**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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