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# Let For how many integers n from 1 to 100, inclusive, does for some number of applications of f?

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Let $$f(n) = \begin{cases} n^2+1 & \text{if }n\text{ is odd} \\ \dfrac{n}{2} & \text{if }n\text{ is even} \end{cases}.$$

For how many integers n from 1 to 100, inclusive, does $$f ( f (\dotsb f (n) \dotsb )) = 1$$ for some number of applications of f?

Jun 9, 2020

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There are 80 integers that work.

Jun 9, 2020