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- W3128300664 abstract "Abstract 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,…,2 n }. We prove that the limit α = lim n→∞ 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." @default.
- W3128300664 created "2021-02-15" @default.
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- W3128300664 date "2021-01-28" @default.
- W3128300664 modified "2023-09-27" @default.
- W3128300664 title "The number of maximum primitive sets of integers" @default.
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- W3128300664 doi "https://doi.org/10.1017/s0963548321000018" @default.
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