On large primitive subsets of $\{1,2,\ldots,2n\}$

Abstract: A subset of ${1,2,\ldots,2n}$ is said to be primitive if it does not contain any pair of elements $(u,v)$ such that $u$ is a divisor of $v$. Let $D(n)$ denote the number of primitive subsets of ${1,2,\ldots,2n}$ with $n$ elements. Numerical evidence suggests that $D(n)$ is roughly $(1.32)^n$. We show that for sufficiently large $n$, $$(1.303...)^n < D(n) < (1.408...)^n$$