In mathematics, an aliquot sequence is a recursive sequence which can be defined in the following way: if we write σ(n) = σ₁(n) to be the divisor function normally, then, the aliquot sequence of k can be written: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ...

s₀ = k

s_n = σ(s_n−1) − s_n−1

For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0.

Many aliquot sequences terminate; all such sequences terminate with 1, 0. (A080907 in the OEIS). 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 (See A000396 in the OEIS). The aliquot sequence of 6, for example, is 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, ....
• It has not been determined whether an aliquot sequence could be infinitely long and aperiodic. There are several numbers whose aliquot sequences have not been fully determined; the first five are called the 'Lehmer Five': 276, 552, 564, 660, and 966.

An important conjecture due to Catalan with respect to aliquot sequences is that every aliquot sequence ends in a prime number, perfect number, or a set of sociable numbers. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ... Eugène Charles Catalan Eugène Charles Catalan (May 30, 1814 - February 14, 1894) was a Belgian mathematician. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ...

There are now 913 open-end sequences in [1, 10⁵] and 9474 OE-sequences in [1, 10⁶]. A reduction in these numbers is possible from further calculations (July 2005).

See also

• Aliquot
• Aliquant

An aliquot part (or simply aliquot), in the context of mathematics, is an integer that is an exact divisor of a quantity. ... An aliquant part (or simply aliquant) is an integer that is not an exact divisor of a given quantity. ...

External links

• Aliquot Pages (W. Creyaufmüller)
• Lehmer Five (W. Creyaufmüller)
• Tables of Aliquot Cycles (J.O.M. Pedersen)
• Catalan's Aliquot Sequence Conjecture from MathWorld (E.W. Weisstein)

MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ...

References

• [1] Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. Aliquot Sequence 3630 Ends After Reaching 100 Digits. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201-206.
• [2] W. Creyaufmüller. Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail. Stuttgart 2000 (3rd ed.), 327p.