A344497 Matching number of the divisor graph of {1,...,n}.

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, 13, 14, 14, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 22, 23, 23, 23, 24, 24, 25, 25, 26, 26, 26, 26, 27, 28, 28, 28, 29, 29, 30, 30, 31, 31

Offset: 1

Author

Paul Revenant, May 21 2021

Keywords

• nonn

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).

Examples

			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.
		

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

Crossrefs

Cf. A002265, A004526.

Formula

floor(n/4) <= a(n) <= floor(n/2).