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

2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347

Offset: 1

Author

N. J. A. Sloane

Keywords

• hard
• more
• nonn

Comments

Prime repunits in base 6.

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).

Also, numbers k such that 6^k-1 is semiprime. - Sean A. Irvine, Oct 16 2023

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
• Paul Bourdelais, A Generalized Repunit Conjecture. - _Paul Bourdelais_, May 24 2010
• H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
• H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
• H. Lifchitz, Mersenne and Fermat primes field
• S. S. Wagstaff, Jr., The Cunningham Project
• Eric Weisstein's World of Mathematics, Repunit
• Index to primes in various ranges, form ((k+1)^n-1)/k

Programs

Mathematica

Select[Range[1000], PrimeQ[(6^# - 1)/5] &] (* Alonso del Arte, Dec 31 2019 *)

PARI

is(n)=isprime((6^n - 1)/5) \\ Charles R Greathouse IV, Apr 28 2015

Extensions

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

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

a(15) corresponds to a probable prime discovered by Paul Bourdelais, May 24 2010

a(16) corresponds to a probable prime discovered by Paul Bourdelais, Dec 31 2019

a(17) corresponds to a probable prime discovered by Ryan Propper, Oct 30 2023