2019/10/06 by Matthew C. Lettington, Lettington, Matthew C., Karl Michael Schmidt +1
Mathematics · Computer Science · #Analytic Number Theory Research #Advanced Mathematical Identities #Graph Labeling and Dimension Problems
paper · doi:10.48550/arxiv.1910.02455
Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions cj(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 cj(-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 Möbius 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.