Primitive weird numbers having more than three distinct prime factors. (English) Zbl 1410.11005

Amato, Gianluca; Hasler, Maximilian; Melfi, Giuseppe; Parton, Maurizio

Riv. Mat. Univ. Parma (N.S.) 7, No. 1, 153-163 (2016).

Summary: 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 \(mp_1\cdots p_k\) for a suitable deficient positive integer \(m\) and primes \(p_1, \ldots, p_k\) 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.

MSC:

11A25

Arithmetic functions; related numbers; inversion formulas

Keywords:

abundant numbers; semiperfect numbers; almost perfect numbers; sum-of-divisor function; Erdős problems