Digit sum numbers

 Published on Saturday, 12th September 2020, 05:00 pm; Solved by 585;
Difficulty rating: 10%

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 $10^{16}$.