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 A320536 a(n) is the least cardinal of a partition of {1..n} into simple paths of its divisorial graph. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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] 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. EXAMPLE 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). CROSSREFS Sequence in context: A103221 A026806 A261348 * A338336 A298783 A053280 Adjacent sequences:  A320533 A320534 A320535 * A320537 A320538 A320539 KEYWORD nonn AUTHOR Michel Marcus, Oct 15 2018 EXTENSIONS More terms from Paul Revenant, Jul 08 2019 STATUS approved

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Last modified June 15 01:49 EDT 2021. Contains 345041 sequences. (Running on oeis4.)