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A004062 Numbers k such that (6^k - 1)/5 is prime.

%I A004062 M0861 #69 Apr 25 2025 04:33:19
%S A004062 2,3,7,29,71,127,271,509,1049,6389,6883,10613,19889,79987,608099,
%T A004062 1365019,3360347
%N A004062 Numbers k such that (6^k - 1)/5 is prime.
%C A004062 Prime repunits in base 6.
%C A004062 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 A004062 Also, numbers k such that 6^k-1 is semiprime. - _Sean A. Irvine_, Oct 16 2023
%D A004062 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 A004062 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
%D A004062 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004062 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 A004062 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 A004062 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 A004062 H. Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
%H A004062 H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>
%H A004062 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>
%H A004062 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H A004062 <a href="/index/Pri#primepop">Index to primes in various ranges</a>, form ((k+1)^n-1)/k
%t A004062 Select[Range[1000], PrimeQ[(6^# - 1)/5] &] (* _Alonso del Arte_, Dec 31 2019 *)
%o A004062 (PARI) is(n)=isprime((6^n - 1)/5) \\ _Charles R Greathouse IV_, Apr 28 2015
%K A004062 hard,more,nonn
%O A004062 1,1
%A A004062 _N. J. A. Sloane_
%E A004062 More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 22 2003
%E A004062 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 A004062 a(15) corresponds to a probable prime discovered by _Paul Bourdelais_, May 24 2010
%E A004062 a(16) corresponds to a probable prime discovered by _Paul Bourdelais_, Dec 31 2019
%E A004062 a(17) corresponds to a probable prime discovered by _Ryan Propper_, Oct 30 2023