Sylvester sums on the Frobenius set in arithmetic progression

TL;DR

This paper derives explicit formulas for Sylvester sums and weighted sums over nonrepresentable integers in arithmetic progressions, extending to various sequence types and providing concrete examples.

Contribution

It provides new explicit expressions for Sylvester and weighted sums for arithmetic progression sequences, including special cases and extensions.

Findings

Explicit formulas for Sylvester sums in arithmetic progressions.

Extensions to weighted sums and other sequence types.

Validation through multiple illustrative examples.

Abstract

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. The concept of the weighted sum $\sum_{n\in{\rm NR}}\lambda^{n}$ is introduced in \cite{KZ0,KZ}, where ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denotes the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. 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 $a_1,a_2,\dots,a_k$ 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.