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

Offset

1,1

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

a(22) > 5000000.

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.

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..23.

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See p. 81.

Paul Bourdelais, A Generalized Repunit Conjecture, Posting in NMBRTHRY@LISTSERV.NODAK.EDU, Jun 25, 2009.

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

Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969 [math.NT], 2012.

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

Mathematica

Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* Farideh Firoozbakht, Feb 09 2005 *)

Prog

(PARI) forprime(p=2, 1e5, if(ispseudoprime(3^p\2), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

Crossrefs

Cf. A076481, A033632, A112646.

Sequence in context: A228209 A176903 A004060 * A137474 A071087 A309775

Adjacent sequences: A028488 A028489 A028490 * A028492 A028493 A028494

Keyword

nonn,more,hard

Author

N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)

Extensions

a(13) from Farideh Firoozbakht, Mar 27 2005

a(14)-a(16) from Robert G. Wilson v, Apr 11 2005

All larger terms only correspond to probable primes.

a(17) from Paul Bourdelais, Feb 08 2010

a(18) from Paul Bourdelais, Jul 06 2010

a(19) from Paul Bourdelais, Feb 05 2019

a(20) and a(21) from Ryan Propper, Dec 29 2021

a(22) from Ryan Propper, Nov 06 2023

a(23) from Ryan Propper, Nov 09 2023

Status

approved