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A028491
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Numbers k such that (3^k - 1)/2 is prime.
(Formerly M2643)
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72
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3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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Table of n, a(n) for n=1..21.
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:1203.3969v1 [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)^n-1)/k
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MATHEMATICA
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Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* Farideh Firoozbakht, Feb 09 2005 *)
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PROG
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(PARI) forprime(p=2, 1e5, if(ispseudoprime(3^p\2), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011
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CROSSREFS
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Cf. A076481, A033632.
Sequence in context: A228209 A176903 A004060 * A137474 A071087 A309775
Adjacent sequences: A028488 A028489 A028490 * A028492 A028493 A028494
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KEYWORD
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nonn,more,hard
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AUTHOR
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N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)
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EXTENSIONS
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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
a(20) and a(21) correspond to probable primes discovered by Ryan Propper, Dec 2021
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STATUS
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approved
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