

A344497


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


1



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
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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 k2k 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: 17, 26, 39, 48, 510, 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



