We consider those natural numbers to be somewhat unlucky whose sum of digits in decimal notation is divisible by 13. What is the maximum difference between consecutive somewhat unlucky numbers?
Let $n_{1}, n_{2}, \ldots, n_{r}$ be natural numbers such that among the sums of their digits, 13 consecutive natural numbers appear, and thus at least one is divisible by 13, meaning that among the numbers $n_{1}, n_{2}, \ldots, n_{r}$, there is a somewhat unlucky number. Therefore, for any non-negative integer $k$, in each of the closed intervals $[100(k+1), 100(k+1)+39]$, $[100 k+60,100 k+99]$, and $[100 k+20,100 k+59]$, there is at least one somewhat unlucky number. Based on this, we will first show that if $n$ and $m$ are such that $m - n > 79$, then the interval $[100 k+20,100 k+59]$ would not contain a somewhat unlucky number, which contradicts the above.

It is easy to see that the numbers $n=\underbrace{9 \ldots 9}_{8 \text{ digits}} 60$ and $m=n+79$ are adjacent somewhat unlucky numbers, so the maximum difference between adjacent somewhat unlucky numbers is 79.