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, 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 \(p+2^{2^k}+m!\) 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 \(\frac{2}{3}\).

中文翻译：

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罗曼诺夫类型问题。

罗曼诺夫证明，可以表示为质数和2的幂的和的正整数比例为正。我们为形式为\（n = p + 2 ^ {2 ^ k} + m！\）和\（n = p + 2 ^ {2 ^ k} + 2 ^ q \）的整数建立相似的结果，其中\（ m，k \ in \ mathbb {N} \）和p，  q是素数。在相反的方向上，Erdős构造了奇数整数的完整算术级数，这些整数都不是素数和2的幂的和。虽然我们在这两种情况下都展示了不包含两种形式的任何整数的完整算术级数，但是对于非这些形式的整数比例，我们证明了更好的结果：（1）非整数形式的正整数比例形成\（p + 2 ^ {2 ^ k} + m！\）大于\（\ frac {3} {4} \）。（2）非整数形式\（p + 2 ^ {2 ^ k} + 2 ^ q \）的正整数比例至少为\（\ frac {2} {3} \）。