Let $t$ be a positive integers. Find all $t$ such that there exists distinct positive integers $k,n < 12$ such that the sum of the digits $t^k$ is the same as $t^n.$
I don't have any idea how to find all such integers... however, I found the example of $3^2$ and $3^3, 3^5.$ I also know that the sum of the digits of $n$ is congruent to $n$ modulo $9$.
Using that, I tried restricting $k,n$ to be such that $$x^k \equiv x^n \pmod 9.$$ I also noticed that a decent amount of these numbers are such that the pair $(x^{n}, x^{n+1})$ works.