Pick a positive integer, list the positive integers that will divide it evenly, add these up, and subtract the number itself:

• 10 is evenly divisible by 10, 5, 2, and 1. (10 + 5 + 2 + 1) – 10 = 8.

Now do the same with that number, and continue:

• 8 is evenly divisible by 8, 4, 2, and 1. (8 + 4 + 2 + 1) – 8 = 7.
• 7 is evenly divisible by 7 and 1. (7 + 1) – 7 = 1.
• 1 is evenly divisible only by 1. (1) – 1 = 0.

Many of these sequences arrive at some resolution — they terminate in a constant, or an alternating pair, or some regular cycle. But it’s an open question whether all of them do this. The fate of the aliquot sequence of 276 is not known; by step 469 it’s reached 149384846598254844243905695992651412919855640, but possibly it reaches some apex and then descends again and finds some conclusion (the sequence for the number 138 reaches a peak of 179931895322 but eventually returns to 1). Do all numbers eventually reach a resolution? For now, no one knows.