I was thinking about the following problem:

Construct an array $$$A$$$ of positive integers such that exactly $$$K$$$ subsets of $$$A$$$ have $$$\gcd = 1$$$.

Formally, let $$$f(A)$$$ be the number of non-empty subsets $$$S \subseteq A$$$ for which $$$\gcd(S) = 1$$$. The task is to design $$$A$$$ so that $$$f(A) = K$$$.

• Shorter arrays are preferable.
• I am not looking for a brute force "search until found" method — I want a clean construction idea.
• For example, if $$$K = 10^9 + 7$$$, how would you construct such an array?

Also, is it true that such an array always exists for any $$$K$$$? If yes, can we formally prove it?

Any insights or clever constructions would be appreciated!