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Concept
Aliquot sequence
Summary
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
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
sn = s(sn−1) = σ1(sn−1) − sn−1 if sn−1 > 0
sn = 0 if sn−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.
For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:
σ1(10) − 10 = 5 + 2 + 1 = 8,
σ1(8) − 8 = 4 + 2 + 1 = 7,
σ1(7) − 7 = 1,
σ1(1) − 1 = 0.
Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:
A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ...
An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ...
A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ...
Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers.
Official source
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