I am working on a problem in which it is important to study the maximal coprime-free subsets of $[n] = \{1,2,\ldots,n\}$. (A set $S\subseteq [n]$ is coprime-free if for all $i,j\in S$ with $i\ne j$, $\gcd(i,j)\ne 1$. I'm counting the maximal ones, namely coprime-free $S$ such that for all $x\in [n]\setminus S$, $S\cup\{x\}$ is not coprime-free.) For a while I was working under the impression that, other than the exceptional case $\{1\}$, these sets were of the form $p{\bf N} \cap [n]$ for all primes $p\in [n]$. But this turns out not to be the case. For example, when $n=15$, there is the maximal subset $\{6,10, 12,15\}$, which is not of the form $p{\bf N}\cap [15]$ for any of the six primes less than $15$.
Would anyone happen to know if these maximal coprime-free subsets admit any kind of simple description? Thank you in advance!