Lexicographic integers
type: Item
instance of: scholarly article
title: Lexicographic integers
zbMATH Open document ID: 0917.11049
DOI: 10.1023/A:1009770126879
author: André Stef
author: Gérald Tenenbaum
published in: The Ramanujan Journal
publication date: 1999-08-02T00:00:00Z
review text: An integer \(n\) is lexicographic if the sequence of its divisors ordered lexicographically (according to the exponents of the increasing sequence of the prime factors of \(n)\) is itself an increasing sequence. Let \(n=\prod^{ \omega (n)}_{j=1} p_j^{\nu_j}\) \((p_1<p_2<\cdots<p_{\omega(n)})\), \(n_j= \prod_{1 \leq i<j} p_i^{\nu_i}\) for \(1\leq j\leq \omega(n)\), \(W(n)=\min_{1\leq j\leq \omega (n)} p_j/n_j \), and \(d_1,\dots,d_{\tau(n)}\) denote the divisors of \(n\) in increasing order. Theorem 1 establishes that \(n\) is lexicographic if and only if \[ W(n)= \min_{1\leq j<\tau(n)}d_{j+1}/d_j. \] \textit{G. Tenenbaum} showed in [Ann. Sci. Éc. Norm. Supér. (4) 19, 1-30 (1986; Zbl 0599.10037)] that the corresponding result with min replaced by max on both sides holds for all \(n\). An asymptotic formula for the number of lexicographic integers \(n\leq x\) is provided as a special case of theorem 2 where a more general weighted sum is considered. The authors go on to establish an Erdős-Kac type of result.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 1203261
zbMATH Keywords: distribution of divisors
zbMATH Keywords: Erdős-Kac theorem
zbMATH Keywords: distribution of prime factors
zbMATH Keywords: sieve
zbMATH Keywords: asymptotic formula
zbMATH Keywords: lexicographic integers
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1203261