Aliquot sequence
Encyclopedia
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:
For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:
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). There are a variety of ways in which an aliquot sequence might not terminate:
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 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.
, there were 902 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9259 such integers less than 1,000,000.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
σ1 in the following way:
- s0 = k
- sn = σ1(sn−1) − sn−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 ; all such sequences necessarily end with a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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 numberPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... - An amicable numberAmicable numberAmicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... - A sociable numberSociable numberSociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself...
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 .
An important conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
due to Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...
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 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
Derrick Henry Lehmer
Derrick Henry "Dick" Lehmer was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes...
): 276
276 (number)
Two hundred seventy-six is the natural number following two hundred seventy-five and preceding two hundred seventy-seven.-Mathematical properties:276 = 15 + 25 + 35...
, 552, 564, 660, and 966.
, there were 902 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9259 such integers less than 1,000,000.
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)