# Problem 343: Fractional Sequences
For any positive integer k, a finite sequence ai of fractions xi/yi is
defined by: a1 = 1/k and ai = (xi-1+1)/(yi-1-1) reduced to lowest terms
for i>1. When ai reaches some integer n, the sequence stops. (That
is, when yi=1.) Define f(k) = n. For example, for k = 20: 1/20 → 2/19 →
3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6 So f(20) = 6. Also f(1) =
1, f(2) = 2, f(3) = 1 and Σf(k3) = 118937 for 1 ≤ k ≤ 100. Find Σf(k3)
for 1 ≤ k ≤ 2×106.