Fix constants $1\leq \alpha<\beta$. What is the density of the set of positive integers $n$ with at least two factors between $\alpha\sqrt{n}$ and $\beta\sqrt{n}$? 


(I am specifically interested when $\alpha=\sqrt{2}$ and $\beta=\sqrt{3}$, and I am hoping the density is zero. I am not an expert in this field, so apologies in advance.)



Density of numbers with multiple factors near square root
Even one such factor gives you zero density. Indeed, if $d \mid n$, $\alpha \sqrt{n} \le d \le \beta \sqrt{n}$ then $\frac{1}{\beta}\sqrt{n} \le \frac{n}{d} \le \frac{1}{\alpha}\sqrt{n}$ therefore $n$ is a product of two numbers not greater than $\max(\frac{1}{\alpha}, \beta)\sqrt{n}$. Therefore amount of desired numbers in the interval $[0, N]$ is not greater than amount of numbers in the multiplicative table of size $\max(\frac{1}{\alpha}, \beta)\sqrt{N} \times \max(\frac{1}{\alpha}, \beta)\sqrt{N}$. And this number is $o(\sqrt{N}^2) = o(N)$, see [this](https://mathoverflow.net/questions/108912/number-of-elements-in-the-set-1-cdots-n-cdot-1-cdots-n/108939#108939) MO question.


