A note on the number of Egyptian fractions

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k\ge 2$, the number of integers $1\le a\le n$ such that the equation $a\mathrm{/}n=1\mathrm{/}{m}_1+\dots +1\mathrm{/}{m}_k$ has a solution in positive integers $m_1,\dots ,m_k$ is bounded above by $n^{1-1\mathrm{/}{2}^{k-2}+o(1)}$ as $n$ goes to infinity. The proof is elementary.