When does d divide a² but not a?

For a prime $d$, if $d$ divides $a^2$ then $d$ divides $a$. For other numbers it can fail. You will find every divisor $d$ from $2$ to $20$ that fails, and look at what they have in common.

The idea. For one $d$, try $a = 1, 2, 3, \ldots$ and look for an $a$ where $d$ divides $a^2$ but not $a$.

Your tasks.

1. In divides, return True when d divides n exactly.
2. In first_failure, return the smallest a from $1$ to $400$ with d dividing a * a but not a. Return None if there is none.
3. Fill failing with the divisors d from $2$ to $20$ that have a failure, in order.