For any positive integer k, a finite sequence a_i of fractions x_i/y_i is defined by:
a₁ = 1/k and
a_i = (x_i-1+1)/(y_i-1-1) reduced to lowest terms for i>1.
When a_i reaches some integer n, the sequence stops. (That is, when y_i=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.