After I have read (though not carefully) some pages about sociable numbers I thought that it would be convenient to ask a question that I "arrived at".
If we denote with $S_k$ the set of all sociable numbers of order $k$, that is, the set of all sociable $k$-tuples, then, for example, $S_1$ is the set of all perfect numbers and $S_2$ is the set of all amicable pairs.
From what I have read I concluded that, of course, it is not known which of the sets $S_k$ are infinite (some are even conjectured to be empty) and it is also stated that "It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.".
But, although this question maybe is a silly one (if there is something obvious of I am not aware of), I would like to know is it at least known is the set $S=\bigcup_{k=1}^{+ \infty}S_k$ infinite?
The set $S$ is clearly infinite if at least one of the sets $S_k$ is infinite (and I think that is not known as of this moment) or if at least an infinite number of sets whose union is $S$ contain at least one member.
What are your thoughts about this, is $S$ known to be infinite?