Ordered Radicals
The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
If we calculate $\operatorname{rad}(n)$ for $1 \le n \le 10$, then sort them on $\operatorname{rad}(n)$, and sorting on $n$ if the radical values are equal, we get:
| Unsorted | Sorted | ||||
|---|---|---|---|---|---|
| n | rad(n) | n | rad(n) | k | |
| 1 | 1 | 1 | 1 | 1 | |
| 2 | 2 | 2 | 2 | 2 | |
| 3 | 3 | 4 | 2 | 3 | |
| 4 | 2 | 8 | 2 | 4 | |
| 5 | 5 | 3 | 3 | 5 | |
| 6 | 6 | 9 | 3 | 6 | |
| 7 | 7 | 5 | 5 | 7 | |
| 8 | 2 | 6 | 6 | 8 | |
| 9 | 3 | 7 | 7 | 9 | |
| 10 | 10 | 10 | 10 | 10 | |
Let $E(k)$ be the $k$-th element in the sorted $n$ column; for example, $E(4) = 8$ and $E(6) = 9$.
If $\operatorname{rad}(n)$ is sorted for $1 \le n \le 100000$, find $E(10000)$.