WEIGHTED SYLVESTER SUMS ON THE FROBENIUS SET IN MORE VARIABLES

Abstract

Let a1, a2, . . . , ak be positive integers with gcd(a1, a2, . . . , ak) = 1. Let NR = NR(a1, a2, . . . , ak) denote the set of positive integers non-representable in terms of a1, a2, . . . , ak. The largest non-representable integer max NR, the number of non-representable positive integers Σn∈NR 1, and the sum of non-representable positive integers Σn∈NR n have been widely studied for a long time as related to the famous Frobenius problem. In this paper, by using Eulerian numbers, we give formulas for the weighted sum Σn∈NR λnnμ, where μ is a non-negative integer and λ is a complex number. We also examine power sums of non-representable numbers and some formulas for three variables. Several examples illustrate and support our results.