Probability of an integer factoring "nicely" into exactly two integers smaller than a certain bound.

Question

Let $N$ be some margin, and $1\leq k\leq N$ some integer. What is the probability, that $k$ factors nicely, in the sense that $\exists x,y\leq\sqrt{N}\in\mathbb{N}:k=xy$. It is obviously bound by the number of primes between $\sqrt{N}$ and $N$, and all the possible numbers, but thats about all, i can think of, to calculate this probability (or more precisely, the approximate number of occurences of such $k$: $$\frac{N}{\ln(N)}-\frac{\sqrt{N}}{\ln(\sqrt{N})}\leq |\{\sqrt{N}\leq k\leq N:\exists x,y\leq\sqrt{N}:k=xy\}|\leq N - \sqrt{N}$$ The left side is the approximate number of primes between $\sqrt{N}$ and $N$, and the right side all numbers between $N$ and $\sqrt{N}$. Some help or guidance would be greatly appreciated.