An Extreme Family of Generalized Frobenius Numbers

Abstract We study a generalization of the Frobenius problem: given k positive relatively prime integers, what is the largest integer g 0 that cannot be represented as a nonnegative integral linear combination of the given integers? More generally, what is the largest integer gs that has exactly s such representations? We construct a family of integers, based on a recent paper by Tripathi, whose generalized Frobenius numbers g 0, g 1, g 2, . . . exhibit unnatural jumps; namely, g 0, g 1, gk , , , . . . form an arithmetic progression, and any integer larger than has at least representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.