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%I #17 Jun 15 2021 07:12:17
%S 0,1,1,2,2,3,3,3,4,5,5,5,5,6,7,7,7,8,8,8,9,10,10,10,11,12,12,13,13,13,
%T 13,14,14,15,16,16,16,17,17,18,18,19,19,19,20,21,21,21,21,22,23,23,23,
%U 24,24,25,25,26,26,26,26,27,28,28,28,29,29,30,30,31,31
%N Matching number of the divisor graph of {1,...,n}.
%C a(n) is the matching number of the graph on vertices {1,...,n} in which two vertices are connected by an edge if one divides another.
%C The maximum matching in a graph can be calculated by the blossom algorithm.
%C By considering the matching k-2k with k = floor(n/4)+1,...,floor(n/2), we obtain the inequality: floor(n/4) <= a(n).
%H Paul Revenant, <a href="/A344497/b344497.txt">Table of n, a(n) for n = 1..8000</a>
%H Paul Revenant, <a href="https://perso.ens-lyon.fr/paul.revenant/Divisor_Graph/Matching_Divisor_Graph.cpp">C++ program using the Blossom algorithm</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Matching_(graph_theory)">Matching (graph theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Blossom_algorithm">Blossom algorithm</a>
%F floor(n/4) <= a(n) <= floor(n/2).
%e a(10) = 5, since the divisor graph of {1,...,10} has a perfect matching: 1-7, 2-6, 3-9, 4-8, 5-10, which is a matching of size 5.
%o (C++) // program available at Revenant link
%Y Cf. A002265, A004526.
%K nonn
%O 1,4
%A _Paul Revenant_, May 21 2021
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