On the conditional infiniteness of primitive weird numbers

Abstract

TextA weird number is a number n for which σ(n)>2n and such that n is not a sum of distinct proper divisors of n. In this paper we prove that n=2kpq is weird for a quite large set of primes p and q. In particular this gives an algorithm to generate very large primitive weird numbers, i.e., weird numbers that are not multiple of other weird numbers. Assuming classical conjectures on the gaps between consecutive primes, this also would prove that there are infinitely many primitive weird numbers, a question raised by Benkoski and Erdős in 1974. VideoFor a video summary of this paper, please visit http://youtu.be/OS93l3a_Mjo.