Erdős, Paul; Freud, R.; Hegyvari, N. Arithmetical properties of permutations of integers. (English) Zbl 0518.10063 Acta Math. Hung. 41, 169-176 (1983). Let \(a_1,\dots,a_n\) be a permutation of \(1,\dots,n\) and let \([a_i,a_j]\) denote the least common multiple of \(a_i\) and \(a_j\). It is shown that \[ \min\max_{1\leq i< n}[a_i,a_{i+1}]=(1+o(1))\frac{n^2}{4\log n}, \] where the minimum is taken over all permutations. This result is best possible since in any permutation there must be an \(a_i\) such that \([a_i,a_{i+1}]\geq(1+o(1))\frac{n^2}{4\log n}\). It is also shown that there is an infinite permutation \(a_1,a_2,\dots\) of the positive integers such that \[ [a_i,a_{i+1}]< ie^{c\sqrt{\log i}\log\log i} \] for all i. Some results are also obtained for the greatest common divisor. See also following review. Reviewer: I.Anderson Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 5 ReviewsCited in 12 Documents MSC: 11B05 Density, gaps, topology 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11B75 Other combinatorial number theory 05A05 Permutations, words, matrices Keywords:permutations; density of sums; least common multiple; greatest common divisor Citations:Zbl 0518.10064 PDFBibTeX XMLCite \textit{P. Erdős} et al., Acta Math. Hung. 41, 169--176 (1983; Zbl 0518.10063) Full Text: DOI Online Encyclopedia of Integer Sequences: List of pairs (m,2m) where m is the least unused positive number. 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). Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies lcm(p_i, p_{i+1}) >= m for some i, 1 <= i <= n-1. Instance of a permutation of the positive integers such that lcm(a(n), a(n+1)) <= c*n*log(n)^2. References: [1] N. G. de Bruijn, On the number of positive integers and free of prime factors , Indag, Math.,13 (1951), 50–60. · Zbl 0042.04204 [2] P. Erdos, R. L. Graham, Old and New Problems and Results In Combinatorial Number Theory, Monographie No 28 deL’Enseignement Mathématique (Genève, 1980). [3] R. Freud, On some of subsequent terms of permutations, to appear inActa Math. Acad. Sci. Hungar. 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.