Fractions with a nice profile - Lycée Arago (Perpignan) Colegiul National B.P. Hasdeu (Buzau, Roumanie)

A positive fraction is said to have a nice profile if it can be written as a sum of other fractions, all different, each one of the form 1/p with p a positive integer.
For instance, the fraction 5/6=1/2+1/3 has a nice profile.
There are a lot of questions that can be asked about that topic. For example, which fractions have a nice profile? Is it possible to find general and automatic methods to write any given fraction in this way? If a fraction has a nice profile, how many ways are there to break it down? What if we are interested in sums of two fractions only? etc…