Sylvester power and weighted sums on the Frobenius set in arithmetic progression

TL;DR

This paper derives formulas for the power and weighted sums of nonrepresentable integers related to Frobenius numbers, especially focusing on sequences forming arithmetic progressions, providing explicit expressions for these sums.

Contribution

It introduces new formulas for sums of nonrepresentable integers in the Frobenius problem for arithmetic progression sequences.

Findings

Explicit formulas for sums of nonrepresentable integers in arithmetic progressions

Applications to calculating Frobenius-related sums

Enhanced understanding of Frobenius numbers in special sequences

Abstract

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Frobenius number is the largest positive integer that is NOT representable in terms of $a_1,a_2,\dots,a_k$. When $k\ge 3$, there is no explicit formula in general, but some formulae may exist for special sequences $a_1,a_2,\dots,a_k$, including, those forming arithmetic progressions and their modifications. In this paper, we give formulae for the power and weighted sum of nonrepresentable positive integers. As applications, we show explicit expressions of these sums for $a_1,a_2,\dots,a_k$ forming arithmetic progressions.