Write all the numbers that are perfect numbers and also a factorial
Attempt:
I know that perfect numbers have the general formula $\ \frac{2^p}{2}(2^p-1) $ where p is prime and the factorial have the general formula of $\ x!$. So the basic attempt is to find an intersection of both graphs which comes at 6 and after that both graphs diverge.
However I want to know whether there is a rigorous method to solve this question which doesnt involve graphing.
For $n \geq 4$, $n!$ is divisible by $24$, so $n/2$, $n/3$, and $n/4$ are proper positive divisors of $n$. Since $$ \frac{n}{2} + \frac{n}{3} + \frac{n}{4} = \frac{13n}{12} > n, $$ $n!$ cannot be a perfect number for $n \geq 4$. So $3! = 6$ is the only perfect number which is also a factorial.