A003306 Numbers k such that 2*3^k + 1 is prime.

0, 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132, 180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225, 7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106, 152529, 165896, 191814, 529680, 1074726, 1086112, 1175232, 1277862, 1346542, 3123036, 3648969, 5570081, 6236772, 10852677

Offset: 1

Author

N. J. A. Sloane, Chris K. Caldwell

Keywords

• nonn

References

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

Links

• C. K. Caldwell, The Prime Pages
• Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019. See also Integers (2020) Vol. 20, Article A11.
• 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 and C. R. Zarnke, Some prime numbers of the forms 2A3^n+1 and 2A3^n-1, Math. Comp., 26 (1972), 995-998.

Crossrefs

Cf. A056802 (k such that 2*9^k + 1 is prime).

Cf. A111974 (primes of the form 2*3^k + 1), A003307 (k such that 2*3^k - 1 is prime).

Programs

Mathematica

lst={};Do[If[PrimeQ[2*3^n+1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

PARI

is(n)=isprime(2*3^n+1) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

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

More terms from David Broadhurst, Feb 14 2010

Another term from David Broadhurst, Feb 22 2010

a(42)-a(45) found by Ryan Propper and Paul S. Vanderveen, Feb 09 2020

a(46) found by Ryan Propper, Feb 14 2020

a(47)-a(48) found by Ryan Propper added by Paul S. Vanderveen, Jan 08 2023