TITLE: Numbers divisible by precisely the same set of primes
QUESTION [5 upvotes]: Say two positive integers are "peers" if they are divisible by precisely the same set of primes, such as 12 and 18 (both divisible by 2 and 3), or 70 and 350 (both divisible by 2, 5 and 7).
What are the best estimates known for the number of pairwise non-peers not greater than an arbitrary positive integer N?

REPLY [8 votes]: Your count equals the number of square-free numbers up to $N$. This is because a set of positive integers are pairwise "non-peers" if any only if their radicals are distinct. This is a well-studied problem in analytic number theory, see in particular Walfisz's estimate here.