An Egyptian number is an integer 
that is the sum of the denominators
in a finite representation of 1 as a sum of positive unit
fractions. 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).
An Egyptian number is called strictly Egyptian when the unit fraction representation of 1 has distinct denominators. 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).
The OEIS sequences also include the trivial case 
as Egyptian and strictly Egyptian, using the single unit
fraction 
.
