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

%I A234310 #19 Sep 26 2015 15:06:23
%S A234310 7,19,31,67,79,127,271,1039,1087,1279,4099,4111,4159,5119,8191,16447,
%T A234310 20479,65539,65551,65599,81919,131071,262147,262399,263167,266239,
%U A234310 524287,1049599,1114111,1310719,4194319,4194559,4195327,16842751,17825791,67108879
%N A234310 Primes of the form 4^k + 4^m - 1, where k and m are positive integers.
%C A234310 Clearly each term is congruent to 1 modulo 6.
%C A234310 By the conjecture in A234309, this sequence should have infinitely many terms.
%C A234310 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.
%H A234310 Zhi-Wei Sun, <a href="/A234310/b234310.txt">Table of n, a(n) for n = 1..800</a>
%e A234310 a(1) = 7 since 7 = 4^1 + 4^1 - 1 is prime.
%e A234310 a(2) = 19 since 19 = 4^1 + 4^2 - 1 is prime.
%e A234310 a(3) = 31 since 31 = 4^2 + 4^2 - 1 is prime.
%t A234310 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}]
%o A234310 (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
%Y A234310 Cf. A000040, A000302, A000668, A233346, A233393, A234309.
%K A234310 nonn
%O A234310 1,1
%A A234310 _Zhi-Wei Sun_, Dec 23 2013