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).

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)
[a1]	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

Comments

Aliquot cycles of length greater than 2 are termed sociable numbers.

The aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]). The cycle of length 28 starts at $n=14316$ (ref [b2]).

References

[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