Average least common multiple

 Published on Sunday, 24th November 2013, 01:00 am; Solved by 313;
Difficulty rating: 80%

Problem 448

The function lcm(a,b) denotes the least common multiple of a and b.
Let A(n) be the average of the values of lcm(n,i) for 1≤i≤n.
E.g: A(2)=(2+2)/2=2 and A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32.

Let S(n)= A(k) for 1≤k≤n.

Find S(99999999019) mod 999999017.