$\frac{1}{n}$ as a difference of Egyptian fractions with all denominators $<n$

Question

Is there a good characterization of the set $S$ of positive integers $n$ such that $\frac{1}{n}$ can be represented as a difference of Egyptian fractions with all denominators $< n$? For example, $44 \in S$ because $$ \dfrac{1}{44} = \left( \frac{1}{33} + \frac{1}{12}\right) - \frac{1}{11} $$

If I'm not mistaken, the first few members of $S$ are $$ 6, 12, 15, 18, 20, 21, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45 $$ This does not appear to be in the OEIS yet; I intend to submit it soon. [ EDIT: It is now in OEIS as A278638.]

Here are some things I know so far:

1. If $n \in S$, then $mn \in S$ for any positive integer $m$.
2. $mn \in S$ for integers $m,n$ with $n < m < 2 n$, because $$\dfrac{1}{mn} = \dfrac{1}{n(m-n)} - \dfrac{1}{m(m-n)}$$
3. $S$ contains no prime or prime power.
4. There are no members of the form $2p^k$ where $p$ is a prime $> 3$.
5. There are no members of the form $3p^k$ where $p$ is a prime $> 11$.