  Given a positive integer $n$ we let $A_k(n)$ be the number of positive
integers $a$ such that
$\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}$ for some
$m_1,m_2,\ldots,m_k\in {\mathbb N}$. We show that $x(\log x)^3\ll \sum_{p\le x}
A_3(p)\ll x(\log x)^5$ as $x\rightarrow\infty$.
