VN-TST 2021 Problem 1

Define the sequence $(a_n)$ as $a_1 = 1$ , $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$ .

a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$ . b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$ .