hihi - it is my vague recollection that for positive integers a, b with gcd(a, b) = 1, then for all n >= a * b, there is a representation n = x * a + y * b with x, y >= 0 (this is trivial for a=1 or b=1, so we can assume a, b >= 2) that would mean density 1 (but this result is even stronger than the original density assertion: it says that all but finitely many positive integers have such a representation; i think i saw a reasonable proof of that once long ago) it is my impression that essentially the same kind of thing could/should/would be true for more than two elements: if we take c = gcd{N_j}, then for any large enough n divisible by c, there is a representation n = sum(x_j * N_j) with all x_j >= 0, so the density is 1/c (only multiples of c can occur, of course, and the stronger assertion is that all but finitely many of them have such a representation) more later, chris On 12/18/20 23:25, Dan Asimov wrote:
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.)
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