Recently I learned about aliquot sequences. Here's how they work: you start with any number. For an example, I'll use $10$. Then you find all the factors of that number, besides itself. So for $10$, that's $1, 2,$ and $5$. Then you add those numbers up - we get $8$.
Now we repeat this process with this sum. So $8$ has factors $1, 2,$ and $4$, which sum to $7.$ Now we do it again with this sum. $7$'s only factor besides itself is $1,$ so our sequence goes to $1.$ The sequence ends at $1.$ The full sequence was $10\rightarrow 8 \rightarrow 7 \rightarrow 1.$
I made a program to calculate these sequences, and I have some questions.
$276$ shoots off to infinity - why does this happen? What makes $276$ special? Is there any way of knowing if it will eventually go back down to $1$ or not?
Is there any way to predict what will happen to a sequence without simply calculating it? Can we tell if it will loop, go down to $1$, or go up to infinity?
Does knowing the aliquot sequence of a number tell you anything about multiples of that number? For example, given the aliquot sequence for $10$, can you derive the aliquot sequence for $20$ from it, without calculating it?
These questions have been bugging me for a little bit, and I would really appreciate some answers. Thank you!