%I M0861 #64 Oct 30 2023 15:23:26

%S 2,3,7,29,71,127,271,509,1049,6389,6883,10613,19889,79987,608099,

%T 1365019,3360347

%N Numbers k such that (6^k - 1)/5 is prime.

%C Prime repunits in base 6.

%C With this 16th prime, the base 6 repunits have an average (best linear fit) occurrence rate of G = 0.4948 which seems to be converging to the conjectured rate of 0.56146 (see ref).

%C Also, numbers k such that 6^k-1 is semiprime. - _Sean A. Irvine_, Oct 16 2023

%D J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Brillhart et al., <a href="http://www.ams.org/books/conm/022/">Cunningham Project</a> [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]

%H Paul Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>. - _Paul Bourdelais_, May 24 2010

%H H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.

%H H. Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]

%H H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>

%H <a href="/index/Pri#primepop">Index to primes in various ranges</a>, form ((k+1)^n-1)/k

%t Select[Range[1000], PrimeQ[(6^# - 1)/5] &] (* _Alonso del Arte_, Dec 31 2019 *)

%o (PARI) is(n)=isprime((6^n - 1)/5) \\ _Charles R Greathouse IV_, Apr 28 2015

%K hard,nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 22 2003

%E a(14) discovered Nov 05 2007, corresponds to a probable prime based on trial factoring to 10^11 and Fermat primality test base 2. - _Paul Bourdelais_

%E a(15) corresponds to a probable prime discovered by _Paul Bourdelais_, May 24 2010

%E a(16) corresponds to a probable prime discovered by _Paul Bourdelais_, Dec 31 2019

%E a(17) corresponds to a probable prime discovered by _Ryan Propper_, Oct 30 2023