RU-ALLRUSSIAN 2015 Problem 104

We denote by $S(k)$ the sum of digits of a positive integer number $k$ . We say that the positive integer $a$ is $n$ -good, if there is a sequence of positive integers $a_0$ , $a_1, \dots , a_n$ , so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$ . Is it true that for any positive integer $n$ there exists a positive integer $b$ , which is $n$ -good, but not $(n + 1)$ -good? A. Antropov