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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:

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

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.


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