Weighted Sylvester sums on the Frobenius set in more variables

Abstract

Let $a_1,a_2,\dots ,a_k$ be positive integers with $\gcd (a_1,a_2,\dots ,a_k)=1$. Let $\mathrm{NR}=\mathrm{NR}(a_1,a_2,\dots ,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots ,a_k$. The largest nonrepresentable integer $\max \mathrm{NR}$, the number of nonrepresentable positive integers ${\sum }_{n\in \mathrm{NR}}1$ and the sum of nonrepresentable positive integers ${\sum }_{n\in \mathrm{NR}}n$ 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 ${\sum }_{n\in \mathrm{NR}}{\lambda }^n{n}^{\mu }$, where $\mu$ is a nonnegative integer and $\lambda$ is a complex number. We also examine power sums of nonrepresentable numbers and some formulae for three variables. Several examples illustrate and support our results.

Cited by in corpus (3)

• The Frobenius number for sequences of triangular numbers associated with number of solutions
• A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)
• Sylvester sums on the Frobenius set in arithmetic progression with initial gaps