论文标题
Sylvester在算术进程中设置的Frobenius汇总
Sylvester sums on the Frobenius set in arithmetic progression
论文作者
论文摘要
令$ a_1，a_2，\ dots，a_k $为正整数，$ \ gcd（a_1，a_2，\ dots，a_k）= 1 $。 The concept of the weighted sum $\sum_{n\in{\rm NR}}λ^{n}$ 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 $ a_1，a_2，\ dots，a_k $。当$λ= 1 $时，这样的总和通常称为sylvester和。本文的主要目的是给出Sylvester和（$λ= 1 $）和称重和（$λ\ ne 1 $）的明确表达式，其中$ a_1，a_2，\ dots，a_k $，a_k $ formation arithmetic进度。作为应用，还考虑了其他各种情况，包括加权总和，几乎算术序列，具有附加项的算术序列以及几何样序列。几个例子说明并确认我们的结果。
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 $\sum_{n\in{\rm NR}}λ^{n}$ 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 $λ=1$, such a sum is often called Sylvester sum. The main purpose of this paper is to give explicit expressions of the Sylvester sum ($λ=1$) and the weighed sum ($λ\ne 1$), 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.
