Ternary Egyptian Fractions

$$\frac{a}{n}=\frac{1}{m_1}+\cdots+\frac{1}{m_k}$$ with positive integers $a,n,m_1,\ldots,m_k$. What is of interest is, given $n$, to count $A^*_k(n)=\{a:(a,n)=1, a/n=1/m_1+\cdots+1/m_k~{\text{for some}}~m_1,\ldots,m_k\}$ as well as $A_k(n)$ which is the same as $A_k^*(n)$ except that without the corporality condition on $a$ and $n$. In my talk, I will survey what is known about this problem for $k=2$ and I will show that $$ x(\log x)^3\ll \sum_{p\le x} A_3^*(p)\ll x(\log x)^5.$$
We believe the lower bound is closer to the truth. The proof uses sieve methods and some results of Elsholtz and Tao on average sum of divisors functions over arithmetic progressions. This is joint work with F. Pappalardi.