====== 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 σ<sub>1</sub> or the aliquot sum function //s// in the following way:\\ //s<sub>0</sub> = k//\\ //s<sub>n</sub> = s(s<sub>n−1</sub>) = σ<sub>1</sub>(s<sub>n−1</sub>) − s<sub>n−1</sub> if s<sub>n−1</sub> > 0//\\ //s<sub>n</sub> = 0 if s<sub>n−1</sub> = 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 : [[https://en.wikipedia.org/wiki/Aliquot_sequence|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, [[https://www.unirioja.es/cu/jvarona/aliquot.html|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.'