Prime Polynomial Pickle

For every prime \(p\) consider all polynomials \(f(x)\) with integer coefficients from \(1\) to \(p\) and degree at most \(p-1\), such that for all integers \(x\) the number \(f(2x)-f(x)\) is divisible by \(p\).

Find the sum of all primes \(p<1000\) such that the number of such polynomials is strictly greater than \(p\cdot2^{p-2}\).

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