|
%I M2643
%S 3,7,13,71,103,541,1091,1367,1627,4177,9011,9551,36913,43063,49681,
%T 57917,483611,877843
%N Numbers n such that (3^n - 1)/2 is prime.
%C 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
%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, n such that 3^n-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 H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Paul Bourdelais,<a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=nmbrthry&T=0&F=&S=&X=7769E632C9622BAE75&Y=paul.bourdelais%40gd-ais.com&P=999">A Generalized Repunit Conjecture</a>
%H J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%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:1203.3969v1 [math.NT], Mar 18, 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="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%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.
%K nonn,more,hard,changed
%O 1,1
%A _N. J. A. Sloane_, Jean-Yves Perrier (nperrj(AT)ascom.ch)
%E 36913 from _Farideh Firoozbakht_, Mar 27 2005
%E a(14), a(15) & a(16) from _Robert G. Wilson v_, Apr 11 2005
%E a(17)=483611 is a probable prime discovered by _Paul Bourdelais_, Feb 08 2010
%E a(18)=877843 is a probable prime discovered by _Paul Bourdelais_, Jul 06 2010
|
