For every positive integer n consider all polynomials f(x) with integer coefficients, such that for some real number a *x(f(x+a)−f(x))=n*f(x) Find the largest possible number of such polynomials f(x) for a fixed n

Question

For every positive integer n consider all polynomials f(x) with integer coefficients, such that for some real number a *x(f(x+a)−f(x))=n*f(x) Find the largest possible number of such polynomials f(x) for a fixed n<1000?