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aliquot sequence (Definition)

For a given $ m$, define the recurrence relation $ a_1 = m$, $ a_n = \sigma(a_{n - 1}) - a_{n - 1}$, where $ \sigma(x)$ is the sum of divisors function. $ a$ is then the aliquot sequence of $ m$.

If $ m$ is an amicable number, its aliquot sequence is periodic, alternating between the abundant and deficient member of the amicable pair. For a prime number $ p$, its aliquot sequence is $ p, 1, 0$. In other cases, the aliquot sequence reaches a fixed point upon 0, or on a perfect number.



"aliquot sequence" is owned by CompositeFan.
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examples of aliquot sequences (Example) by PrimeFan
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Cross-references: perfect number, fixed point, prime number, alternating, periodic, amicable number, sum of divisors function, recurrence relation
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This is version 2 of aliquot sequence, born on 2006-07-28, modified 2006-08-09.
Object id is 8189, canonical name is AliquotSequence.
Accessed 972 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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