Antichains

Given an integer $N$, let $F_N$ be the set of factors of $N$. e.g., $F_6 = \{1,2,3,6\}$. The radical of an integer $N$, denoted by $\text{rad}(N)$, is defined as the product of the distinct prime factors of $N$. e.g., $\text{rad}(12) = 2 \times 3 = 6$. Define an **antichain** of a set $S$ of integers to be a subset of $S$ such that for any two elements $x$ and $y$ in the antichain, $\text{rad}(x