Mazet, Pierre; Saias, Eric Study of the divisor graph. IV. (Étude du graphe divisoriel. 4.) (French. English summary) Zbl 1458.11049 Ann. Fac. Sci. Toulouse, Math. (6) 29, No. 4, 971-975 (2020). A chain-permutation of the divisor graph of \(\mathbb{N}^*\) is a one-to-one mapping \(f: \mathbb{N}^*\rightarrow \mathbb{N}^*\) such that \(f(n)\) is a divisor or a multiple of \(f(n+1)\) for any positive integer \(n\). The main result of the paper says that there is a constant \(c_1\) and a chain-permutation of the divisor graph of \(\mathbb{N}^*\) such that \(f(n)\leq c_1n(\log n)^2\) for every \(n\geq 2\). This implies the existence of a constant \(c_2\) and a permutation of \(\mathbb{N}^*\) such that \(\operatorname{lcm}[f(n),f(n+1)]\leq c_2 n(\log n)^2\) for every \(n\geq 2\). This improves previous results of P. Erdős et al. [Acta Math. Hung. 41, 169–176 (1983; Zbl 0518.10063)] and Y. G. Chen and C. S. Ji [Acta Math. Hung. 132, No. 4, 307–309 (2011; Zbl 1249.11001)]. Reviewer: Štefan Porubský (Praha) MSC: 11B75 Other combinatorial number theory 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 05C38 Paths and cycles 05C63 Infinite graphs Keywords:chain-permutation; divisor graph of positive integers; least commnon multiple Citations:Zbl 1177.11021; Zbl 0518.10063; Zbl 1249.11001 PDFBibTeX XMLCite \textit{P. Mazet} and \textit{E. Saias}, Ann. Fac. Sci. Toulouse, Math. (6) 29, No. 4, 971--975 (2020; Zbl 1458.11049) Full Text: DOI Online Encyclopedia of Integer Sequences: a(1)=1, a(2)=2; for n>0, a(2*n+2) = smallest number missing from {a(1), ... ,a(2*n)}, and a(2*n+1) = a(2*n)*a(2*n+2). a(1) = 1; for n > 1, a(n) = max(d*lpf(d) : d|n, d > 1), where lpf is the least prime factor function (A020639). Instance of a permutation of the positive integers such that lcm(a(n), a(n+1)) <= c*n*log(n)^2. Divisor-or-multiple permutation of natural numbers constructed from two-dimensional Hilbert curve (A163357) and Fermi-Dirac primes (A050376). References: [1] Chen, Yong Gao; Ji, C. S., The permutation of integers with small least common multiple of two subsequent terms, Acta Math. Hung., 132, 4, 307-309 (2011) · Zbl 1249.11001 · doi:10.1007/s10474-011-0099-x [2] Erdös, Paul; Freud, Robert; Hegyvári, Norbert, Arithmetical properties of permutations of integers, Acta Math. Hung., 41, 1-2, 169-176 (1983) · Zbl 0518.10063 · doi:10.1007/BF01994075 [3] Saias, Eric, Entiers à diviseurs denses. I, J. Number Theory, 62, 1, 163-191 (1997) · Zbl 0872.11039 · doi:10.1006/jnth.1997.2057 [4] Saias, Eric, Applications des entiers à diviseurs denses, Acta Arith., 83, 3, 225-240 (1998) · Zbl 0893.11038 · doi:10.4064/aa-83-3-225-240 [5] Tenenbaum, Gérald, Sur un problème de crible et ses applications, Ann. Sci. Éc. Norm. Supér., 19, 1, 1-30 (1986) · Zbl 0599.10037 · doi:10.24033/asens.1502 [6] Tenenbaum, Gérald, Sur un problème de crible et ses applications II. Corrigendum et étude du graphe divisoriel, Ann. Sci. Éc. Norm. Supér., 28, 2, 115-127 (1995) · Zbl 0852.11048 · doi:10.24033/asens.1710 [7] Weingartner, Andreas, Practical numbers and the distribution of divisors, Q. J. Math, 66, 2, 743-758 (2015) · Zbl 1338.11087 · doi:10.1093/qmath/hav006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.