Aliquot sequence

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

starting from $n$

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule $a_1 = n$, $a_{k+1} = s(a_k)$ where $s(a)$ is the sum of aliquot divisors function $$s(a) = \sum_{d|a}d - a \ .$$

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 all $n$ 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): aliquot cycles of length greater than 2 are also termed sociable numbers.

An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Longer cycles are known; e.g., a sequence with cycle length $28$, starting at $n=14316$ (ref [b2]).

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. The aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]).

References

[a1]	H.J.J. te Riele, "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam (1976)
[a2]	H.J.J. te Riele, "A Note on the Catalan–Dickson Conjecture" , Mathematics of Computation 27 No.121 (1973) 189-192. DOI 10.2307/2005261
[b1]	Benito, Manuel; Varona, Juan L. "Advances in aliquot sequences", Mathematics of Computation 68, No.225 (1999) 389-393. DOI 10.1090/S0025-5718-99-00991-6 Zbl 0957.11060
[b2]	P. Poulet, "Question 4865", L'interméd. des Math. 25 (1918) 100–101
[b3]	Richard K. Guy, Unsolved Problems in Number Theory 3rd ed. Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001