S-sets (even though this title is too short)

Question

I say a finite sequence of positive integers is 'autistic' if and only if the sum of the reciprocals of its elements is itself an element of the sequence. That way, the 'sequence' just made of the number 1 is autistic, but so is the sequence of the divisors of an even perfect number arranged in increasing order, as well as any permutation thereof. Are there other examples of autistic sequences ?

Thanks in advance.

Edit: title changed because of some comments.

Answer 1

As tomi pointed out in the comments, there's nothing about the sequence that you need to keep in order, so it's more basic to just talk about sets.

Any Egyptian fraction sequence can be indefinitely prolonged, so we can write, for instance,

\begin{align} 2 & = 1+\frac{1}{2}+\frac{1}{2} \\ & = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \\ & = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{42} \\ & = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1806} \end{align}

or alternatively

$$ 2 = 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{12}+\frac{1}{6} $$

by expanding $1/3$, and that can extend indefinitely as well. That means that the sets $\{1, 2, 3, 6\}, \{1, 2, 3, 7, 42\}, \{1, 2, 3, 7, 43, 1806\}, \{1, 2, 4, 6, 12\}$ (among an infinity of others) all satisfy the condition.

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ETA: If we allow multisets, then $\{1, 2, 2\}$, and the like, also work.

Answer 2

Do the sequences have to increase in order? Otherwise there are surely infinitely many (e.g. 1, 2, 2)

For example, to make the sequence out of any integer $n$ you just need $n^2$ of $n$.