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A028491 Numbers n such that (3^n - 1)/2 is prime.
(Formerly M2643)
59
3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If n is in the sequence and m=3^(n-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, n such that 3^n-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.

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..18.

Paul Bourdelais,A Generalized Repunit Conjecture

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. Lifchitz, Mersenne and Fermat primes field

Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969v1 [math.NT], Mar 18, 2012

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Repunit

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.

Sequence in context: A228209 A176903 A004060 * A137474 A071087 A038691

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

36913 from Farideh Firoozbakht, Mar 27 2005

a(14), a(15) & a(16) from Robert G. Wilson v, Apr 11 2005

a(17) = 483611 is a probable prime discovered by Paul Bourdelais, Feb 08 2010

a(18) = 877843 is a probable prime discovered by Paul Bourdelais, Jul 06 2010

STATUS

approved

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Last modified September 23 07:37 EDT 2014. Contains 247114 sequences.