Aliquot sequence

starting from $n$

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule

$$a_1=n,\quad a_k=\left(\sum_{d|a_{k-1}}d\right)-a_{k-1}.$$

The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a perfect number, while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also Amicable numbers).

An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Larger cycles are possible; e.g., a sequence with cycle length $28$ is known.

The Catalan–Dickson conjecture states that all aliquot sequences either terminate or are eventually periodic. This conjecture is still (1996) open, but generally thought to be false.

References

[a1]	H.J.J. te Riele, "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam (1976)