Let be a non-trivial set of integers i.e. and We call such a set a "remarkable set of type " if it has the following properties:
If is an element of then is also an element of .
If is an element of , then is also an elements of .
If are elements of (not necessarily different), then is also an element of .
How many (non-trivial) remarkable sets of type 18 are there?
Bonus: Generalize. If = is the prime factorization of , how many non-trivial remarkable sets of type exist? What do they look like?