A003307 Numbers k such that 2*3^k - 1 is prime. (Formerly M0823)

1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104

Offset

1,2

Comments

2234430 and 2834778 are also terms (found by Propper). - Hermann Stamm-Wilbrandt, Feb 12 2026

References

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

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

Links

Table of n, a(n) for n=1..37.

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Steven Harvey, Williams Primes.

Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.

H. C. Williams, The primality of certain integers of the form 2*A*r^n - 1, Acta Arith. 39 (1981), 7-17.

H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*A*3^n+1 and 2*A*3^n-1, Math. Comp., 26 (1972), 995-998.

Prog

(PARI) for(n=1, 1e4, if(ispseudoprime(2*3^n-1), print1(n", "))) \\ Charles R Greathouse IV, Jul 16 2011

Crossrefs

Cf. A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Cf. A079363 (primes of the form 2*3^k - 1), A003306 (k such that 2*3^k + 1 is prime).

Sequence in context: A184860 A242655 A145489 * A105601 A390917 A199971

Adjacent sequences: A003304 A003305 A003306 * A003308 A003309 A003310

Keyword

nonn,hard,nice,more

Author

N. J. A. Sloane

Extensions

More terms from Douglas Burke (dburke(AT)nevada.edu)

More terms from T. D. Noe, Aug 24 2005

Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008

a(35) from Borys Jaworski, Sep 02 2011

a(36) from Borys Jaworski, Feb 13 2012

a(37) from Jeppe Stig Nielsen, Sep 28 2018

Status

approved