I am not a mathematician nor a cientist, I'm just a curious person. My math background is always trying to be "back there", as anything you learn tends to be. So, there is a risk that I post silly questions here. I apologize to those that get a little mad about this. Thank you.
Here they go...
1) A perfect number can be defined as $$N\ is\ perfect\ if\ \sum id = N - 1$$
where $\Bbb id$ are the internal divisors of N (1 and N are not included). Define the following: $$N\ is\ imperfect\ if \sum id = N - 2$$ $$N\ is\ pluperfect\ if\ \sum id = N$$
I did a search in a sequence from 2 to 10 million and: Found only 4 perfects (ok, confirmed by Wikipedia), zero pluperfects and 22 imperfects.
It is known that perfects have the form $2^{n-1}(2^n - 1)$ where n is prime and $2^n - 1$ is a Marsenne prime.
Those that I call 'imperfects' have the form $2^n$. I don't know the form of those I call 'pluperfect'. I don't know if the exist, either.
I found that a perfect number can have the following form also: $2^n(2^{n+1} - 1)$ for $n = (Marsenne\ prime) - 1$.
2) If this $1 + \sum di = N$ is called a perfect number, how shoud we call this?: $$ P = \prod di = N$$
This question is because for some N, we can find that $P = N^2$; for some N, we can find that $P = N$ (my question falls here) and for some N, we can find that $P \div N = N^2$
It seems that there are more $P=N$ than $P \div N=N^2$ and more $P \div N=N^2$ than $P=N^2$.