# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a028491 Showing 1-1 of 1 %I A028491 M2643 #116 Aug 06 2024 02:36:17 %S A028491 3,7,13,71,103,541,1091,1367,1627,4177,9011,9551,36913,43063,49681, %T A028491 57917,483611,877843,2215303,2704981,3598867,7973131,8530117 %N A028491 Numbers k such that (3^k - 1)/2 is prime. %C A028491 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 %C A028491 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). %C A028491 Also, k such that 3^k-1 is a semiprime - see also A080892. - _M. F. Hasler_, Mar 19 2013 %C A028491 a(22) > 5000000. %D A028491 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements. %D A028491 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A028491 Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See p. 81. %H A028491 Paul Bourdelais, A Generalized Repunit Conjecture, Posting in [email protected], Jun 25, 2009. %H A028491 J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. %H A028491 H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. %H A028491 H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy] %H A028491 H. Lifchitz, Mersenne and Fermat primes field %H A028491 Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969 [math.NT], 2012. %H A028491 S. S. Wagstaff, Jr., The Cunningham Project %H A028491 Eric Weisstein's World of Mathematics, Repunit %H A028491 Index to primes in various ranges, form ((k+1)^n-1)/k %t A028491 Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* _Farideh Firoozbakht_, Feb 09 2005 *) %o A028491 (PARI) forprime(p=2,1e5,if(ispseudoprime(3^p\2),print1(p", "))) \\ _Charles R Greathouse IV_, Jul 15 2011 %Y A028491 Cf. A076481, A033632, A112646. %K A028491 nonn,more,hard %O A028491 1,1 %A A028491 _N. J. A. Sloane_, Jean-Yves Perrier (nperrj(AT)ascom.ch) %E A028491 a(13) from _Farideh Firoozbakht_, Mar 27 2005 %E A028491 a(14)-a(16) from _Robert G. Wilson v_, Apr 11 2005 %E A028491 All larger terms only correspond to probable primes. %E A028491 a(17) from _Paul Bourdelais_, Feb 08 2010 %E A028491 a(18) from _Paul Bourdelais_, Jul 06 2010 %E A028491 a(19) from _Paul Bourdelais_, Feb 05 2019 %E A028491 a(20) and a(21) from _Ryan Propper_, Dec 29 2021 %E A028491 a(22) from _Ryan Propper_, Nov 06 2023 %E A028491 a(23) from _Ryan Propper_, Nov 09 2023 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE