## Fractional Sequences

### Problem 343

Published on Saturday, 18th June 2011, 04:00 pm; Solved by 736

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 2106.