Strongly Pseudoperfect Numbers

A positive integer $n$ is said to be strongly pseudoperfect if there is a subset $S$ of the positive divisors of $n$ such that $d\in S$ if and only if $n\mathrm{/}d\in S$, and ${\sum }_{d\in S}d=2n$. We construct new infinite families of strongly pseudoperfect numbers, and also prove new restrictions on when a positive integer $n$ can be strongly pseudoperfect.