There is a family of recursive sequences that I came up with: $$a_n=\text{sum of factors less than $a_{n-1}$ of } a_{n-1}, a_0\in\mathbb{N}$$ First question: Has it been studied before?

Here are some observations I've made about it:

Starting from successive positive integers, and defining the sum of factors of $0$ less than $0$ as $0$, the first few sequences are: $$1,0,0\dots$$ $$2,1,0,0\dots$$ $$3,1,0,0\dots$$ $$4,2,1,0,0\dots$$ $$5,1,0,0\dots$$ $$6,6,6\dots$$ etc. Setting $a_0$ equal to a perfect number $n$ produces the sequence $n,n,n\dots$ Starting from $a_0=n_1$, where $n_1\text{ and }n_2$ are a pair of amicable numbers, produces the sequence $n_1, n_2, n_1, n_2\dots$

If $a_0$ is a prime number, $a_1=1$.

There are several conjectures I've made about this family of sequences, but haven't been able to prove or disprove:

  1. No sequence diverges to infinity.
  2. For any positive integer $>2$ defined as $a_n$, there is at least one possible $a_{n-1}$ that fits the rule of the sequence. Edit: Never mind
  3. For every $j\in\mathbb{N}$, there is at least one $a_0$ fso that $a_j=a_0$ but for all $a_k$ where $1\leq k<j$, $a_k\neq a_0$.
  4. (this might be easier than the other conjectures) No $k\in\mathbb{N}=a_n$ (greater than one) generates an infinite number of positive integers that can be set equal to $a_{n-1}$ and have these two terms follow the rule of the sequence.

Second question: Does anyone know how to prove any of these conjectures?

P.S. I do not necessarily want a proof to all four of these conjectures, although that would be good. It's just that these seem so difficult to prove that I feel like a proof of any one, two, or three of these conjecturs also has a place here.

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    $\begingroup$ The divisor function counts what you want plus the $a_{n-1}$ term. There are published bounds for it so your claim number 1 might not be hard to prove. $\endgroup$ – Sandeep Silwal Jun 9 '14 at 19:23
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    $\begingroup$ For conjecture #4, isn't it sufficient to observe that $a_{n-1}$ is either a prime (in which case $a_n=1$) or has at least one proper divisor not smaller $\sqrt{a_{n-1}}$ (so $a_n\geq \sqrt{a_{n-1}}$ or, equivalently, $a_n^2\geq a_{n-1}$) $\endgroup$ – Peter Košinár Jun 14 '14 at 22:42
  • $\begingroup$ This might be an useful reference too. $\endgroup$ – Peter Košinár Jun 15 '14 at 7:38
  • $\begingroup$ @PeterKošinár, I believe your logic is correct. Would you please write that up as an answer? $\endgroup$ – Aidan F. Pierce Jun 15 '14 at 13:31

The sequence in question is known as aliquot sequence. Based on the references I was able to find, question #1 and #3 are open problems at the moment; we don't even know whether :

Question #4 turns out to be an easy one. If $a_{n-1}$ was a prime, we'd have $a_n=1$. Otherwise, $a_{n-1}$ must have at least one divisor greater or equal to $\sqrt{a_{n-1}}$, so $a_n\geq \sqrt{a_{n-1}}$. Thus, $a_{n-1}\leq a_n^2$ which means that any particular value $a_n>1$ can have at most $a_n^2$ immediate predecessors (the actual number of possible predecessors tends to be considerably smaller).

This upper bound can be used to check if a particular number is untouchable (which means it can only ever appear at the beginning of the sequence) and find out that $5$ is the smallest counter-example to question #2.

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