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{\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 $k&lt;100,000$, 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://oeis.org}.]




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\begin{thebibliography}{1}


\bibitem{RB92}
R.~P.~Brent, H.~J.~J.~te Riele, \emph{Factorizations of 
$a^n\pm~1,\ \ 13\le a&lt;100$}, 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 
$a^n\pm~1,\ \ 13~\le~a~&lt;~100$}, 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
&amp; &amp; max $n$\\
$b$ &amp; \qquad $n$ for which $Q$ is prime or prp(*) &amp;  tested\\
\hline
  2 &amp; 3 5 7 11 13 17 19 23 31 43 61 79 101 127 167 191 &amp;   32000\\
    &amp; 199 313 347 701 1709 2617 3539 5807* 10501* &amp;\\
    &amp; 10691* 11279* 12391* 14479* [\htmladdnormallink{A978}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000978}] &amp;\\
  3 &amp; 3 5 7 13 23 43 281 359 487 577 1579 1663 1741    &amp; 25000\\
    &amp; 3191 9209* 11257* 12743* 13093* 17027* [\htmladdnormallink{A7658}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007658}]&amp;\\
  4 &amp; 3 Algebraic &amp;\\
  5 &amp; 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}]                          &amp; 20000\\
  6 &amp; 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}]         &amp; 20000\\
  7 &amp; 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}]                       &amp; 20000\\
  8 &amp; Algebraic &amp;\\
  9 &amp; 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}]                  &amp; 20000\\
 10 &amp; 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}]                 &amp; 20000\\
 11 &amp; 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}]                           &amp; 10000\\
 12 &amp; 5 11 109 193 1483* [\htmladdnormallink{A57178}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057178}]                                  &amp; 10000\\
 13 &amp; 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}]                 &amp; 10000\\
 14 &amp; 7 53 503 1229 [\htmladdnormallink{A57180}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057180}]                                      &amp; 10000\\
 15 &amp; 3 7 29 1091* 2423* [\htmladdnormallink{A57181}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057181}]                                  &amp; 10000\\
 16 &amp; 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}]                 &amp; 10000\\
 17 &amp; 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}]                         &amp; 10000\\
 18 &amp; 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}]               &amp; 10000\\
 19 &amp; 17 37 157 163 631 7351* [\htmladdnormallink{A57185}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057185}]                             &amp; 10000\\
 20 &amp; 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}]                       &amp; 10000\\
 21 &amp; 3 5 7 13 37 347 [\htmladdnormallink{A57187}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057187}]                                     &amp; 10000\\
 22 &amp; 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}]                  &amp; 10000\\
 23 &amp; 11 13 67 109 331 587 [\htmladdnormallink{A57189}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057189}]                                &amp; 10000\\
 24 &amp; 7 11 19 2207* 2477* 4951* [\htmladdnormallink{A57190}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=057190}]                           &amp; 10000\\
 25 &amp; 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}]    &amp; 10000\\
 26 &amp; 11 109 227 277 347 857 2297* 9043*                 &amp; 10000\\
 27 &amp; Algebraic &amp;\\
 28 &amp; 3 19 373 419 491 1031*                              &amp; 10000\\
 29 &amp; 7                                                   &amp; 10000\\
 30 &amp; 139 173 547 829 2087* 2719* 3109*                  &amp; 10000\\
 31 &amp; 109 461 1061*                                       &amp; 10000\\
 32 &amp; Algebraic &amp;\\
 33 &amp; 5 67 157                                            &amp; 10000\\
 34 &amp; 3                                                   &amp; 10000\\
 35 &amp; 11 13 79 127 503 617 709 857 1499* 3823*         &amp; 10000\\
 36 &amp; 31 191 257 367 3061*                                &amp; 10000\\
 37 &amp; 5 7 2707*                                           &amp; 10000\\
 38 &amp; 5 167 1063* 1597* 2749* 3373*                        &amp; 8000\\
