cp's OEIS Frontend

PrimePi[ a(n) ] = {4, 7, 24, 200, 724, 1361, 1389, 1503, 2578, 3868, 5368, 5400, 11814, 31200, ...}.

3 terms found by David Broadhurst in Nov 2006: {36479, 52567, 52919}.

Only 2 terms found by Jean Penne in Nov 2006 belong to a(n): {125929, 365689}.

5 other numbers found by Jean Penne in Nov 2006 belong to related sequence of primes p such that (2^p - 2^((p+1)/2) + 1)/5 is prime: {221891, 235099, 305867, 311027, 333227}.

All terms belong to A124112 = {5, 7, 9, 11, 13, 17, 29, 43, 53, 89, 283, 557, 563, 613, 691, 1223, 2731, ...} (numbers k such that ((1+i)^k+1)/(2+i) is a Gaussian prime).

The terms 1127239 and 1148729 were found by Borys Jaworski in 2006-2007. - Alexander Adamchuk, Jun 20 2007

PrimePi[ a(n) ] = {3, 5, 6, 10, 14, 16, 61, 102, 103, 112, 125, 399, 686, 705, 1180, 1312, 19768, 20843, 26482, 26882, 28656, ...}. (2^p - 2^((p+1)/2) + 1) is the Aurifeuillan cofactor of 4^p + 1, where p is odd prime. All a(n) belong to A124112(n) = {5, 7, 9, 11, 13, 17, 29, 43, 53, 89, 283, 557, 563, 613, 691, 1223, 2731, ...} Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime. 5 largest currently known terms found by Jean Penne in Nov 2006: {221891, 235099, 305867, 311027, 333227}.

These numbers are sometimes called Eisenstein-Mersenne cofactors EQ(n).

The p-th Eisenstein-Mersenne cofactor can be written as EQ(p) = (3^p + Legendre(3, p) * 3^((p + 1)/2) + 1)/7.

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).

These numbers have been proved prime only up to exponent a(19) = 20047.

Next term a(28) > 1500000.