A number 
is called an Egyptian number if it is the sum of the denominators
in some unit fraction representation of a positive
whole number not consisting entirely of 1s. For example,
 |
so 
is an Egyptian number. The numbers that are not Egyptian are 2, 3, 5, 6, 7,
8, 12, 13, 14, 15, 19, 21, and 23 (OEIS A028229;
Konhauser et al. 1996, p. 147).
If 
is the sum of denominators of a unit fraction representation composed of distinct
denominators which are not all 1s, then it is called a strictly Egyptian number.
For example, by virtue of
 |
is Egyptian, but it is not
strictly Egyptian. Graham (1963) proved that every number 
is strictly Egyptian. Numbers that are strictly Egyptian
are 11, 24, 30, 31, 32, 37, 38, 43, ... (OEIS A052428),
and those which are not are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ... (OEIS A051882).
