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:^[1]

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.^[2] 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^[update] 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.^[3]

As of April 2012^[update], there were 901 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9231 such integers less than 1,000,000.^[4]

External links

• Tables of Aliquot Cycles (J.O.M. Pedersen)
• Aliquot Page (Wolfgang Creyaufmüller)
• Aliquot sequences (Christophe Clavier)
• Aliquot Sequences from the Trenches (Clifford Stern)
• Forum on calculating aliquot sequences (MersenneForum)
• Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges) (Karsten Bonath)
• Active research site on aliquot sequences (Jean-Luc Garambois)

References

v t e Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Forms of factorization Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number Constrained divisor sums Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Primitive semiperfect number Practical number With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Related to aliquot sequences Untouchable number Amicable number Sociable number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number