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Equisum Partitions of Sets of Positive Integers

School of Mathematical and Physical Sciences, University of Newcastle, Newcastle 2308, Australia
Algorithms 2019, 12(8), 164; https://doi.org/10.3390/a12080164
Received: 12 July 2019 / Revised: 5 August 2019 / Accepted: 7 August 2019 / Published: 11 August 2019
PDF [902 KB, uploaded 12 August 2019]

Abstract

Let V be a finite set of positive integers with sum equal to a multiple of the integer b. When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V, where n is a given positive integer: (1) an initial interval a∈Z+:a≤n; (2) an initial interval of primes p∈P:p≤n, where P is the set of primes; (3) a divisor set d∈Z+:d|n; (4) an aliquot set d∈Z+:d|n, d<n. Open general questions and conjectures are included for each of these classes.
Keywords: integer set partitions; equisum sets; twin primes; perfect numbers; aliquot divisors integer set partitions; equisum sets; twin primes; perfect numbers; aliquot divisors
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Eggleton, R.B. Equisum Partitions of Sets of Positive Integers. Algorithms 2019, 12, 164.

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