Aliquot sequence
From Encyclopedia of Mathematics
starting from
The sequence of natural numbers defined by the rule
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The sequence is said to be terminating if for some
and eventually periodic if there is a
such that
for
sufficiently large. If
, then
is a perfect number, while if
, then
and
form an amicable pair (cf. also Amicable numbers).
An example of an eventually periodic aliquot sequence is the sequence . Larger cycles are possible; e.g., a sequence with cycle length
is known.
The Catalan–Dickson conjecture states that all aliquot sequences either terminate or are eventually periodic. This conjecture is still (1996) open, but generally thought to be false.
References
[a1] | H.J.J. te Riele, "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam (1976) |
How to Cite This Entry:
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=33283
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=33283
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article