Divisor Functions and the Number of Sum Systems

TL;DR

This paper introduces new divisor functions related to ordered factorizations, explores their properties, and demonstrates their application in counting sum systems with specific properties.

Contribution

It defines and analyzes novel divisor functions, providing explicit formulas and recurrence relations, and connects these functions to the enumeration of sum systems.

Findings

Derived explicit formulas for the new divisor functions.

Established recurrence relations for these functions.

Connected divisor functions to counting sum systems.

Abstract

Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions $c_j^{(-j)}(n)$ count, up to a sign, the number of ordered factorisations of $n$ into $j$ square-free non-trivial factors. These functions are related to a modified version of the M\"obius function and turn out to play a central role in counting the number of sum systems of given dimensions. \par Sum systems are finite collections of finite sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. Using a recently established bijection between sum systems and joint ordered factorisations of their component set cardinalities, we prove a formula expressing the number of different sum systems in terms of associated divisor functions.