How many polynomials?

For every positive integer \(n,\) consider all monic polynomials \(f(x)\) with integer coefficients, such that for some real number \(a\) \[x\left[ f(x+a)-f(x) \right]=nf(x)\] Find the largest possible number of such polynomials \(f(x)\) for a fixed \(n<1000.\)

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

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