OME 2020 Problem 2

Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as: $\bullet$ $f(1) = 1$ . $\bullet$ $f(n) = f(n/2)$ if $n$ is even. $\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$ . $\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$ .

a) Compute $f(2^{2020}-1)$ .

b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$ .