A125744 -id:A125744 - cp's OEIS frontend

A125743 Primes p such that (3^p - 3^((p + 1)/2) + 1)/7 is prime.

5, 31, 53, 163, 509, 1061, 13627, 20047, 28411, 50993, 71453, 272141, 1353449

Offset: 1

Alexander Adamchuk, Dec 04 2006

PrimePi[ a(n) ] = {3, 11, 16, 38, 97, 178,...}.

Henri & Renaud Lifchitz, PRP Records.

Cf. A125744 = Primes p such that (3^p + 3^((p + 1)/2) + 1)/7 is prime. Cf. A125738 = Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime. Cf. A125739 = Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime. Cf. A007670, A007671, A124165, A125742.

Do[p=Prime[n];f=(3^p-3^((p+1)/2)+1)/7;If[PrimeQ[f],Print[{n,p}]],{n,1,178}]

a(6)-a(11) from Lelio R Paula (lelio(AT)sknet.com.br), May 08 2008
a(12) from Serge Batalov, Mar 07 2014
a(13) from Serge Batalov, Mar 25 2014

A239842 Numbers n such that the Eisenstein integer ((1-ω)^n+1)/(2-ω) has prime norm, where ω = - 1/2 + sqrt(-3)/2.

Original entry on oeis.org

5, 11, 31, 37, 47, 53, 97, 163, 167, 509, 877, 1061, 2027, 2293, 3011, 6803, 8423, 13627, 20047, 28411, 50221, 50993, 71453, 152809, 272141, 505823, 1353449

Offset: 1

Serge Batalov, Mar 27 2014

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.

			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.

Henri Lifchitz & Renaud Lifchitz: PRP Records. Probable Primes Top 10000.

Cf. A125743 = Primes p such that (3^p - 3^((p+1)/2) + 1)/7 is prime.
Cf. A125744 = Primes p such that (3^p + 3^((p+1)/2) + 1)/7 is prime.
Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.
Cf. A124112 = Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime.

forprime(n=3,2300,if(ispseudoprime((3^n+kronecker(3,n)*3^((n+1)/2)+1)/7),print1(n ", "))); /* Serge Batalov, Mar 29 2014 */

cp's OEIS Frontend

A125743 Primes p such that (3^p - 3^((p + 1)/2) + 1)/7 is prime.

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