A176008 Numbers n such that 3^(2n-1)+3^n+1 is prime.

1, 2, 3, 4, 5, 9, 10, 40, 82, 159, 177, 525, 880, 2577, 3771, 11872

Offset

1,2

Comments

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

h=2n-1 must be a power of 3 or a prime congruent to 5 or 7 (mod 12). For all other h, there are algebraic factorizations: for prime p>3, M(3,pq) are divisible by L(3,p) or M(3,p).

No other terms up to 41000 exist.

a(17) > 10^5. - Robert Price, Mar 20 2014

Links

Table of n, a(n) for n=1..16.

Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.

Example

5 is a term because 3^(5*2-1) + 3^5 + 1 = 19927 is prime.

Mathematica

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

Prog

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

Crossrefs

Cf. A176007 for Aurifeuillean co-factor L(3, 6n-3).

Sequence in context: A075177 A062096 A360687 * A143827 A100797 A139441

Adjacent sequences: A176005 A176006 A176007 * A176009 A176010 A176011

Keyword

hard,more,nonn

Author

Serge Batalov, Apr 09 2010

Status

approved