Multiplicative representations of integers
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.
number of multiplicative representations
scientific article; zbMATH DE number 4006365