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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.

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    We can assume that $t$ is not divisible by $10$ , since if $10t$ has the property , then also $t$. Are you sure that under this assumption, only finite many such $t$ exist ? Or that they can at least be completely classified ? – Peter Apr 10 '21 at 13:43
  • I am trying to classify all such $t$. –  Apr 10 '21 at 13:59
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    I suggest doing a search first. For small $t$ there are lots of examples. $(6^2,6^3), (7,7^4), (8,8^3),(9,9^2), (15^3,15^4), (18^3,18^6),(19,19^2)$ for example. No idea if that is an accident for small numbers or not. – lulu Apr 10 '21 at 14:03
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    Please edit your post to indicate your efforts and to supply some context for the question. What have you tried? Why should anyone imagine that this has a sensible answer? – lulu Apr 10 '21 at 14:22

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