Perfect numbers, please help Perfect numbers:6, 28, 496, 8128, ...
There is some interesting patterns, I just want to share it with you guy and ask some questions about.
(1)
$1+2+3=6$, perfect number 6 has 3 terms
$\frac{3-1}{2}=1$
$$6=1\cdot2\cdot3$$
(2)
$1+2+4+7+14=28$, has 5 terms
$\frac{5-1}{2}=2$
$$28^2=1\cdot2\cdot4\cdot7\cdot14$$
(3)
$1+2+4+8+16+31+62+124+248=496$, has 9 terms
$\frac{9-1}{2}=4$
$$496^4=1\cdot2\cdot4\cdot8\cdot16\cdot31\cdot62\cdot124\cdot248$$
This is resembling the factorial numbers. 
I want to know does this pattern continue for all perfect numbers? If it does how you go about justifying it.

You can say :


*

*even perfect numbers are of the form $n=2^{p−1}(2^p − 1)$ where $p$ and $2^p-1$ are prime

*$n$ then has $2p-1$ divisors other than $n$, and the product of those divisors is $n^{p-1}$
The basis for the first of these is the Euler-Euclid theorem and the second is a simple derivation from the first  
