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If there are only n primes dividing A(k) as k varies, there are only 2^n possibilities for squarefree values of A(k), and then you can use the fact that the polynomial is unbounded.
I was not familiar with this term up until now but yes, that's what I meant. Perhaps a better formulation would be: is there a curve in the plane such that all of its points have at least one rational coordinate
