Eccoci.

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 $i>1$.
When $a_i$ reaches some integer $n$, the sequence stops. (That is, when $y_i=1$.)
Define $f(k)=n$.
For example, for $k=20$:

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

So $f(20)=6$.

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

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