Aliquot sequences

Add up a number's divisors, leaving out the number itself, and you get the next number. Then do it again. 12 goes to 1 + 2 + 3 + 4 + 6 = 16, then 15, 9, 4, 3, 1 and 0. Most numbers fall to 0 like that, and a few loop forever. 276 climbs and climbs, and nobody knows if it ever comes down.

Point at the line to read off a step.

Any whole number up to 15 digits. Each step means factoring a number: past 30 digits or so a step can take seconds, then minutes, then far longer.

    The first thousand

    Every start from 1 to 1,000, followed until it reaches 0, falls into a loop, or passes 25 digits. Point at a square to see where it goes, and tap it to follow it in the machine.

    They all walk together, faster and faster. Watch which squares stay empty.

    • 0 within 5 steps
    • Within 10
    • Within 20
    • Within 50
    • Within 100
    • More than 100
    • A loop
    • Too high to follow here
    • Nobody knows

    276 is the first start nobody has followed to the end.

    Where they go

    Most sequences hit a prime, whose only divisor below itself is 1, and 1 goes to 0. Some land on a perfect number like 6 and stay there. Some fall into a pair: the divisors of 220 add up to 284, and those of 284 add up to 220. Amicable pairs like that were known to the Pythagoreans, yet the second smallest, 1,184 and 1,210, was overlooked until a 16-year-old, Nicolò Paganini (not the violinist), found it in 1867. In 1918 Paul Poulet found longer loops: 12,496 comes back to itself after 5 steps, and 14,316 after 28.

    Below 1,000, every start ends one of those ways except twelve: the Lehmer five, 276, 552, 564, 660 and 966, and seven more that run into them: 306, 396 and 696 into 276, 780 into 564, 828 and 996 into 660, and 888 into 552. By October 2026, computers had followed 276 for 2,162 steps, to a 215-digit number. The next step needs a 210-digit number factored, and nobody has managed it yet. 840 climbs past 25 digits too, but it comes back down: it reaches 0 after 749 steps.

    And nothing leads to 276: no number's divisors add up to it, which makes it one of the untouchable numbers, like 2, 5, 52, 88 and 96.

    The Catalan–Dickson conjecture says every aliquot sequence ends or loops. Richard Guy and John Selfridge doubted it, and thought some climb forever. Nobody has proved it either way.