I would like to determine the probability that a natural number is a k-th power. It is quite straightforward to see that the probability for a natural number less than N to be a I-that power is $$N^{1/k-1}$$

However I do not understand what to do to extend it for N going to infinity. I essentially though of cutting dyadically or so, and sum the « local » probability above...

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    $\begingroup$ There is no such thing as a uniformly random natural number. $\endgroup$ – Hagen von Eitzen Dec 9 '18 at 5:14
  • $\begingroup$ I think that by probability, you mean the limit of #$\{ n \leq N | n \text{ is a $k$-th power}\} / N$ for $N \to \infty$? You should specify this in your post. $\endgroup$ – Tki Deneb Dec 9 '18 at 11:01

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