Mysterious Asymptotic Behavior

Let $C(N)$ be the number of positive integers $n\ge 2$ such that $n\cdot \text{(largest prime factor of }n\text{)}\le N$.

You are given that $C(10)=3,C({10}^6)=9107,C({10}^{11})=34113192$ and $C({10}^{12})=188014194$.

Find $C({10}^{16})$.