Let f be a polynomial with integer coefficients. Let B(f) be the set of all values of f on positive integers. B(f) = {f(n)| n is a positive integer} = {f(1), f(2), ...}

A positive integer k is called "good" if it is a sum of distinct members of B(f). Otherwise we say k is "bad".

For instance, if f(x)=x^3, then it is known that there are finitely many bad numbers. In other words, all but finitely many natural numbers can be written as sum of distinct cubes.

If f(x)=2x^2 + 2, then there are infinitely many bad numbers since every odd number is bad.

My question : if gcd of coefficients of f is 1, are there only finitely many bad numbers?