Question 133
A natural number is cool (bacana) if the sum of all its positive divisors (including $1$ and $n$) is greater than or equal to twice the number. For example, $12$ is cool because $1+2+3+4+6+12=28 \geq 24=2 \cdot 12$, while $4$ is not cool because $1+2+4\lt 8=2 \cdot 4$. Prove that there are infinitely many numbers that are cool and infinitely many numbers that are not cool.