

A107360


Numbers p (necessarily prime) such that 2^p  1 is a Mersenne prime and (2^p+1)/3 is a Wagstaff prime.


4




OFFSET

1,1


COMMENTS

Intersection of A000043 and A000978.
'The New Mersenne Conjecture' (BatemanSelfridgeWagstaff) states that if two of the following statements about an odd positive integer p are true, then the third one is also true: (a) p = 2^k + 1 or p = 4^k + 3, (b) 2^p  1 is prime, (c) (2^p + 1)/3 is prime. (Amer Math Monthly, 96 (1989) p. 125.)  R. K. Guy, May 20 2005
The next term, if it exists, is not any currently known Mersenne prime exponent or Wagstaff prime exponent: it must be larger than A000043(47) = 43112609 and cannot be 57885161, 74207281, 77232917, or 82589933. See Caldwell and both Wanless links. The New Mersenne Conjecture would require this sequence to be a subsequence of A122834, in which case the next term could not be less than A122834(28) = 2305843009213693951. See Caldwell and Höglund links.  Gord Palameta, Jun 28 2019
p either has the form 2^k + 1 or the form 4^k + 3, according to the New Mersenne Conjecture.  Lekraj Beedassy, Sep 20 2006
Primes p such that (4^p  1)/3 is a semiprime.  Arkadiusz Wesolowski, Jun 01 2013
Numbers m != 4 such that (4^m  1)/3 is a semiprime.  Thomas Ordowski, Sep 25 2015
The indices of Wagstaff primes relating to the new Mersenne conjecture A122834 in a list of Jacobsthal numbers A001045.  Steve Homewood, Dec 01 2020


LINKS

Table of n, a(n) for n=1..9.
P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125128, 1989.
Chris K. Caldwell, Prime Pages, The New Mersenne Prime Conjecture
Andreas Höglund, New Mersenne Conjecture (Wagstaff p = 268435459 has a factor)
Carlos Rivera, Conjecture 15  The New Mersenne Conjecture
James Wanless, Mersenneplustwo Factorizations (for p in known Mersenneprime exponents, all Wagstaff numbers have factors except p = 19937, 30402457, 42643801, 74207281)
James Wanless, posts to Google group "Mersenneplustwo" show that Wagstaff p = 30402457, 42643801 are composite by PRP test and Wagstaff p = 74207281 is composite by PRP test
Wikipedia, New Mersenne conjecture


MATHEMATICA

Select[Prime@Range[31], PrimeQ[(2^# + 1)/3] && PrimeQ[2^#  1] &] (* Arkadiusz Wesolowski, Jun 01 2013 *)


PROG

(MAGMA) [p: p in PrimesUpTo(500)  IsPrime(2^p1) and IsPrime((2^p+1) div 3)]; // Vincenzo Librandi, Sep 25 2015
(PARI) forprime(p=2, 1e3, if (!((2^p+1) % 3) && isprime((2^p+1)/3) && isprime(2^p1), print1(p, ", "))); \\ Altug Alkan, Sep 25 2015


CROSSREFS

Cf. A000043, A000978, A122834.
Sequence in context: A045399 A122834 A174265 * A058341 A116036 A304122
Adjacent sequences: A107357 A107358 A107359 * A107361 A107362 A107363


KEYWORD

nonn,hard,more


AUTHOR

Lekraj Beedassy, May 23 2005


STATUS

approved



