The First Number Nobody Can Follow

Take any whole number. Add up every divisor except itself. Do it again to the result. Do it again. Every trajectory ends the same way, unless it doesn't. For most starts under a hundred thousand the story runs to its close in under thirty steps; a handful curl back to a fixed point (the perfect numbers), a few pair off (the amicable pairs), a hundred and thirty enter one of two cycles (a five-step at 12,496 and a twenty-eight-step at 14,316). Fourteen thousand are still climbing when this page's budget runs out. Five of those live under a thousand. Nobody has ever been able to follow them home.

Every number this page prints comes from research/aliquot-frontier/verify.mjs, which runs the whole census over the first 100,000 starts from scratch, plus a deep pass over the five Lehmer starts and one clean counter-example (start 138, which peaks near 10^11 and lands): .../... checks passed.

The operation

The rule is a hundred and fifty years old and takes half a line to write. Pick a whole number n. Sum its proper divisors, the ones strictly less than itself. Call the sum s(n). Do it again to s(n). Repeat. The trajectory is what happens next.

Nothing forces a trajectory to terminate. The four earliest exceptions live under ten thousand.

• 6 is its own sum: 1 + 2 + 3 = 6. The trajectory is stationary. Numbers that do this are called perfect.
• 220 sums to 284, and 284 sums to 220. A two-step cycle: an amicable pair.
• 12,496 sums to 14,288 sums to 15,472 sums to 14,536 sums to 14,264 sums back to 12,496. A five-step cycle: a sociable chain.
• 276 keeps growing.

The trajectory of a number is what happens when you keep applying the operation. There are only four ways for a trajectory to close: reach a prime and terminate, land on a fixed point, land in a two-cycle, land in a longer cycle. And then, for some starts, there is a fifth thing that has never yet been ruled out: the trajectory grows forever. Nobody has proved that any single sequence does, and nobody has proved that no sequence does.

Try one

Type any positive number under a few million. The trajectory is computed here, live, from its prime factorisation. If it terminates, joins one of the small basins, or enters a cycle, the ending is colour-coded to match the census below. If it hits the browser's honest budget without settling, it is marked open, the same way the census marks its own frontier.

The in-browser budget is 400 steps or the moment a term exceeds 10¹⁵, whichever comes first; every larger n is factored by naive trial division and Pollard's rho over BigInt. That is enough to reach the sociable-28 cycle from 14,316 and to show every Lehmer start climbing past its second horizon in a few seconds.

The census, every start from 1 to 100,000

What the operation does to every whole number is not something you argue about. You compute it. The census below runs s(n) on every start from 1 to 100,000, iterated to a budget of 500 steps or a term-size of 10¹⁵, and sorts every trajectory into one of five bins by fate. The whole run takes a little over a minute on a laptop; the JSON it writes is what this page reads. Nothing here is estimated.

The four small fates absorb the overwhelming majority: eighty-four percent terminate outright, and the perfect, amicable, and sociable numbers of the census together account for a further two percent. What is left over, the fourteen and a third percent still climbing when the budget cap fires, is a mixture of two things this page is careful to distinguish. Some of those trajectories would settle at a bigger budget; some would not settle at any budget the world has ever tried. The census cannot tell the two apart on its own. What it can do is name where in the number line to look for the pure case, and that turns out to be everywhere but especially under a thousand.

The small basins, in full

Every fixed point, every amicable pair, and every cycle whose smallest member is under 100,000, along with the count of starts whose trajectory ends there. Click any number to load its trajectory into the operator above.

The twenty-eight-step chain at 14,316

The longest sociable cycle inside the census, and the first one anyone found. Paul Poulet published both known cycles of length greater than two in a 1918 note in L'Intermediaire des Mathematiciens; the five-step chain and the twenty-eight-step chain both come from that paper, and no cycle of length between three and four inclusive has ever been found (no length-3 cycle can exist below 10¹⁵, per computer search). The twenty-eight cycle at 14,316 is smaller than every known sociable cycle of any length not equal to 2, and its longest member sits above six hundred thousand: the trajectory leaves the census before it comes back.

The Lehmer five, watched from the outside

Under a thousand the census marks seventeen starts still climbing at its budget. Twelve of those have been shown, by public projects that carry aliquot sequences to depths this page cannot match (aliquot.de; FactorDB), to terminate at greater depth. Five have not. In 1953 Derrick Henry Lehmer named the ones that were open then: 276, 552, 564, 660, 966. Seventy-three years on the list has not changed. The plot below shows how far the deep runner on this laptop gets before its budget fires (200 steps, or a term above 10⁴⁰); the classical counter-example, the start 138, whose trajectory peaks near 10¹¹ before turning around and terminating at step 178, is drawn beside them.

What is open, and what is only open here

The census budget on this page ends every open trajectory at term size 10¹⁵. That budget was chosen so the whole run finishes in under two minutes and every number the page prints can be reproduced from scratch by a reader who has this repository. It is not the budget the world has tried. Public projects have tracked the Lehmer five to depths well past two thousand, term sizes past 10²⁰⁰; they are still climbing. Every other start under a thousand has, at some depth, been shown to settle. Nothing about this is a proof either way; both directions have been conjectured, and neither has been settled. The Catalan-Dickson conjecture says every trajectory eventually terminates or cycles; the Guy-Selfridge conjecture, made after long computation, says many trajectories diverge. Whichever conjecture is correct, 276 is where the argument starts.

Perfect numbers appear to be an even more one-sided open question. Every one found has been of Euclid's form 2^p-1(2^p-1) with 2^p-1 prime, and nobody has ever produced an odd perfect number nor proved they cannot exist. The four found in this census (6, 28, 496, 8128) are the first four terms of an infinite conjectural sequence, all of whose members are believed to correspond to Mersenne primes.

Primary sources. D. H. Lehmer, "On a Conjecture of Catalan", American Journal of Mathematics, vol. 75 (1953), 89-104, is where the five open starts under a thousand are named. Paul Poulet, L'Intermediaire des Mathematiciens 25 (1918), 100-101, gives both sociable cycles found here (length 5 at 12,496; length 28 at 14,316). R. K. Guy, Unsolved Problems in Number Theory, section B6, and J. L. Selfridge's follow-up (1976) are where the divergence conjecture is worked out. The OEIS entries A005114 (untouched by the aliquot map) and A216072 (open aliquot sequence starts) hold the community's running lists. The current best public traces of the Lehmer five, past two thousand terms with the top term past 10²⁰⁰, live at aliquotes.com, hosted by Jean-Luc Garambois and colleagues; a longer-running effort at aliquot.de ceased in 2019 and its traces are preserved.

Reproducibility. One command, offline: node research/aliquot-frontier/verify.mjs. It runs the full census (~75s), the deep Lehmer pass (~75s), and asserts every headline number the page prints. Every number here reads from public/strata/the-first-number-nobody-can-follow/data.json, written by research/aliquot-frontier/page-data.mjs from files the verifier itself writes.