## Ordered radicals

### Problem 124

Published on Friday, 14th July 2006, 06:00 pm; Solved by 7220

The radical of n, rad(n), is the product of distinct prime factors of n. For example, 504 = 23 32 7, so rad(504) = 2 3 7 = 42.

If we calculate rad(n) for 1 n 10, then sort them on 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 kth element in the sorted n column; for example, E(4) = 8 and E(6) = 9.

If rad(n) is sorted for 1 n 100000, find E(10000).