KURSCHAK 2012 Problem 2

Denote by $E(n)$ the number of $1$ 's in the binary representation of a positive integer $n$ . Call $n$ interesting if $E(n)$ divides $n$ . Prove that (a) there cannot be five consecutive interesting numbers, and (b) there are infinitely many positive integers $n$ such that $n$ , $n+1$ and $n+2$ are each interesting.