# 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 $11$.

• You missed $10=2+4+4$. This is sequence A125726 in OEIS. – TonyK Jul 7 '16 at 18:33
• And $9=3+3+3$.  – kccu Jul 7 '16 at 18:35
• @kccu: No, $9$ is there :-) – joriki Jul 7 '16 at 18:35
• This is OEIS sequence A125726. The entry includes two references but not much other information. – joriki Jul 7 '16 at 18:36
• @joriki Whoops I was looking at the "for instance." – kccu Jul 7 '16 at 18:36

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.