A176007 - OEIS

%I #12 Feb 16 2025 08:33:12

%S 2,5,6,97,120,330,355,552,1015,4525,5227

%N Numbers n such that 3^(2n-1)-3^n+1 is prime.

%C 3^(2n-1)-3^n+1 is an Aurifeuillean factor of 3^(6n-3)+1, sometimes written as L(3,6n-3).

%C h=2n-1 must be a power of 3 or a prime congruent to 1 or 11 (mod 12). For all other h, there are algebraic factorizations: for prime p>3, L(3,pq) are divisible by L(3,p) or M(3,p).

%C No other terms a(n)<55800 exist.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AurifeuilleanFactorization.html">Aurifeuillean Factorization</a>.

%e For n = 2 the a(2) = 5, because 3^9-3^5+1 = 19441 is prime.

%t Do[ If[ PrimeQ[3^(2*n-1)-3^n+1], Print[n]], {n, 0, 10000}]

%o (PARI) for(k=1, 1000, if(isprime(3^(2*k-1)-3^k+1), print(k)))

%Y Cf. A176008 for Aurifeuillean co-factor M(3, 6n-3).

%K hard,more,nonn

%O 1,1

%A _Serge Batalov_, Apr 09 2010