Can an odd perfect number be divisible by either $2049$ or $2051$?
Note that $2049 = 3 \cdot {683}$, and that $2051 = 7 \cdot {293}$.
Added July 15 2016
It is known that an odd perfect number cannot be divisible by 105, 825, and 5313. It is also tempting to conjecture that an odd perfect number cannot be divisible by 165, but this has not yet been completely proved.