I was working on a totally different problem relating to divisors recently, and I drifted on to this question: What is the sum of the first $n$ perfect numbers?
One pattern I have noticed regarding this question: \begin{alignat*}{3} 6 + 28 &= 34 &= 2 \cdot 17\\ 6 + 28 + 496 &= 530 \ &= 2 \cdot 5\cdot 53\\ 6 + 28 + 496 + 8128 &= 8658\ &= 2\cdot 3^2\cdot 13\cdot 37\\ \vdots \end{alignat*} It seems at first that the sum of the first $n$ perfect numbers yields a number with $n$ distinct prime factors, but this pattern soon breaks down once we find $6 + 28 + 496 + 8128 + 33550336 = 33558994 = 2 \cdot 7 \cdot 2397071$. I don't know how this is helpful to the problem of finding the sum of the first $n$ perfect numbers, or whether it even has any significance to solving the problem. I'm just really curious as to how the sum of perfect numbers behave.
I recognize an important question to ask on the way to answering this question is whether or not there are infinitely many perfect numbers, and this is still an open question (although intuition implies that there are). Regardless, can we find a sum of the first $n$ perfect numbers or at least a satisfactory relation regarding this sum? Are there other patterns for this perfect number series? I'd appreciate your responses!