Aliquot sequence

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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:

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. (sequence in 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 (). 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, .... Numbers that like 95 are not perfect, but eventually their aliquot sequence becomes a repeating sequence of period 1 are called aspiring numbers. ().
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.

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).

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)

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.

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Aliquot sequence

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