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

Melfi Me15 has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1} - p_n < \frac{1}{10} \sqrt{p_n}$ for all large $n$, which in turn would follow from well-known conjectures concerning prime gaps.