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Really Cool Numbers

Define a *cool* number as a number whose proper divisors (all except for the number itself) have an integral mean. Define a *really cool* number as a number whose divisors (including itself) are all cool. (We explicitly define 1 to be both cool and really cool.) Given a positive integer, determine whether or not it is really cool.
# Examples
Prime numbers are both cool and really cool, since 1 is defined as cool.
15 is really cool, because $\frac{1+3+5}{3} = 3$ and primes/1 are cool.
30 is cool, since $\frac{1+2+3+5+6+10+15}{7} = 6$, but not really cool, since 10 is not cool.
Here is a short list of really cool numbers for testing: $2, 5, 6, 9, 25, 207$
~~This is code-golf, so shortest code wins.~~

Define a *cool* number as a number whose proper divisors (all except for the number itself) have an integral mean. Define a *really cool* number as a number whose divisors (including itself) are all cool. (We explicitly define 1 to be both cool and really cool.) Given a positive integer, determine whether or not it is really cool.
# Examples
Prime numbers are both cool and really cool, since 1 is defined as cool.
15 is really cool, because $\frac{1+3+5}{3} = 3$ and primes/1 are cool.
30 is cool, since $\frac{1+2+3+5+6+10+15}{7} = 6$, but not really cool, since 10 is not cool.
Here is a short list of really cool numbers for testing: $2, 5, 6, 9, 25, 207$
This is <a class="badge is-tag">code-golf</a>, so shortest code wins.

code-golf math