OPEN
This is open, and cannot be resolved with a finite computation.
Let $\epsilon>0$. Is there some $r\ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\epsilon$?
Romanoff
[Ro34] proved that the set of integers of the form $2^k+p$ (where $p$ is prime) has positive lower density.
See also
[205].
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