Ordered radicals
Problem 124
Published on Friday, 14th July 2006, 06:00 pm; Solved by 8652The radical of n, rad(n), is the product of the distinct prime factors of n. For example, 504 = 2^{3} × 3^{2} × 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).