Suppose one is tasked to find the longest sequence of unique unit fractions (1/x) that sums to 1, given that 2 <= x <= 100.
Currently, I am going with a decomposition approach. For example, by noticing that 1/3 = 1/4 + 1/12, we can increase the number of terms without a change in value. This procedure can be repeated for all other fractions. However, it is inevitable that a collision will occur, meaning that a unit fraction in the decomposition is already in use. And this is where I am stuck.
After stumbling across this paper, I learnt that it is possible to look for different decompositions of the same term to avoid collision. For example, 1/30 = 1/50 + 1/75 = 1/55 + 1/66. Furthermore, I have created the following identity for when the denominator is odd, $$\frac{2}{2a+1} = \frac{1}{a+1} + \frac{1}{(a+1)(2a+1)}$$ and if it's even, divide the denominator by 2 and resolve further collisions using the same procedure. And if there are still duplicates, I have no choice but to give up that decomposition.
Using this algorithm, I am able to obtain a sequence of length 38, just 4 less than the maximum 42. However, I believe this gap will become larger and larger as the bound increases, so my question is that what are some ideas of improvement for the current algorithm? Or even better, are there any alternative approaches to this problem?