Remarkable subsets

Let AZA \subset \mathbb Z be a non-trivial set of integers ((i.e. AA \not= \emptyset and AZ).A \not= \mathbb Z). We call such a set AA a "remarkable set of type NN" if it has the following properties:

  • If aa is an element of A,A, then a-a is also an element of AA.

  • If aa is an element of AA, then a+Na+N is also an elements of AA.

  • If a,ba,b are elements of AA (not necessarily different), then a+2ba + 2b is also an element of AA.

How many (non-trivial) remarkable sets of type 18 are there?

Bonus: Generalize. If NN = 2e23e35e52^{e_2}\cdot 3^{e_3}\cdot 5^{e_5}\cdots is the prime factorization of NN, how many non-trivial remarkable sets of type NN exist? What do they look like?


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