A234310 Primes of the form 4^k + 4^m - 1, where k and m are positive integers.

7, 19, 31, 67, 79, 127, 271, 1039, 1087, 1279, 4099, 4111, 4159, 5119, 8191, 16447, 20479, 65539, 65551, 65599, 81919, 131071, 262147, 262399, 263167, 266239, 524287, 1049599, 1114111, 1310719, 4194319, 4194559, 4195327, 16842751, 17825791, 67108879

Offset: 1

Author

Zhi-Wei Sun, Dec 23 2013

Keywords

• nonn

Comments

Clearly each term is congruent to 1 modulo 6.

By the conjecture in A234309, this sequence should have infinitely many terms.

Note that any Mersenne prime greater than 3 has the form 2^{2*k+1} - 1 = 4^k + 4^k - 1, where k is a positive integer.

Examples

			a(1) = 7 since 7 = 4^1 + 4^1 - 1 is prime.
a(2) = 19 since 19 = 4^1 + 4^2 - 1 is prime.
a(3) = 31 since 31 = 4^2 + 4^2 - 1 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..800

Crossrefs

Cf. A000040, A000302, A000668, A233346, A233393, A234309.

Programs

Mathematica

n=0;Do[If[PrimeQ[4^k+4^m-1],n=n+1;Print[n," ",4^m+4^k-1]],{m,1,250},{k,1,m}]

PARI

for(k=1,30,for(m=1,k,if(ispseudoprime(t=4^k+4^m-1),print1(t", ")))) \\ Charles R Greathouse IV, Dec 23 2013