Question 96
Define the sequence $(a_n)$ by $a_1=1$, $a_{2n}=a_n$, and $a_{2n+1}=a_n+1$ for every positive integer $n$.
(a) Find all positive integers $n$ such that $a_{kn}=a_n$ for all integers $1\leq k\leq n$.
(b) Prove that there are infinitely many positive integers $m$ such that $a_{km}\geq a_m$ for every positive integer $k$.
(a) Find all positive integers $n$ such that $a_{kn}=a_n$ for all integers $1\leq k\leq n$.
(b) Prove that there are infinitely many positive integers $m$ such that $a_{km}\geq a_m$ for every positive integer $k$.