A122834 Primes in the new Mersenne conjecture; odd primes of the form 2^k+-1 or 4^k+-3.

3, 5, 7, 13, 17, 19, 31, 61, 67, 127, 257, 1021, 4093, 4099, 8191, 16381, 65537, 65539, 131071, 262147, 524287, 1048573, 4194301, 16777213, 268435459, 1073741827, 2147483647, 2305843009213693951, 19342813113834066795298819

Offset: 1

Author

T. D. Noe, Sep 12 2006

Keywords

• nonn

Comments

Let p be a prime in this sequence. Call q=2^p-1 and r=(2^p+1)/3. The new Mersenne conjecture implies that either q and r are both prime or both composite.

References

• R. K. Guy, Unsolved Problems in Number Theory, Section A3.

Links

• Gord Palameta, Table of n, a(n) for n = 1..40
• P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125-128, 1989.
• John Renze and Eric Weisstein's World of Mathematics, MathWorld: New Mersenne Prime Conjecture.

Crossrefs

Superset of: A000668, A019434, A228026.

Cf. A000043 (n such that 2^n-1 is prime), A000978 (n such that (2^n+1)/3 is prime), A107360 (the intersection of these).

Programs

Mathematica

nn=100; Union[Select[1+2^Range[16],PrimeQ], Select[ -1+2^Range[2nn],PrimeQ], Select[3+4^Range[nn],PrimeQ], Select[ -3+4^Range[nn],PrimeQ]]