A003307 Numbers k such that 2*3^k - 1 is prime.

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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

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

• 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
• W. Keller and J. 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 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.
• H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

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).

Programs

PARI

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

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