Romanov type problems

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form (n=p+2^{2^k}+m!) and (n=p+2^{2^k}+2^q) where (m,k in mathbb {N}) and p, q are primes. In the opposite direction, Erd?s constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove a much better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form (p+2^{2^k}+m!) is larger than (frac{3}{4}). (2) The proportion of positive integers not of the form (p+2^{2^k}+2^q) is at least (frac{2}{3}).