On a Special Case of the Frobenius Problem

摘要

For any set of positive and relatively prime integers A, the set of positive integers that are not representable as a nonnegative integral linear combination of elements of A is always a non-empty finite set. Thus we may define g(A), n(A), s(A) to denote the largest integer in, the number of integers in, and the sum of integers in this finite set, respectively. We determine g(A), n(A), s(A) when A = {a, b, c} with a | lcm(b, c). A particular case of this is when A = {kl, lm, mk}, with k, l, m pairwise coprime. We also solve a related problem when a | lcm(b, c), thereby providing another proof of the formula for g(A).