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+22k+m! and n=p+22k+2q where m,k∈N and p, q are primes. In the opposite direction, Erdős 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+22k+m! is larger than 34. (2) The proportion of positive integers not of the form p+22k+2q is at least 23.