This is a naive question about number theory.
Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a remarkable structure that on some lines there seems to be a higher density of prime values. This is addressed in Hardy-Littlewood's conjecture F.
My question is: Can we find sequences of consecutive prime number
$an^2+bn+c, a(n+1)^2+b(n+1)+c,\ldots, a(n+k)^2+b(n+k)+c$
for arbitrary large $k$ if we allow $a,b,c$ to vary? Or, maybe more restrictively, varying $c$ for fixed $a$ and $b$ (for example $a=4$, $b=2$ as in the Ural spiral)? Does this follow from any other conjectures?