Erdős Problem #9 — Density of odd integers not of the form $2^{k_1}+2^{k_2}+p$

Crocker [16] proved that there are infinitely many odd integers not of the form $2^{k_1} + 2^{k_2} + p$, but his proof only gives that the number of integers $\leq x$ not of the form $2^{k_1} + 2^{k_2} + p$ is $> c \log\log x$. This surely can be improved but I am not at all sure if the upper density of the integers not of the form $2^{k_1} + 2^{k_2} + p$ is positive.