Aliquot sequences

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

For some small starting numbers it has been demonstrated that the sequences can eventually terminate (e.g. the one beginning from 138) - but there is disagreement in the mathematics community about whether all aliquot sequences will eventually come to an end - or whether some go on to infinity, i.e. are 'unbound'.

For further information on the sequences, see the work of Dr. Juan Luis Varona of the Department of Mathematics and Computer Science, University of La Rioja, Spain.

Note : In mathematics, the word 'Aliquot' means: 'Signifying, or relating to an exact divisor of a quantity or number.'