My joint paper with Keneth Adrian Precillas Dagal was published in NNTDM 27/2 last June 12, 2021. Here is the abstract:
In this note, we show that if $N=q^k n^2$ is an odd perfect number with special prime $q$, and $N$ is not divisible by $3$, then the inequality $q < n$ holds. We then give another unconditional proof for the inequality $q < n$ which is independent of the results of Brown and Starni.
