An [[wp:Aliquot sequence|aliquot sequence]] of a positive integer K is defined recursively as the first member
being K and subsequent members being the sum of the [[Proper divisors]] of the previous term.

:* If the terms eventually reach 0 then the series for K is said to '''terminate'''.

:<br>There are several classifications for non termination:
:* If the second term is K then all future terms are also K and so the sequence repeats from the first term with period 1 and K is called '''perfect'''.
:* If the third term ''would'' be repeating K then the sequence repeats with period 2 and K is called '''amicable'''.
:* If the N<sup>th</sup> term ''would'' be repeating K for the first time, with N > 3 then the sequence repeats with period N - 1 and K is called '''sociable'''.
:<br>Perfect, amicable and sociable numbers eventually repeat the original number K; there are other repetitions...
:* Some K have a sequence that eventually forms a periodic repetition of period 1 but of a number other than K, for example 95 which forms the sequence <code>95, 25, 6, 6, 6, ...</code> such K are called '''aspiring'''.
:* K that have a sequence that eventually forms a periodic repetition of period >= 2 but of a number other than K, for example 562 which forms the sequence <code>562, 284, 220, 284, 220, ...</code> such K are called '''cyclic'''.

:<br>And finally:
:* Some K form aliquot sequences that are not known to be either terminating or periodic; these K are to be called '''non-terminating'''. <br>For the purposes of this task, K is to be classed as non-terminating if it has not been otherwise classed after generating '''16''' terms or if any term of the sequence is greater than 2**47 = 140,737,488,355,328.


;Task:
# Create routine(s) to generate the aliquot sequence of a positive integer enough to classify it according to the classifications given above.
# Use it to display the classification and sequences of the numbers one to ten inclusive.
# Use it to show the classification and sequences of the following integers, in order:
:: 11, 12, 28, 496, 220, 1184,  12496, 1264460, 790, 909, 562, 1064, 1488, and optionally 15355717786080.

Show all output on this page.


;Related tasks:
* &nbsp; [[Abundant, deficient and perfect number classifications]]. (Classifications from only the first two members of the whole sequence).
* &nbsp; [[Proper divisors]]
* &nbsp; [[Amicable pairs]]
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