VN-TST 1993 Problem 5

Let an integer $k > 1$ be given. For each integer $n > 1$ , we put

$$f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)$$

where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$ . Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$ .