A sieve problem and its application

Abstract

Let $\theta$ be an arithmetic function and let $\mathcal{B}$ be the set of positive integers $n=p_1^{{\alpha }_1}\cdots p_k^{{\alpha }_k}$, which satisfy $p_{j+1}\le \theta (p_1^{{\alpha }_1}\cdots p_j^{{\alpha }_j})$ for $0\le j<k$. We show that $\mathcal{B}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of $\mathcal{B}$. When $\theta (n)/n$ is non-decreasing, the set $\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1<d_2<\dots <d_{\tau (n)}=n$ satisfy $d_{j+1}\le \theta (d_j)$ for $1\le j<\tau (n)$.

17 pages, 1 figure