A sieve problem and its application

Let $θ$ be an arithmetic function and let $\mathcal{B}$ be the set of positive integers $n=p_1^{α_1} \cdots p_k^{α_k}$, which satisfy $p_{j+1} \le θ( p_1^{α_1}\cdots p_{j}^{α_{j}})$ 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 $θ(n)/n$ is non-decreasing, the set $\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1< d_2 < \ldots <d_{τ(n)}=n$ satisfy $d_{j+1} \le θ( d_j )$ for $1\le j <τ(n)$.

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