Aliquot sequences are defined thus :

The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ₁ or the aliquot sum function s in the following way:
s₀ = k
s_n = s(s_n−1) = σ₁(s_n−1) − s_n−1 if s_n−1 > 0
s_n = 0 if s_n−1 = 0
(if we add this condition, then the terms after 0 are all 0, and all aliquot sequences would be infinite sequence, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6) and s(0) is undefined.

Source : Wikipedia

It's conjectured - but not yet proved - that all aliquot sequences will eventually terminate.

All sequences so far tested do terminate - but it nevertheless remains unknown if all examples will.

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