With $N$ integers, how many different GCDs can you make by taking subsets of them? More formally:

Let $S$ be a set of non-negative integers.

Define $G(S)$ as $\{gcd(T) : T \subseteq S \}$

Define $f(N)$ as $\max_{|S| = N} \{|G(S)|\}$

For example:

- $f(0) = 1$
- $f(1) = 2$
- $f(2) = 4$ (e.g. $S = \{4, 6\}$, $G(S) = \{0, 2, 4, 6\}$)
- $f(3) = 8$ (e.g. $S = \{12, 20, 30\}$, $G(S) = \{0, 2, 4, 6, 10, 12, 20, 30\}$)

Is it the case that $f(N) = 2^N$?