Graded lexicographic order

Consider the integers from $1$ to $n$. Define the weight of a number as the sum of its digits. Let the weight of $x$ be denoted as $w(x)$.

Arrange these numbers in graded lexicographic order. For two numbers $a$ and $b$: if w(a) < w(b), then $a$ comes before $b$ in graded lexicographic order. If w(a) = w(b), then $a$ comes before $b$ if and only if the decimal representation of $a$ is lexicographically less than the decimal representation of $b$.

For example, in this order:

• $120$ comes before $4$;

• $555$ comes before $78$;

• $20$ comes before $200$.

Given $n$ and $k$, find the number at the $k$-th position in the graded lexicographic ordering of natural numbers from $1$ to $n$.