%I M2643 #121 Feb 16 2025 08:32:35

%S 3,7,13,71,103,541,1091,1367,1627,4177,9011,9551,36913,43063,49681,

%T 57917,483611,877843,2215303,2704981,3598867,7973131,8530117

%N Numbers k such that (3^k - 1)/2 is prime.

%C 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 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 Also, k such that 3^k-1 is a semiprime - see also A080892. - _M. F. Hasler_, Mar 19 2013

%D 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Antal Bege and Kinga Fogarasi, <a href="https://arxiv.org/abs/1008.0155">Generalized perfect numbers</a>, arXiv:1008.0155 [math.NT], 2010. See p. 81.

%H Paul Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>, Posting in [email protected], Jun 25, 2009.

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.

%H H. Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]

%H H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>

%H Christian Salas, <a href="http://arxiv.org/abs/1203.3969">Cantor Primes as Prime-Valued Cyclotomic Polynomials</a>, arXiv:1203.3969 [math.NT], 2012.

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>

%H <a href="/index/Pri#primepop">Index to primes in various ranges</a>, form ((k+1)^n-1)/k

%t Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* _Farideh Firoozbakht_, Feb 09 2005 *)

%o (PARI) forprime(p=2,1e5,if(ispseudoprime(3^p\2),print1(p", "))) \\ _Charles R Greathouse IV_, Jul 15 2011

%Y Cf. A076481, A033632, A112646.

%K nonn,more,hard,changed

%O 1,1

%A _N. J. A. Sloane_, Jean-Yves Perrier (nperrj(AT)ascom.ch)

%E a(13) from _Farideh Firoozbakht_, Mar 27 2005

%E a(14)-a(16) from _Robert G. Wilson v_, Apr 11 2005

%E All larger terms only correspond to probable primes.

%E a(17) from _Paul Bourdelais_, Feb 08 2010

%E a(18) from _Paul Bourdelais_, Jul 06 2010

%E a(19) from _Paul Bourdelais_, Feb 05 2019

%E a(20) and a(21) from _Ryan Propper_, Dec 29 2021

%E a(22) from _Ryan Propper_, Nov 06 2023

%E a(23) from _Ryan Propper_, Nov 09 2023