TITLE: Diophantine equation: Egyptian fraction representations of 1
QUESTION [14 upvotes]: According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation
$$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there are 159330691 solutions.
My question: What are they? Is there a way of counting them without knowing them?
The bound for $x_n$ for $n$ summands is double exponential and I could only compute the solutions up to $n=6$ with Maple.

REPLY [2 votes]: A related interesting problem is  the number of partitions of 1 into n distinct positive Egyptian fractions, the first few terms of which are given in A006585.
These values were counted laboriously by brute force in the early 1990's, and although they have been reconfirmed (by Jud McCranie) the sequence has only been extended one further term (by John Dethridge) in 2004.
Additional terms, cleverer algorithms and/or a generating function, would be most welcome!