A necessary condition for a Romanoff type theorem

Let $\{a_k{\}}_{k\ge 1}$ be an infinite strictly increasing sequence of positive integers with $a_k\mid a_{k+1}$ for every integer $k\ge 1$. Dai and Pan proved that the set of integers of the form $p^2+q^2+a_k+a_l$, where $p,q$ are primes and $k,l\ge 1$, has positive lower asymptotic density if ${\mathrm{lim\hspace{0.17em}sup}}_{k\to \mathrm{\infty }}\frac{\log a_k}{k}<\mathrm{\infty }$ and ${\sup }_{k\ge 1}{\sum }_{d\mid a_k}\frac{1}{d}<\mathrm{\infty }$. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if ${\sup }_{k\ge 1}{\sum }_{d\mid a_k}\frac{1}{d}=\mathrm{\infty }$, then this set has asymptotic density zero.