OPEN
Let $A$ be the set of all integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?
Crocker
[Cr71] has proved there are are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$. Pan
[Pa11] improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. Erdős believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.
The sequence of such numbers is A006286 in the OEIS.
See also [10], [11], and [16].