Gianluca Amato,^[1] Maximilian Hasler,^[2] Giuseppe Melfi^[3] and Maurizio Parton^[4]

Primitive weird numbers having more than three distinct prime factors

Pages: 153-163
Received: 21 December 2015
Accepted: 13 April 2016
Mathematics Subject Classification (2010): 11A25, 11B83.
Keywords: Abundant numbers, semiperfect numbers, almost perfect numbers, sum-of-divisor function, Erdős problems.
Authors address:
[1],[4] : Università "G. D'Annunzio" di Chieti-Pescara, Dipartimento di Economia, Viale della Pineta 4, I-65129, Pescara, Italy
[2] : Université des Antilles, Département Scientifique Interfacultaire, B.P. 7209 Campus de Schoelcher, F-97275 Schoelcher cedex, Martinique, French West Indies
[3] : University of Applied Sciences of Western Switzerland, HEG-Arc, Espace de l'Europe 21, CH-2000 Neuchâtel, Switzerland

Abstract: In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a given number of distinct prime factors are presented. These algorithms yield primitive weird numbers of the form \(mp_1\dots p_k\) for a suitable deficient positive integer \(m\) and primes \(p_1,\dots,p_k\) and generalize a recent technique developed for generating primitive weird numbers of the form \(2^np_1p_2\). The same techniques can be used to search for odd weird numbers, whose existence is still an open question.

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