OPEN
Are there infinitely many solutions to
\[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\]
where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?
For example,
\[\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}\]
and
\[\frac{1}{2}+\frac{1}{3}+\frac{1}{7}=1-\frac{1}{42}.\]
It is clear that we must have $m=p_1\cdots p_k$, and hence in particular (up to ordering) there is at most one solution for each $m$. The integers $m$ for which there is such a solution are known as
primary pseudoperfect numbers, and there are $8$ known, listed in
A054377 at the OEIS.
