Median divisor of even perfect numbers I noticed that when divisors of even perfect numbers are listed in ascending order, the middle divisor (I guess the median), is always of the form $2^n$, some power of 2. If true is there a proof for this, or does it happen all the time? I only checked up to the 8th perfect number. Thank you and apologies for the possible silliness of the question.

Every even perfect number is of the form $n=2^{p-1}(2^p-1)$ with $2^p-1$ prime.  
Therefore, the factors of $n$ are 
$1, 2, 2^2, 2^3, ..., 2^{p-1}, 2^p-1, 2(2^p-1), 2^2(2^p-1), 2^3(2^p-1), ...2^{p-2}(2^p-1),$ 
and the middle one is $2^{p-1}.$ 