 39 &amp; 3 13 149                                             &amp; 8000\\
 40 &amp; 53 67 1217* 5867* 6143*                              &amp; 8000\\
 41 &amp; 17 691                                              &amp; 8000\\
 42 &amp; 3 709 1637*                                         &amp; 8000\\
 43 &amp; 5 7 19 251 277 383 503 3019* 4517*                  &amp; 8000\\
 44 &amp; 7                                                    &amp; 8000\\
 45 &amp; 103 157                                              &amp; 8000\\
 46 &amp; 7 23 59 71 107 223 331 2207* 6841*                   &amp; 8000\\
 47 &amp; 5 19 23 79 1783* 7681*                               &amp; 8000\\
 48 &amp; 5 17 131                                             &amp; 8000\\
 49 &amp; 7 19 37 83 1481*                                     &amp; 8000\\
 50 &amp; 1153*                                                &amp; 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
&amp; &amp; max $n$\\
$b$ &amp; \quad $n$ for which $Q$ is prime or prp(*) &amp;  tested\\
\hline
 51 &amp; 3 149 3253*                                          &amp; 6000\\
 52 &amp; 7 163 197 223 467 5281*                              &amp; 6000\\
 53 &amp;                                                      &amp; 6000\\
 54 &amp; 7 19 67 197 991*                                     &amp; 6000\\
 55 &amp; 3 5 179 229 1129* 1321* 2251*                        &amp; 6000\\
 56 &amp; 37 107 1063* 4019*                                   &amp; 6000\\
 57 &amp; 53 227                                               &amp; 6000\\
 58 &amp; 3 17 1447*                                           &amp; 6000\\
 59 &amp; 17 43 991*                                           &amp; 6000\\
 60 &amp; 3 937* 1667* 3917*                                   &amp; 6000\\
 61 &amp; 7 41 359                                             &amp; 6000\\
 62 &amp; 11 29 167 313                                        &amp; 6000\\
 63 &amp; 3 37 41 2131* 4027*                                  &amp; 6000\\
 64 &amp; Algebraic &amp;\\
 65 &amp; 19 31                                                &amp; 6000\\
 66 &amp; 7 17 211 643                                        &amp; 6000\\
 67 &amp; 3 2347* 2909* 3203*                                  &amp; 6000\\
 68 &amp; 757* 773*                                            &amp; 6000\\
 69 &amp; 11 211 239 389 503 4649*                            &amp; 6000\\
 70 &amp; 3 61 97                                              &amp; 6000\\
 71 &amp; 5 37 5351*                                           &amp; 6000\\
 72 &amp; 3 7 79 277 3119*                                     &amp; 6000\\
 73 &amp; 7                                                    &amp; 6000\\
 74 &amp; 13 31 37 109                                         &amp; 6000\\
 75 &amp; 5 83                                                 &amp; 6000\\
 76 &amp; 3 5 191 269                                          &amp; 6000\\
 77 &amp; 37 317                                               &amp; 6000\\
 78 &amp; 3 7 31 661* 4217*                                    &amp; 6000\\
 79 &amp; 3 107 457 491 2011*                                &amp; 6000\\
 80 &amp; 5 13 227 439                                        &amp; 6000\\
 81 &amp; 3 5 701* 829* 1031* 1033*                            &amp; 6000\\
 82 &amp; 293 1279*                                            &amp; 6000\\
 83 &amp; 19 31 37 43 421 547 3037*                           &amp; 6000\\
 84 &amp; 7 13 139 359 971* 1087* 3527*                        &amp; 6000\\
 85 &amp; 167 3533*                                            &amp; 6000\\
 86 &amp; 7 17 397                                             &amp; 6000\\
 87 &amp; 7 467                                               &amp; 6000\\
 88 &amp; 709* 1373*                                           &amp; 6000\\
 89 &amp; 13 59 137 1103* 4423*                                &amp; 6000\\
 90 &amp; 3 47                                                 &amp; 6000\\
 91 &amp; 3 11 43 397                                          &amp; 6000\\
 92 &amp; 37 59 113                                            &amp; 6000\\
 93 &amp; 89 571 601 3877*                                   &amp; 6000\\
 94 &amp; 71 307 613 1787* 3793*                              &amp; 6000\\
 95 &amp; 43                                                   &amp; 6000\\
 96 &amp; 37 103 131 263                                       &amp; 6000\\
