Primitive abundant and weird numbers with many prime factors

Amato, Gianluca (Università di Chieti-Pescara, Italy) ; Hasler, Maximilian F. (Université des Antilles, France) ; Melfi, Giuseppe (Haute école de gestion Arc, HES-SO // Haute Ecole Spécialisée de Suisse Occidentale) ; Parton, Maurizio (University of teacher education BEJUNE, Biel/Bienne, Suisse)

We give an algorithm to enumerate all primitive abundant numbers (PAN) with a fixed Ω, the number of prime factors counted with their multiplicity. We explicitly find all PAN up to Ω=6, count all PAN and square-free PAN up to Ω =7 and count all odd PAN and odd square-free PAN up to Ω =8. We find primitive weird numbers (PWN) with up to 16 prime factors, the largest of which is a number with 14712 digits. We find hundreds of PWN with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWN with at least one odd prime factor with multiplicity greater than one and Ω =7 and prove that there are none with Ω <7. Regarding PWN with a cubic (or higher power) odd prime factor, we prove that there are none with Ω ≤7. We find several PWN with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples. We finally observe that these results are in favor of the existence of PWN with arbitrarily many prime factors.


Keywords:
Article Type:
scientifique
Faculty:
Economie et Services
School:
HEG Arc
Institute:
IMVT - Institut du Management des villes et du territoire
Subject(s):
Economie/gestion
Date:
2019-08
Pagination:
23 p.
Published in:
Journal of number theory
Numeration (vol. no.):
2019, vol. 201, pp. 436-459
DOI:
ISSN:
0022314X
Appears in Collection:

Note: The file is under embargo until: 2020-08-01


 Record created 2019-09-25, last modified 2019-11-28

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