Catalan–Dickson conjecture on aliquot sequences

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\in\mathbb N$ define $s(n)=\sum_{d\mid n,\ 1\le d<n}d$, the sum of positive proper divisors. Starting at $a_0\in\mathbb N$, iterate $a_{j+1}=s(a_j)$ until a zero is reached, at which point the sequence terminates. A cycle means $a_r=a_t>0$ for some $0\le r<t$, which forces repetition thereafter.

Statement recorded in the supplied source.

For every $a_0\in\mathbb N$, does this aliquot sequence either terminate at zero or eventually enter a cycle? Equivalently, is there no infinite nonrepeating aliquot sequence? [S2]

Short English statement

Must repeated summation of proper divisors always terminate or eventually repeat?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400003 (AMR-093-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Snehal Shekatkar and Tian An Wong, Shifts of the prime divisor function of Alladi and Erdős, arXiv:1710.10875 (2017) Introduction, p. 2, Catalan--Dickson discussion. https://arxiv.org/pdf/1710.10875.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Catalan–Dickson conjecture on aliquot sequences. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.