Please use Mathematica.

The lcm(a,b) represents the least common multiple of integers a and b. For a given number n, we can calculate the average of the lcm of n and each number less or equal to it. That is, we can calculate the average of lcm(n,k) with 1<=k<=n.

Example: Let's consider n=8. Then: lcm(8,1)=8; lcm(8,2)=8; lcm(8,3)=24; lcm(8,4)=8; lcm(8,5)=40; lcm(8,6)=24; lcm(8,7)=56. The average of these numbers is 168/8=21. So, for n=8, we associate the number 21.

Your task: identify the largest difference for the average of lcm() between two consecutive numbers, for numbers from 1 to 10000. For which two numbers does this occur for the first time?

Example. For n=9, we have the average lcm equal to 31. Then, the difference between the average lcm of 8 and 9 is 10.