Extending an Erdős result on a Romanov type problem (Q2140597)

\textit{N. P. Romanov} [Math. Ann. 109, 668--678 (1934; Zbl 0009.00801)] has shown that the set of positive integers of the form \(p+2^k\) (with prime \(p\)) has positive density. Later \textit{P. Erdős} [Summa Brasil. Math. 2, 113--123 (1950; Zbl 0041.36808)] proved that for the number \(f(n)\) of solutions of \(n=p+2^k\) one has \[ \limsup_{n\to\infty} \frac{f(n)}{\log\log n} > 0, \] and conjectured that if \(a_1< a_2 < \cdots\) is a sequence of positive integers with \(\sum_{a_i\le x}1>\log x\), then the number \(f(n)\) of representations \(n=p+a_i\) with prime \(p\) is unbounded. The author shows that this holds for sequences satisfying \(a_i\mid a_{i+1}\) for \(i=1,2,\dots\) by showing that this condition implies \[ \limsup_{n\to\infty} \frac{f(n)}{\sqrt{\log\log n}} > 0. \]