Inventory001:  Egyptian numbers (proposed by Paul Yiu, 11/1/01).

A positive integer N is Egyptian if it can be partitioned in the form N = a_1 + a_2 +  ... + a_k in which a_1, a_2, ..., a_k are positive integers, not necessarily distinct, such that 1/a_1 + 1/a_2 + ... +  1/a_k = 1. For example, the number 10 is Egyptian: 10 = 2+4+4, and 1/2 + 1/4 + 1/4 = 1. Find all Egyptian integers.

Discussions. [FvL, 11/8/01]: I think this problem is already known and settled. See for instance the Mathworld page:

http://mathworld.wolfram.com/EgyptianNumber.html

[PY, 11/8/01]: Thank you for this, and it is good to see Eric Weisstein's Mathworld again! I checked up

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028229

and found that 2,3,5,6,7,8,12,13,14,15,19,21,23 are the only numbers that are not Egyptian. So the answer to my question is that the Egyptian integers are precisely 1,4,9,10,11,16,17,18,20,22, and all integers >= 24.
What led me to this question was Problem 3 of the 7th British Mathematical Olympiad (1978). Here is a paraphrase: Given that the numbers 33 to 73 are all Egyptian, to prove that every number >= 33 is Egyptian. And I wondered what the set of all Egyptian numbers are.

Bibliography.