An Extreme Family of Generalized Frobenius Numbers

We study a generalization of the Frobenius problem: given k positive relatively prime integers, what is the largest integer g0 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 g0, g1, g2, . . . exhibit unnatural jumps; namely, g0, g1, gk, g(k+1 k−1) , g ( k+2 k−1) , . . . form an arithmetic progression, and any integer larger than g ( k+j k−1) has at least �k+j+1 k−1 � representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.