A positive natural number is said to be accessible if it can be written as sum of poisitive integers whose sum of reciprocals adds up to $1$. It is easy to show that $3, 5 $ are not accessible while $4$ is. Is there a way to characterize all accessible positive integers, for example, an inductive process? It is clear that for any number $n$, the number of summands is at most $\lfloor n/2 \rfloor$ as all of these summands are greater than or equal to $2$. Is it possible to answer the problem in two cases: distinct and not necessarily distinct positive integers?
Please give some hint or reference.
Thank you.
These are called Egyptian numbers in OEIS A125726. The first $73$ are given $$1, 4, 9, 10, 11, 16, 17, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,\\ 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,\\ 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,\\ 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86$$ There are a couple references. All the squares are there, as $n^2$ can be represented as $n\ n$'s. If $n$ is in the list, so is $2n+2$ as you can double all the numbers in an expansion and add a $2$. If $n$ is in the list with a $2$ in the expansion, so is $n+6$ because you can replace the $2$ with $4,4$. The last one my eye sees missing is $23$. Mathworld states that every number $\ge 78$ can be expressed this way with distinct numbers in the expansion and calls those numbers strictly Egyptian. There is a link to the paper in the OEIS entry.