Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ is $1$-perfect, since $1=\sigma(1)=1\cdot{1}=1$.
If a number $P$ is $2$-perfect, then $P$ can simply be called a perfect number.
If a number $T$ is $3$-perfect, then $T$ is called a triperfect number.
The following excerpt is taken verbatim from the Introduction of Cohen and Sorli's paper titled "ON ODD PERFECT NUMBERS AND EVEN 3-PERFECT NUMBERS":
There are six known $3$-perfect numbers, all found more than $360$ years ago, the last in $1643$. They are $$120 = 2^3 \cdot 3 \cdot 5,$$ $$672 = 2^5 \cdot 3 \cdot 7,$$ $$523776 = 2^9 \cdot 3 \cdot {11} \cdot {31},$$ $$459818240 = 2^8 \cdot 5 \cdot 7 \cdot {19} \cdot {37} \cdot {73},$$ $$1476304896 = 2^{13} \cdot 3 \cdot {11} \cdot {43} \cdot {127},$$ $$51001180160 = 2^{14} \cdot 5 \cdot 7 \cdot {19} \cdot {31} \cdot {151}.$$ At a Western Number Theory Conference some years ago, John Selfridge quickly wrote these up by considering even $3$-perfect numbers in the form $2^a M$, $M$ odd, and factorizing $2^{a+1} − 1$ for $a \leq 14$; he went on to ask what was known otherwise of such numbers. It is generally assumed, for example by Achim Flammenkamp in his Multiply Perfect Numbers Page (http://wwwhomes.uni-bielefeld.de/achim/mpn.html), that there are no others, but this presupposes that no odd perfect numbers exist. This is a consequence of the following result; the proof is easy.
Lemma 1 An odd number $N$ is perfect if and only if $2N$ is $3$-perfect.
Here then is my question:
Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
MY OWN THOUGHTS
(I would like to state at the outset that I understand the proof of Lemma 1.)
Surely, if the list of six ($6$) even $3$-perfect numbers as given above is complete, and if there are no odd perfect numbers, then there are no other $3$-perfect numbers.
My point is that: What if the list of six ($6$) even $3$-perfect numbers as given above is incomplete? Or is the fact that the six ($6$) known even $3$-perfect numbers complete the list of even $3$-perfect numbers derivable from known properties of even $3$-perfect numbers?