On natural densities of sets of some type integers

Let $a_0=b_0=0$ and $0<a_1\le b_1<a_2\le b_2<\dots \le b_n$ be integers. Let $Q(x;{\bigcup }_{j=1}^n[a_j,b_j])$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in ${\bigcup }_{j=1}^n[a_j,b_j]$. The following formula holds: $$\underset{x\to \mathrm{\infty }}{\lim }\frac{Q(x;\bigcup _{j=1}^n[a_j,b_j])}{x}=\prod _p\sum _{i=0}^n(\frac{1}{p^{a_i}}-\frac{1}{p^{b_i+1}})\mathrm{.}$$ In this paper, we prove this result and then generalize it.