Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density.

Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density.

HasPosDensity is the right reading here, rather than the positive *lower* density that Erdős' "positive density" usually abbreviates. Their Theorem 5 is stated as "the density of weird numbers is positive", and they first establish that the density exists at all: "It is clear that the weird numbers have a density since both the abundant numbers and the pseudoperfect numbers have a density. (A weird number is abundant and not pseudoperfect.)" The work in the paper goes into showing that density is not 0.