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Study of the divisor graph. IV. (Étude du graphe divisoriel. 4.) (French. English summary) Zbl 1458.11049

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)].

MSC:

11B75 Other combinatorial number theory
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
05C38 Paths and cycles
05C63 Infinite graphs
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[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
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