Sylvester sums on the Frobenius set in arithmetic progression with initial gaps

TL;DR

This paper derives explicit formulas for Sylvester sums, Frobenius numbers, and counts of nonrepresentable integers for sequences forming arithmetic progressions with initial gaps, expanding understanding of Frobenius problems in special cases.

Contribution

It provides new explicit formulas for Frobenius-related quantities for arithmetic progression sequences with initial gaps, a case previously lacking such formulas.

Findings

Explicit formulas for Sylvester sums and Frobenius numbers.

Formulas applicable to sequences with initial gaps in arithmetic progressions.

Enhanced understanding of Frobenius problem in special sequence cases.

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 explicit formulae for the sum of nonrepresentable positive integers (Sylvester sum) as well as Frobenius numbers and the number of nonrepresentable positive integers (Sylverster number) for $a_1,a_2,\dots,a_k$ forming arithmetic progressions with initial gaps.