# Amicable chains of numbers

Let amicable chain of $n$ numbers be (a$_{1}$, a$_{2}$, a$_{3},\dotsc$, a$_{n}$) such that for $i < n$ sum of proper divisors of a$_{i}$ is a$_{i+1}$. For sum of proper divisors of a$_{n}\rightarrow$ a$_{1}$. All numbers in the chain are different.

Case $n = 1$ has small solutions ($\{6\}, \{28\},\dotsc$).

Case $n=2$ has ($\{220, 284\},\{1184,1210\},\dotsc$) examples below $1000$.

What about $n=3$? It doesn't seem to have any examples below $364$ million. Are examples known?

Note that this is different from definition given in "Amicable Number Triples" L. E. Dickson.

Update: I checked up to Amicable$_{5}$ (up to $50$M) and up to Amicable$_{32}$ (up to $10$M). Here are some Amicable$_{n}$ with the smallest number in a cycle shown:

Amicable$_{12}$: $4256$ Amicable$_{5}$: $12496$ Amicable$_{28}$: $14316$

Amicable$_{4}$ are common:
$1264460$; $2115324$; $2784580$; $4938136$; $7169104$; $18048976$; $18656380$; $28158165$; $46722700$

Amicable$_{3}$ - no hits.

• A script to verify such a property should not be too complex to write, I presume. What happens with larger numbers above one million? And as a side note, are there numbers for which your process "converges" to known cycles? Commented Nov 7, 2016 at 23:02
• I just wrote an script (complexity O(n^2) - not very efficient) and checked cases up to 1M. I am going to improve it but I was curious if someone have already studied it. I will write an update once I have more data. Commented Nov 8, 2016 at 0:46
• You can obtain $n^{3/2}$ without much trouble - just check the divisors up to $\sqrt n$. Commented Nov 8, 2016 at 10:02