Divisor functions and the number of sum systems

Lettington, Matthew C. ORCID: https://orcid.org/0000-0001-9327-143X and Schmidt, Karl Michael ORCID: https://orcid.org/0000-0002-0227-3024 2020. Divisor functions and the number of sum systems. Integers 20 , A61.

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Abstract

Divisor functions have attracted the attention of number theorists from Dirichletto the present day. Here we consider associated divisor functionsc(r)j(n) which fornon-negative integersj, rcount the number of ways of representingnas an orderedproduct ofj+rfactors, of which the firstjmust be non-trivial, and their naturalextension to negative integersr.We give recurrence properties and explicit formulaefor these novel arithmetic functions. Specifically, the functionsc(−j)j(n) count, upto a sign, the number of ordered factorisations ofnintojsquare-free non-trivialfactors. These functions are related to a modified version of the M ̈obius functionand turn out to play a central role in counting the number of sum systems of givendimensions.Sum systems are finite collections of finite sets of non-negative integers, of pre-scribed cardinalities, such that their set sum generates consecutive integers with-out repetitions. Using a recently established bijection between sumsystems andjoint ordered factorisations of their component set cardinalities,we prove a for-mula expressing the number of different sum systems in terms of associated divisorfunctions.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Walter de Gruyter
ISSN:	1867-0652
Date of First Compliant Deposit:	31 July 2020
Date of Acceptance:	4 August 2020
Last Modified:	08 May 2023 23:27
URI:	https://orca.cardiff.ac.uk/id/eprint/133889

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