scholarly journals The probability of choosing primitive sets

2007 ◽  
Vol 125 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Sergi Elizalde ◽  
Kevin Woods
Keyword(s):  
Primitive Sets

scholarly journals The number of maximum primitive sets of integers

2021 ◽  
pp. 1-15
Author(s):  
Hong Liu ◽  
Péter Pál Pach ◽  
Richárd Palincza
Keyword(s):  
Related Problem ◽  
Maximum Size ◽  
Multiplicative Error ◽  
Coprime Integers ◽  
Primitive Sets

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,…,2n}. 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.

scholarly journals On the Critical Exponent for k-Primitive Sets

COMBINATORICA ◽  
2021 ◽  
Author(s):  
Tsz Ho Chan ◽  
Jared Duker Lichtman ◽  
Carl Pomerance
Keyword(s):  
Critical Exponent ◽  
Primitive Sets
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