59 79 811* & 6000\\ 110 & 23 101 & 6000\\ 111 & 3 5 23 53 383 2039* & 6000\\ 112 & 3 & 6000\\ 113 & & 6000\\ 114 & 7 13 1801* & 6000\\ 115 & 7 31 293 & 6000\\ 116 & 113 1481* 2089* & 6000\\ 117 & 271 & 6000\\ 118 & 3 23 109 2357* & 6000\\ 119 & 29 53 797* & 6000\\ 120 & 3 31 43 263 4919* & 6000\\ 121 & 5 13 97 1499* & 6000\\ 122 & 293 3877* & 6000\\ 123 & 29 739* & 6000\\ 124 & & 6000\\ 125 & Algebraic &\\ 126 & 5 13 47 163 239 4523* & 6000\\ 127 & 317 1061* & 6000\\ 128 & 7 Algebraic &\\ 129 & 17 227 1753* & 6000\\ 130 & 467 & 6000\\ 131 & 5 101 3389* 3581* & 6000\\ 132 & 3 101 157 1303* & 6000\\ 133 & 5 7 17 59 79 157 & 6000\\ 134 & 13 1171* & 6000\\ 135 & 5 7 2671* & 6000\\ 136 & 5 7 23 59 199 2053* & 6000\\ 137 & 101 241 353 1999* & 6000\\ 138 & 103 577* & 6000\\ 139 & 3 17 47 2683* 2719* & 6000\\ 140 & 59 & 6000\\ 141 & 5 1471* & 6000\\ 142 & 3 & 6000\\ 143 & 7 17 19 47 103 4423* & 6000\\ 144 & 3 23 41 317 3371* & 6000\\ 145 & 7 23 281 & 6000\\ 146 & 17 1439* & 6000\\ 147 & 11 151 & 6000\\ 148 & 3 7 31 43 163 317 1933* 5669* & 6000\\ 149 & 17 769* & 6000\\ 150 & & 6000\\ \hline \end{tabular} } \end{table} \begin{table} \caption{Primes of form $Q(b,n)=(b^n+1)/(b+1)$ - continued} {\fontsize 8 9 \selectfont \begin{tabular}{|r|l|r|} \hline & & max $n$\\ $b$ & \quad $n$ for which $Q$ is prime or prp(*) & tested\\ \hline 151 & 3 367 3203* & 6000\\ 152 & 13 19 & 6000\\ 153 & 13 1063* 5749* & 6000\\ 154 & 3 29 263 601* 619* 809* 1217* 2267* & 6000\\ 155 & 5 & 6000\\ 156 & 3 1301* & 6000\\ 157 & 5 157 809* 1861* 2203* & 6000\\ 158 & 5 769* 5023* & 6000\\ 159 & 283 449 1949* & 6000\\ 160 & 11 37 1907* & 6000\\ 161 & 31 331 1483* & 6000\\ 162 & 3 1823* & 6000\\ 163 & 3 11 31 661* 1999* 4079* & 6000\\ 164 & 7 103 541 1109* & 6000\\ 165 & 3 5 383 & 6000\\ 166 & 17 5437* & 6000\\ 167 & 17 59 1301* 3167* & 6000\\ 168 & 3 31 1741* 2099* & 6000\\ 169 & 3 7 109 & 6000\\ 170 & 7 & 6000\\ 171 & 13 149 257 4967* & 6000\\ 172 & 37 283 647* 4483* 5417* & 6000\\ 173 & 7 59 569* 2647* & 6000\\ 174 & 3 3191* & 6000\\ 175 & & 6000\\ 176 & 5 31 269 479 599* 809* 1307* & 6000\\ 177 & 3 5 19 419 & 6000\\ 178 & 61 167 227 & 6000\\ 179 & 827* 5011* & 6000\\ 180 & 5 13 & 6000\\ 181 & 449 2687* 4877* & 6000\\ 182 & 1487* & 6000\\ 183 & 11 & 6000\\ 184 & 19 79 149 & 6000\\ 185 & 11 & 6000\\ 186 & & 6000\\ 187 & & 6000\\ 188 & & 6000\\ 189 & 3 31 71 & 6000\\ 190 & 3 19 1153* & 6000\\ 191 & 479 1163* & 6000\\ 192 & 109 197 587 727* 1997* 2441* & 6000\\ 193 & 3 11 67 3253* & 6000\\ 194 & 19 31 & 6000\\ 195 & 3 13 19 43 89 1087* 1949* 2939* & 6000\\ 196 & 43 1049* 5441* & 6000\\ 197 & 31 37 101 163 & 6000\\ 198 & 37 151 937* & 6000\\ 199 & 313 2579* 5387* & 6000\\ 200 & 7 277 & 6000\\ \hline \end{tabular} } \end{table} \clearpage \centerline{\rule{6.5in}{.01in}} \vspace*{+.1in} \noindent {\small (Concerned with sequences \htmladdnormallink{A000978}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000978}, \htmladdnormallink{A007658}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007658} \htmladdnormallink{A057171}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057171} \htmladdnormallink{A057172}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057172} \htmladdnormallink{A057173}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057173} \htmladdnormallink{A057175}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057175} \htmladdnormallink{A057176}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057176} \htmladdnormallink{A057177}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057177} \htmladdnormallink{A057178}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057178} \htmladdnormallink{A057179}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057179} \htmladdnormallink{A057180}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057180} \htmladdnormallink{A057181}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057181} \htmladdnormallink{A057182}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057182} \htmladdnormallink{A057183}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057183} \htmladdnormallink{A057184}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057184} \htmladdnormallink{A057185}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057185} \htmladdnormallink{A057186}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057186} \htmladdnormallink{A057187}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057187} \htmladdnormallink{A057188}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057188} \htmladdnormallink{A057189}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057189} \htmladdnormallink{A057190}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057190} and \htmladdnormallink{A057191}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057191}.) } \centerline{\rule{6.5in}{.01in}} \vspace*{+.1in} \noindent Received Sept. 10, 2000; published in Journal of Integer Sequences Nov. 28, 2000. \centerline{\rule{6.5in}{.01in}} \vspace*{+.1in} \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www. research.att.com/~njas/sequences/JIS/}. \centerline{\rule{6.5in}{.01in}} \end{document}