Numbers whose reciprocals sum to $1$ What are all the numbers that can be written as $a_1+a_2+\dots+a_n$, where $a_1,\dots,a_n$ are positive integers such that $\frac{1}{a_1}+\dots+\frac{1}{a_n}=1$? For instance, such numbers include $4=2+2$, $11=2+3+6$, and $16=4+4+4+4$. 
Is there a characterization of such numbers? The first few are $1, 4, 9$ and $11$.

These are called Egyptian numbers. It is known that all numbers greater than $23$ are Egyptian, so you get a characterization by listing a finite list of non-Egyptian numbers.
