A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).

3, 5, 9, 17, 65, 1025, 65537, 16777217, 67108865, 34359738369, 4503599627370497, 36028797018963969, 39614081257132168796771975169, 22300745198530623141535718272648361505980417

Offset: 1

Author

Jaroslav Krizek, Jan 28 2016

Keywords

• nonn

Comments

Conjecture: subsequence of prime terms (3, 5, 17, 65537, ...) is not the same as A249759.

Corresponding values of numbers k are in A098855 (numbers n such that 4^n + 2^n - 1 is prime).

Corresponding values of primes q: 5, 19, 71, 271, 4159, 1049599, 4295032831, ...

4 out of 5 known Fermat primes (3, 5, 17, 65537) are terms; corresponding values of primes q: 5, 19, 271, 4295032831.

Examples

			17  = 2^4 + 1 is a term because 17^2 - 17 - 1 = 271 (prime).
		

Crossrefs

Intersection of A002328 and A000051.

Cf. A019434, A091567, A098855, A249759.

Programs

Magma

[2^n + 1: n in [0..300] | IsPrime((2^n + 1)^2 - 2^n - 2)]

Mathematica

2^# + 1 &@ Select[Range[0, 300], PrimeQ[#^2 - # - 1 &@ (2^# + 1)] &] (* Michael De Vlieger, Jan 29 2016 *)

PARI

lista(nn) = {for (k=0, nn, n = 2^k+1; if (isprime(n^2-n-1), print1(n, ", ")););} \\ Michel Marcus, Mar 06 2016