A004061 Numbers k such that (5^k - 1)/4 is prime.

3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593

Offset: 1

Author

N. J. A. Sloane

Keywords

• hard
• nonn
• more

Comments

With the addition of the 19th prime in the sequence, the new best linear fit to the sequence has G=0.4723, which is slightly closer to the conjectured limit of G=0.56145948, A080130 (see link for Generalized Repunit Conjecture). - Paul Bourdelais, Apr 30 2018

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

• Paul Bourdelais, A Generalized Repunit Conjecture [From _Paul Bourdelais_, Jun 01 2010]
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• 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

Crossrefs

Cf. A080130.

Programs

Mathematica

lst={};Do[If[PrimeQ[(5^n-1)/4], AppendTo[lst, n]], {n, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 20 2008 *)

PARI

forprime(p=2,1e4,if(ispseudoprime(5^p\4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

Extensions

a(13)-a(15) from Kamil Duszenko (kdusz(AT)wp.pl), Mar 25 2003

a(16) corresponds to a probable prime based on trial factoring to 4*10^13 and Fermat primality testing base 2. - Paul Bourdelais, Dec 11 2008

a(17) corresponds to a probable prime discovered by Paul Bourdelais, Jun 01 2010

a(18) corresponds to a probable prime discovered by Paul Bourdelais, Apr 30 2018

a(19) corresponds to a probable prime discovered by Ryan Propper, Jan 02 2022