OFFSET
1,4
COMMENTS
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.
The maximum matching in a graph can be calculated by the blossom algorithm.
By considering the matching k-2k with k = floor(n/4)+1,...,floor(n/2), we obtain the inequality: floor(n/4) <= a(n).
LINKS
Paul Revenant, Table of n, a(n) for n = 1..8000
Paul Revenant, C++ program using the Blossom algorithm
Wikipedia, Matching (graph theory)
Wikipedia, Blossom algorithm
FORMULA
floor(n/4) <= a(n) <= floor(n/2).
EXAMPLE
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.
PROG
(C++) // program available at Revenant link
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Revenant, May 21 2021
STATUS
approved