Primitive weird numbers having more than three distinct prime factors

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 mp1…pk for a suitable deficient positive integer m and primes p1,…,pk and generalize a recent technique developed for generating primitive weird numbers of the form 2n*p1*p2. The same techniques can be used to search for odd weird numbers, whose existence is still an open question.