[math-fun] Question about nonnegative integral linear combinations of a set of positive integers

Let N_1, ..., N_r be a set of positive integers ≥ 2 whose prime factorizations are known. Let X = {∑ c_j N_j} be the set of all linear combinations of the N_j with nonnegative integer coefficients c_j. Is it easy to determine what the (asymptotic) density of X is in Z+ ??? (An N_j may have repeated prime factors, and several N_j's may have common factors.) —Dan