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Difference between revisions of "Aliquot sequence"

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''starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105202.png" />''
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''starting from $n$''
  
The sequence of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105203.png" /> defined by the rule
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The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105204.png" /></td> </tr></table>
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$$a_1=n,\quad a_k=\left(\sum_{d|a_{k-1}}d\right)-a_k.$$
  
The sequence is said to be terminating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105205.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105206.png" /> and eventually periodic if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105207.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105208.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a1105209.png" /> sufficiently large. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052011.png" /> is a [[Perfect number|perfect number]], while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052014.png" /> form an amicable pair (cf. also [[Amicable numbers|Amicable numbers]]).
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The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a [[Perfect number|perfect number]], while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also [[Amicable numbers|Amicable numbers]]).
  
An example of an eventually periodic aliquot sequence is the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052015.png" />. Larger cycles are possible; e.g., a sequence with cycle length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110520/a11052016.png" /> is known.
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An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Larger cycles are possible; e.g., a sequence with cycle length $28$ 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.
 
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.

Revision as of 13:21, 14 September 2014

starting from $n$

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule

$$a_1=n,\quad a_k=\left(\sum_{d|a_{k-1}}d\right)-a_k.$$

The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a perfect number, while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also Amicable numbers).

An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Larger cycles are possible; e.g., a sequence with cycle length $28$ 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
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article