On the number of $n$-perfect numbers

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A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. (e.g. $6$ is a perfect number) Call them $1$ - perfect numbers.

Similarly amicable numbers are two different numbers so that the sum of the proper divisors of each is equal to the other number. (e.g. $220, 284$) Let's call them $2$ - perfect numbers.

Now consider the directed graph $\langle G, E\rangle$ such that $G=\mathbb{N}$ and $m~E~n$ iff $n$ is the sum of proper positive divisors of $m$. We call this, the perfectness graph.

For example if $k$ is a perfect number it is connected to itself in the perfectness graph and forms a $1$ - cycle. Similarly a tuple of $2$ - perfect numbers form a $2$ - cycle in $G$. We call the n - tuple of numbers that form a $n$ - cycle in $G$, $n$ - perfect numbers.

My questions are:

(1) Does the perfectness graph $G$ have a $n$ - cycle for arbitrary large number $n$? In other words, is there a $n$ - perfect tuple of numbers for each $n\geq 1$?

(2) Define $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)$ is the number of distinct $n$ - cycles in perfectness graph. Is $f$ (strictly) increasing/decreasing? In other words, will it become easier/harder to find tuples of $n$ - perfect numbers when we increase $n$?