Suppose $ \{ 2,3,4,...,k \}$ is such that no element in it is a factor of N. Then is there a way of determining how large $k$ has to be, in order to generate at least half of $(\mathbb{Z}/ \text{N}\mathbb{Z})^\times$?
For smaller N, having just $\{ 2 \}$ works fine, however I'm not sure how this extends to larger N.
N can be assumed either prime or composite, and is not Carmichael.