A320536 a(n) is the least cardinal of a partition of {1..n} into simple paths of its divisorial graph.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 14, 15, 15, 14, 14, 15, 14, 15, 15, 15, 15, 16, 16

Offset: 1

Author

Michel Marcus, Oct 15 2018

Keywords

• nonn

Comments

Saias proves that n/6 <= a(n) for all positive integers, and a(n) < n/4 for n large enough. [clarified by Paul Revenant, Jul 08 2019]

Examples

			a(30) = 5 with (13, 26, 1, 11, 22, 2, 14, 28, 7, 21, 3, 27, 9, 18, 6, 12, 24, 8, 16, 4, 20, 10, 30, 15, 5, 25), (17), (19), (23) and (29).
		

Links

• Paul Revenant, Table of n, a(n) for n = 1..3210
• P. Erdos, and E. Saias, Sur le graphe divisoriel, Acta Arithmetica 73, 2 (1995), 189-198.
• Paul Melotti and Eric Saias, On path partitions of the divisor graph, arXiv:1807.07783 [math.NT], 2018.
• Carl Pomerance, On the longest simple path in the divisor graph, Proc. Southeastern Conf. Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983, Cong. Num. 40 (1983), 291-304.
• Eric Saias, Etude Du Graphe Divisoriel 3, Preprint 849, Laboratoire de Probabilités et Modèles Aléatoires, October 2003.
• Eric Saias, Etude Du Graphe Divisoriel 3, Rend. Circ. Mat. Palermo (2003) 52: 481.

Formula

a(n) = floor((n+1)/2) - floor(n/3) for n <=35.

Extensions

More terms from Paul Revenant, Jul 08 2019