Erdős problem 470

Melfi [Me15](https://mathscinet.ams.org/mathscinet/relay-station?mr=3276337) 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.