Perfect numbers

Coprimality. $\gcd(2^{p-1},2^p-1)=1$ since $2^{p-1}$ is a power of $2$ and $2^p-1$ is odd.

Multiplicativity.
$$\sigma(n)=\sigma(2^{p-1})\sigma(2^p-1)=(2^p-1)\cdot(1+(2^p-1))=(2^p-1)2^p.$$
(The second factor uses $\sigma(q)=1+q$ since $q=2^p-1$ is prime.)

Conclusion. $\sigma(n)=2^p(2^p-1)=2\cdot2^{p-1}(2^p-1)=2n$, so $n$ is perfect.

Example. $p=2$: $n=6$; $p=3$: $n=28$; $p=5$: $n=496$.