Let's define an infinite sequence of positive integers as :
$a_n=k^2+(2n-1)k+2n-1 $ , where $ k,n \in \mathbf{Z^{+}}$
Suppose that one can prove that this sequence contains infinitely many prime numbers for any particular $k$. The first term of the sequence for any $k$ is of the form :
$a_1=k^2+k+1$
My question is : what is sufficient condition to prove that this polynomial produces primes for infinitely many values of $k$ ?
Note that there is strong experimental evidence that $k^2+k+1$ is prime for infinitely many values of $k$ and that this polynomial satisfies conditions of Bunyakovsky's conjecture.