I've been looking into Egyptian fractions now, but information on certain topics seems scarce. Can you answer any of these questions that intrigue me:
1) What is known about the Egyptian fraction representation (by the greedy algorithm) of irrational numbers? Are Egyptian fractions known to be interesting in any similar sense as the continued fractions? I have not found any references to these expansions; OEIS lists a few terms for some numbers though.
2) What would be the "average" growth of the n:th term in the sequence? The slowest possible would be 2, 3, 7 etc. but for a random real, what is the expected size of the n:th term?