A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

中文翻译：

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最大原始整数集的数量

一组整数是原始如果它不包含分割另一个的元素。让F(n) 表示 {1,…,2 的最大尺寸基元子集的数量n}。我们证明了极限 α = limn→∞F(n)1/n存在。此外，我们提出了一种近似算法α与 (1 +ε) 乘法误差ñ(ε) 步骤，特别表明α≈ 1.318。我们的算法可以适应于估计 {1,…,n} 以及。我们解决了 Cameron 和 Erdős 的另一个相关问题。他们表明在 {1,... 中包含成对互质整数的集合的数量n} 在。。。之间${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$和${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. 我们证明了这些界限都不是严格的：事实上${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$这样的套装。