An improved lower bound for odd integers not of the form $p+2^a+2^b$

Let $x$ be sufficiently large and $$N(x)=\mid \{n\le x:n\text{ is odd and }n\mathrm{\ne }p+2^a+2^b\text{ with }p\text{ a prime and }a,b\in \mathbb{N}\}\mid \mathrm{.}$$ Motivated by Crocker’s result $$N(x)\gg \log \log x,$$ Erd\H os repeatedly asked whether there is an absolute constant $c_0$ such that $N(x)>c_{0}x$. Pan \cite{Pan} proved in 2011 that $$N(x)\gg x\exp \text{\hspace{0.17em}⁣}(-C_0\frac{\log \log \log \log x}{\log \log \log x}\log x),$$ where $C_0>0$ is an absolute constant. We improve on Pan’s result by showing that, given any $\eta >0$, for all sufficiently large $x$, $$N(x){\gg }_{\eta }x\exp (-(4+\eta )\frac{\log \log \log x}{\log \log x}\log x)\mathrm{.}$$