I suppose the answer is $n=16$ but my solution is super-ugly - nothing but a brute force (at least it can be done in less than an hour ). Maybe somebody will come up with a nice idea.

The 14 prime numbers $< \sqrt{2005}$ are : $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43$ If we take 15, or more, numbers relatively prime they can't be all of the form $\prod^n p_i^{a_i}$ with $n=1$ : one must be prime (since it must be prime with all prime numbers $< \sqrt{2005}$).

tµtµ: you forgot about $1$ . Well, my idea is pretty much the same but I complicated it much more

hint