Let $Q(n)$ denote the count of the primitive subsets of the integers $\{ 1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{ (1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $\alpha$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
Categories: math.NT math.CO