$n$ such that the digits immediately after the decimal point of $\pi^n$ give $n$ again

Question

I was doing something with value of $\pi$ as I know that the beauty of numbers will always exist , doesn't matter either number is real or complex it must be beautiful. I observe something strange by using Wolfram alpha's calculator:

$$ \pi^4 = 97.40909103400243723644033268870511124972758567268542169146... $$

as we can see the power of $\pi$ is $4$ and on right hand side the number after decimal place is also $4$.

Now a question arise in my mind:

"Is there any value also exist which can show the below relation as $4$ did?"

$$ \pi^n = ....ABCD.nPQRST....... $$ ( where n is the number after decimal which can be of any digits)

I still don't get any clue.Any hint or solution will helpful for me. Thanks.

Answer 1

Next ones are $75$, $9\,424$ and $12\,669$. I do not see any pattern.