Fractional Sequences

For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by: $a_1 = 1/k$ and $a_i = (x_{i-1}+1)/(y_{i-1}-1)$ reduced to lowest terms for $i\gt 1$. When $a_i$ reaches some integer $n$, the sequence stops. (That is, when $y_i=1$.) Define $f(k)=n$. For example, for $k=20$:

$1/20\rightarrow 2/19\rightarrow 3/18=1/6 \rightarrow 2/5\rightarrow 3/4\rightarrow 4/3\rightarrow 5/2\rightarrow 6/1 = 6$

So $f(20)=6$.

Also $f(1)=1, f(2)=1, f(3)=1$ and $\sum f(k^3) = 118937$ for $1\le k\le 100$.

Find $\sum f(k^3)$ for $1\le k\le 2\times 10^6$.