While digging through some old notebooks today, I found a problem from a long time ago that I was never able to solve. It involves a sequence of positive integers called the “self numbers” defined such that $s_n$ is the nth positive integer for which the equation $$k+\text{digit sum of }k = s_n$$ has no solution (this is is base 10, by the way). This sequence is A003052 on OEIS.
It appears that this sequence grows asymptotically linearly, and I tried (but failed) to find the asymptotic rate at which it grows:
$$r=\lim_{n\to\infty}\frac{s_n}{n}$$
Can anyone figure out how to calculate $r$, even in series or integral form?
Note: The best I was able to prove is that $r < 25/2$.
U. Zannier in his paper `On the Distribution of Self-Numbers' proves a theorem about the number of self numbers less than $x$, specifically that $A(x)= Lx + O(\log^2(x))$ for some positive $L$. You can invert this function to get a statement about the $n$th self number growing linearly (in the same way that one can invert the prime number theorem $\pi(x) \sim x/\log(x)$ into $p_n\sim n\log(n)$, being careful with analytic asymptotic inverses). I'm not sure if an exact way to compute the constant is mentioned but that would be somewhere to look to get a sense of how to do what you want to do. His $L$ for decimal (or what is referred to in the paper as scale $g=10$) appears to be $\approx 0.0977$, which you invert to get the $r$ in your formula to be $\approx 10.24$. I don't know if there's a nice analytic form.