dc.contributor.author Broughan, Kevin A. en_NZ dc.contributor.author Delbourgo, Daniel en_NZ dc.contributor.author Zhou, Qizhi en_NZ dc.date.accessioned 2017-09-12T23:17:49Z dc.date.available 2014-04-01 en_NZ dc.date.available 2017-09-12T23:17:49Z dc.date.issued 2014 en_NZ dc.identifier.citation Broughan, K. A., Delbourgo, D., & Zhou, Q. (2014). A conjecture of De Koninck regarding particular square values of the sum of divisors function. Journal of Number Theory, 137, 50–66. https://doi.org/10.1016/j.jnt.2013.10.011 en dc.identifier.issn 0022-314X en_NZ dc.identifier.uri https://hdl.handle.net/10289/11326 dc.description.abstract We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that a + 1 is a prime power. Lastly we prove that the number of solutions up to x > 1 is at most x¹/⁶⁺є, for any є > 0 and all x > xє. dc.format.mimetype application/pdf dc.language.iso en en_NZ dc.publisher Elsevier Inc en_NZ dc.rights This is an author’s accepted version of an article published in the journal: Journal of Number Theory. © 2013 Elsevier Inc. dc.subject Science & Technology en_NZ dc.subject Physical Sciences en_NZ dc.subject Mathematics en_NZ dc.subject Sum of divisors en_NZ dc.subject Squarefrce core en_NZ dc.subject De Koninck's conjecture en_NZ dc.subject Compactification en_NZ dc.subject Perfect numbers en_NZ dc.title A conjecture of De Koninck regarding particular square values of the sum of divisors function en_NZ dc.type Journal Article dc.identifier.doi 10.1016/j.jnt.2013.10.011 en_NZ dc.relation.isPartOf Journal of Number Theory en_NZ pubs.begin-page 50 pubs.elements-id 39151 pubs.end-page 66 pubs.organisational-group /Waikato pubs.organisational-group /Waikato/2018 PBRF pubs.organisational-group /Waikato/FCMS pubs.organisational-group /Waikato/FCMS/2018 PBRF - FCMS pubs.publication-status Published en_NZ pubs.volume 137 en_NZ dc.identifier.eissn 1096-1658 en_NZ
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