The Erdös-Strauss conjecture states that $$\frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ has a solution in positive integers for every integer $n>1$.
What is known about this extension : The equation $$\frac{k}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ with $k=5,6$ or $7$ and any integer $n\ge 3$ has a solution in positive integers.
In the case $k=5$, I worked out that a counterexample is only possible, if $n$ is of the form $20l+1$ and in this case, the conjecure holds upto $n\le 2\cdot 10^6$. I neither found counterexamples for $k=6$ and $k=7$. $k=8$ is no more possible , since $\frac{8}{11}$ has not a representation of the desired form. What is known about this extension ? Any ideas or references ?