Let $S(m)$ be the sum of digits in decimal notation of a positive integer $m$, and $f(n)$ be the number of ways to represent $n$ in the form $n=a+b+c$, where $a,b,c$ are different positive integers such that $S(a)=S(b)=S(c)$.
Find all $n\in\mathbb{Z}^+$ such that $f(n)>n$.
I have a hypothetical answer: $153$, as well as all numbers that are multiples of three, starting with $159$. There are many ways, and there are more and more of them. It is necessary to get an estimate for some number, and everything less is checked by computer search.