 97 &amp;                                                      &amp; 6000\\
 98 &amp; 19 101                                               &amp; 6000\\
 99 &amp; 7 37 41 71                                           &amp; 6000\\
100 &amp; 3 293 461                                           &amp; 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
&amp; &amp; max $n$\\
$b$ &amp; \quad $n$ for which $Q$ is prime or prp(*) &amp;  tested\\
\hline
101 &amp; 7 229                                                &amp; 6000\\
102 &amp; 3                                                    &amp; 6000\\
103 &amp;                                                      &amp; 6000\\
104 &amp; 673* 839* 1031*                                      &amp; 6000\\
105 &amp; 11 149 1187* 1627*                                   &amp; 6000\\
106 &amp; 3 7 19 23 31 3989*                                   &amp; 6000\\
107 &amp; 103 983*                                             &amp; 6000\\
108 &amp; 13 223                                               &amp; 6000\\
109 &amp; 59 79 811*                                           &amp; 6000\\
110 &amp; 23 101                                               &amp; 6000\\
111 &amp; 3 5 23 53 383 2039*                                  &amp; 6000\\
112 &amp; 3                                                    &amp; 6000\\
113 &amp;                                                      &amp; 6000\\
114 &amp; 7 13 1801*                                           &amp; 6000\\
115 &amp; 7 31 293                                             &amp; 6000\\
116 &amp; 113 1481* 2089*                                      &amp; 6000\\
117 &amp; 271                                                  &amp; 6000\\
118 &amp; 3 23 109 2357*                                       &amp; 6000\\
119 &amp; 29 53 797*                                           &amp; 6000\\
120 &amp; 3 31 43 263 4919*                                    &amp; 6000\\
121 &amp; 5 13 97 1499*                                        &amp; 6000\\
122 &amp; 293 3877*                                            &amp; 6000\\
123 &amp; 29 739*                                              &amp; 6000\\
124 &amp;                                                      &amp; 6000\\
125 &amp; Algebraic &amp;\\
126 &amp; 5 13 47 163 239 4523*                                &amp; 6000\\
127 &amp; 317 1061*                                            &amp; 6000\\
128 &amp; 7 Algebraic &amp;\\
129 &amp; 17 227 1753*                                         &amp; 6000\\
130 &amp; 467                                                 &amp; 6000\\
131 &amp; 5 101 3389* 3581*                                    &amp; 6000\\
132 &amp; 3 101 157 1303*                                      &amp; 6000\\
133 &amp; 5 7 17 59 79 157                                     &amp; 6000\\
134 &amp; 13 1171*                                             &amp; 6000\\
135 &amp; 5 7 2671*                                            &amp; 6000\\
136 &amp; 5 7 23 59 199 2053*                                  &amp; 6000\\
137 &amp; 101 241 353 1999*                                    &amp; 6000\\
138 &amp; 103 577*                                             &amp; 6000\\
139 &amp; 3 17 47 2683* 2719*                                  &amp; 6000\\
140 &amp; 59                                                   &amp; 6000\\
141 &amp; 5 1471*                                              &amp; 6000\\
142 &amp; 3                                                    &amp; 6000\\
143 &amp; 7 17 19 47 103 4423*                                 &amp; 6000\\
144 &amp; 3 23 41 317 3371*                                    &amp; 6000\\
145 &amp; 7 23 281                                             &amp; 6000\\
146 &amp; 17 1439*                                             &amp; 6000\\
147 &amp; 11 151                                               &amp; 6000\\
148 &amp; 3 7 31 43 163 317 1933* 5669*                        &amp; 6000\\
149 &amp; 17 769*                                              &amp; 6000\\
150 &amp;                                                      &amp; 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
&amp; &amp; max $n$\\
$b$ &amp; \quad $n$ for which $Q$ is prime or prp(*) &amp;  tested\\
\hline
151 &amp; 3 367 3203*                                          &amp; 6000\\
152 &amp; 13 19                                                &amp; 6000\\
153 &amp; 13 1063* 5749*                                       &amp; 6000\\
