content:mathematics:aliquot_sequences

# Wikenigma - an Encyclopedia of Unknowns Wikenigma - an Encyclopedia of the Unknown

# Aliquot sequences

Aliquot sequences are defined thus :

The aliquot sequence starting with a positive integerkcan be defined formally in terms of the sum-of-divisors function ฯ_{1}or the aliquot sum functionsin the following way:

s_{0}= k

s_{n}= s(s_{nโ1}) = ฯ_{1}(s_{nโ1}) โ s_{nโ1}if s_{nโ1}> 0

s_{n}= 0 if s_{nโ1}= 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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