Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that $$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$ On the other hand, in 1950 van der Corput showed that $$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1\not=p+2^k\ \text{for any prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$ These two results have stimulated many further researches.
For any prime $p$, let $p'$ denote the first prime after $p$, which is smaller than $2p$ by the proved Bertrand Postulate. Motivated by the results of Romanoff and van der Corput, here I ask the following question.
QUESTION. Let $$d_1:=\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n=p+p'+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x$$ and $$d_2:=\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n\not=p+p'+2^k\ \text{for any prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x.$$ Is it true that $d_1$ and $d_2$ are both positive?
I guess that the two constants $d_1$ and $d_2$ are indeed positive. Can one prove $d_1>0$ or $d_2>0$?
Your comments are welcome!
