Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?
Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?
Melfi [Me15] has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1}-p_n<\frac{1}{10}p_n^{1/2}$ for all large $n$, which in turn would follow from well-known conjectures concerning prime gaps.
The sequence of weird numbers is A006037 in the OEIS. Fang [Fa22] has shown there are no odd weird numbers below $10^{21}$, and Liddy and Riedl [LiRi18] have shown that an odd weird number must have at least 6 prime divisors.