Aliquot Sequence

In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ₁ in the following way:

s₀ = k

s_n = σ₁(s_n−1) − s_n−1.

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 (sequence A080907 in OEIS); 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). 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 ( A063769).

An important conjecture due to Catalan with respect to aliquot sequences is that every aliquot sequence ends in one of the above ways–with a prime number, a perfect number, or a set of amicable or sociable numbers. The alternative would be that a number exists whose aliquot sequence is infinite, yet aperiodic. There are several numbers whose aliquot sequences as of 2010 have not been fully determined, and thus might be such a number. The first five candidate numbers are called the Lehmer five (named after Dick Lehmer): 276, 552, 564, 660, and 966.

As of November 2012, there were 900 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9224 such integers less than 1,000,000.