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

1,4

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

Cf. A002265, A004526.

Sequence in context: A166079 A269381 A080677 * A316628 A153112 A005350

Adjacent sequences: A344494 A344495 A344496 * A344498 A344499 A344500

Keyword

nonn

Author

Paul Revenant, May 21 2021

Status

approved