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

1,1

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.

Links

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

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A216532 A249375 A212492 * A141338 A237366 A216531

Adjacent sequences: A234307 A234308 A234309 * A234311 A234312 A234313

Keyword

nonn

Author

Zhi-Wei Sun, Dec 23 2013

Status

approved