Lettington, Matthew C. and Schmidt, Karl Michael
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 |
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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: | 07 Jan 2021 11:30 |
URI: | http://orca.cf.ac.uk/id/eprint/133889 |
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