Sylvester Sums on the Frobenius Set in Arithmetic Progression

Abstract

Let $a_1,a_2,\ldots ,a_k$ be positive integers with $\gcd (a_1,a_2,\ldots ,a_k)=1$. The concept of the weighted sum $\sum _{n\in \mathrm{NR}}\lambda ^{n}n$ is introduced in [1, 2], where $\mathrm{NR}=\mathrm{NR}(a_1,a_2,\ldots ,a_k)$ denotes the set of positive integers nonrepresentable in terms of $a_1,a_2,\ldots ,a_k$. When $\lambda =1$, such a sum is often called Sylvester sum. The main purpose of this paper is to give explicit expressions of the Sylvester sum ($\lambda =1$) and the weighed sum ($\lambda \ne 1$), where $a_1,a_2,\ldots ,a_k$ forms arithmetic progressions. As applications, various other cases are also considered, including weighted sums, almost arithmetic sequences, arithmetic sequences with an additional term, and geometric-like sequences. Several examples illustrate and confirm our results.