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

Counting primitive subsets and other statistics of the divisor graph of $\{1,2, \ldots n\}$