## Digit sum numbers

### Problem 725

A number where one digit is the sum of the **other** digits is called a *digit sum number* or DS-number for short. For example, 352, 3003 and 32812 are DS-numbers.

We define $S(n)$ to be the sum of all DS-numbers of $n$ digits or less.

You are given $S(3) = 63270$ and $S(7) = 85499991450$.

Find $S(2020)$. Give your answer modulo $10^{16}$.