154 &amp; 3 29 263 601* 619* 809* 1217* 2267*                  &amp; 6000\\
155 &amp; 5                                                    &amp; 6000\\
156 &amp; 3 1301*                                              &amp; 6000\\
157 &amp; 5 157 809* 1861* 2203*                               &amp; 6000\\
158 &amp; 5 769* 5023*                                         &amp; 6000\\
159 &amp; 283 449 1949*                                       &amp; 6000\\
160 &amp; 11 37 1907*                                          &amp; 6000\\
161 &amp; 31 331 1483*                                         &amp; 6000\\
162 &amp; 3 1823*                                              &amp; 6000\\
163 &amp; 3 11 31 661* 1999* 4079*                             &amp; 6000\\
164 &amp; 7 103 541 1109*                                     &amp; 6000\\
165 &amp; 3 5 383                                             &amp; 6000\\
166 &amp; 17 5437*                                             &amp; 6000\\
167 &amp; 17 59 1301* 3167*                                    &amp; 6000\\
168 &amp; 3 31 1741* 2099*                                     &amp; 6000\\
169 &amp; 3 7 109                                              &amp; 6000\\
170 &amp; 7                                                    &amp; 6000\\
171 &amp; 13 149 257 4967*                                     &amp; 6000\\
172 &amp; 37 283 647* 4483* 5417*                              &amp; 6000\\
173 &amp; 7 59 569* 2647*                                      &amp; 6000\\
174 &amp; 3 3191*                                              &amp; 6000\\
175 &amp;                                                      &amp; 6000\\
176 &amp; 5 31 269 479 599* 809* 1307*                        &amp; 6000\\
177 &amp; 3 5 19 419                                          &amp; 6000\\
178 &amp; 61 167 227                                           &amp; 6000\\
179 &amp; 827* 5011*                                           &amp; 6000\\
180 &amp; 5 13                                                 &amp; 6000\\
181 &amp; 449 2687* 4877*                                     &amp; 6000\\
182 &amp; 1487*                                                &amp; 6000\\
183 &amp; 11                                                   &amp; 6000\\
184 &amp; 19 79 149                                            &amp; 6000\\
185 &amp; 11                                                   &amp; 6000\\
186 &amp;                                                      &amp; 6000\\
187 &amp;                                                      &amp; 6000\\
188 &amp;                                                      &amp; 6000\\
189 &amp; 3 31 71                                              &amp; 6000\\
190 &amp; 3 19 1153*                                           &amp; 6000\\
191 &amp; 479 1163*                                           &amp; 6000\\
192 &amp; 109 197 587 727* 1997* 2441*                        &amp; 6000\\
193 &amp; 3 11 67 3253*                                        &amp; 6000\\
194 &amp; 19 31                                                &amp; 6000\\
195 &amp; 3 13 19 43 89 1087* 1949* 2939*                      &amp; 6000\\
196 &amp; 43 1049* 5441*                                       &amp; 6000\\
197 &amp; 31 37 101 163                                        &amp; 6000\\
198 &amp; 37 151 937*                                          &amp; 6000\\
199 &amp; 313 2579* 5387*                                      &amp; 6000\\
200 &amp; 7 277                                                &amp; 6000\\
\hline
\end{tabular}
}
\end{table} 

\clearpage
\centerline{\rule{6.5in}{.01in}}

\vspace*{+.1in}
\noindent
(Concerned with sequences
\seqnum{A000978},
\seqnum{A007658},
\seqnum{A057171},
\seqnum{A057172},
\seqnum{A057173},
\seqnum{A057175},
\seqnum{A057176},
\seqnum{A057177},
\seqnum{A057178},
\seqnum{A057179},
\seqnum{A057180},
\seqnum{A057181},
\seqnum{A057182},
\seqnum{A057183},
\seqnum{A057184},
\seqnum{A057185},
\seqnum{A057186},
\seqnum{A057187},
\seqnum{A057188},
\seqnum{A057189},
\seqnum{A057190}, and
\seqnum{A057191}.)

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\vspace*{+.1in}
\noindent
Received September 10, 2000;
published in Journal of Integer Sequences November 28, 2000.

\centerline{\rule{6.5in}{.01in}}

\vspace*{+.1in}
\noindent
Return to \htmladdnormallink{Journal of Integer Sequences home
page}{http://oeis.org}

\centerline{\rule{6.5in}{.01in}}

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