Weird numbers were investigated by Benkoski and Erd\H{o}s \cite{BeEr74}, who proved that the set of weird numbers has positive density. The smallest weird number is $70$.

Melfi \cite{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 \cite{Fa22} has shown there are no odd weird numbers below $10^{21}$, and Liddy and Riedl \cite{LiRi18} have shown that an odd weird number must have at least 6 prime divisors.

If there are no odd weird numbers then every weird number has abundancy index $<4$ (see [825]).

This is problem B2 in Guy's collection \cite{Gu04} (the \$10 is reported by Guy, offered by Erd\H{o}s for a solution to the question of whether any odd weird numbers exist).


References


[BeEr74] Benkoski, S. J. and Erd\H{o}s, P., On weird and pseudoperfect numbers. Math. Comp. (1974), 617-623.

[Fa22] Searching on the boundary of abundance for odd weird numbers, W. Fang. arXiv:2207.12906 (2022).

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

[LiRi18] J. Liddy and J. Riedl, An algorithm to determine all odd primitive abundant numbers with $d$ prime divisors. Honors Research Projects. 728 (2018).

[Me15] Melfi, Giuseppe, On the conditional infiniteness of primitive weird numbers. J. Number Theory (2015), 508-514.