Multiplicative representations of integers
type: Item
instance of: scholarly article
title: Multiplicative representations of integers
zbMATH Open document ID: 0621.10040
DOI: 10.1007/BF02772173
author: Melvyn B. Nathanson
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(h\geq 2\), and let \({\mathcal B}=(B_ 1,...,B_ h)\), where \(B_ i\subseteq {\mathbb{N}}=\{1,2,3,...\}\) for \(i=1,...,h\). Denote by \(g_{{\mathcal B}}(n)\) the number of representations of n in the form \(n=b_ 1,...,b_ h\), where \(b_ i\in B_ i\). If \(g_{{\mathcal B}}(n)>0\) for all \(n>n_ 0\), then \({\mathcal B}\) is an asymptotic multiplicative system of order h. The set B is an asymptotic multiplicative basis of order h if \(n=b_ 1,...,b_ n\) is solvable with \(b_ i\in B\) for all \(n>n_ 0\). Denote by g(n) the number of such representations of n. Let M(h) be the set of all pairs (s,t), where \(s=\liminf_{n\to \infty}g_{{\mathcal B}}(n)\) and \(t=\limsup_{n\to \infty}g_{{\mathcal B}}(n)\) for some multiplicative system \({\mathcal B}\) of order h. It is proved that \[ M(h) = \{(1,t)| \quad t\in {\mathbb{N}}\}\quad \cup \quad \{(s,\infty)| \quad s=1,...,h\}\quad. \] In particular, it follows that \(s\geq 2\) implies \(t=\infty\). A corollary is a theorem of Erdős that if B is a multiplicative basis of order \(h\geq 2\), then \(\limsup_{n\to \infty}g(n)=\infty\). Similar results are obtained for asymptotic union bases of finite subsets of \({\mathbb{N}}\) and for asymptotic least common multiple bases of integers.
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 4006365
zbMATH Keywords: number of multiplicative representations
zbMATH Keywords: asymptotic multiplicative system
zbMATH Keywords: asymptotic multiplicative basis
zbMATH Keywords: asymptotic union bases
zbMATH Keywords: asymptotic least common multiple bases
MaRDI profile type: Publication
cites work: Extension of some theorems on the density of series of elements of \(N\) to series of finite subsets of \(N\).
cites work: On the multiplicative representation of integers
cites work: Nonexistence of maximal asymptotic union nonbases
cites work: Q4145801
cites work: Two Proofs in Combinatorial Number Theory
full work available at URL: https://doi.org/10.1007/bf02772173
OpenAlex ID: W2062563700
description: scientific article; zbMATH DE number 4006365