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%I #25 Apr 02 2014 17:01:42
%S 5,11,31,37,47,53,97,163,167,509,877,1061,2027,2293,3011,6803,8423,
%T 13627,20047,28411,50221,50993,71453,152809,272141,505823,1353449
%N Numbers n such that the Eisenstein integer ((1-ω)^n+1)/(2-ω) has prime norm, where ω = - 1/2 + sqrt(-3)/2.
%C These numbers are sometimes called Eisenstein-Mersenne cofactors EQ(n).
%C The p-th Eisenstein-Mersenne cofactor can be written as EQ(p) = (3^p + Legendre(3, p) * 3^((p + 1)/2) + 1)/7.
%C Following an idea of Harsh Aggarwal, some of these numbers have been discovered as by-products of the search for prime Eisenstein-Mersenne norms. The reason of that is the Aurifeuillan factorization of T(k) = 3^(3k) + 1 with k odd. These numbers can be written as T(k) = (3^k + 1)*EM(k)*EQ(k)*7, EM(k) is the norm of the Eisenstein-Mersenne (1-ω)^k-1, while EQ(k) is the norm of ((1-ω)^a[n]+1)/(2-ω).
%C These numbers have been proved prime only up to exponent a(19) = 20047.
%C Next term a(28) > 1500000.
%H Henri Lifchitz & Renaud Lifchitz: <a href="http://www.primenumbers.net/prptop/searchform.php?form=%283%5E%3F%2B1%29%2F7&action=Search">PRP Records. Probable Primes Top 10000</a>.
%e For n = 3: ((1-ω)^31+1)/(2-ω) is an Eisenstein prime because its norm, (3^31-3^16+1)/7 = 88239050462461, is prime.
%o (PARI) forprime(n=3,2300,if(ispseudoprime((3^n+kronecker(3,n)*3^((n+1)/2)+1)/7),print1(n ", "))); /* _Serge Batalov_, Mar 29 2014 */
%Y Cf. A125743 = Primes p such that (3^p - 3^((p+1)/2) + 1)/7 is prime.
%Y Cf. A125744 = Primes p such that (3^p + 3^((p+1)/2) + 1)/7 is prime.
%Y Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.
%Y Cf. A124112 = Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime.
%K more,nonn,hard
%O 1,1
%A _Serge Batalov_, Mar 27 2014
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