A028491 - OEIS

A028491

Numbers k such that (3^k - 1)/2 is prime.
(Formerly M2643)

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117

COMMENTS

If k is in the sequence and m=3^(k-1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m))), so 3^(A028491-1) is a subsequence of A033632. For example since 9551 is in the sequence, phi(sigma(3^9550)) = sigma(phi(3^9550)). - Farideh Firoozbakht, Feb 09 2005

Salas lists these, except 3, in "Open Problems" p. 6 [March 2012], and proves that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).

Also, k such that 3^k-1 is a semiprime - see also A080892. - M. F. Hasler, Mar 19 2013

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

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Eric Weisstein's World of Mathematics, Repunit

EXTENSIONS

All larger terms only correspond to probable primes.