

A028491


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


70



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

1,1


COMMENTS

If n is in the sequence and m=3^(n1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m)), so 3^(A0284911) 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 primevalued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).
Also, n such that 3^n1 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..19.
Antal Bege, 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), 927930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Christian Salas, Cantor Primes as PrimeValued Cyclotomic Polynomials, arXiv:1203.3969v1 [math.NT], Mar 18, 2012.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n1)/k


MATHEMATICA

Do[If[PrimeQ[(3^n1)/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, JeanYves 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
a(17) corresponds to a probable prime discovered by Paul Bourdelais, Feb 08 2010
a(18) corresponds to a probable prime discovered by Paul Bourdelais, Jul 06 2010
a(19) corresponds to a probable prime discovered by Paul Bourdelais, Feb 05 2019


STATUS

approved



