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(cite te Riele (1973))
(Aliquot cycles of length greater than 2 are termed ''sociable 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 all $n$ 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]]).
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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 all $n$ 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|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.
 
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.
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<table>
 
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J.J. te Riele,  "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam  (1976)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J.J. te Riele,  "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam  (1976)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J.J. te Riele,  "A Note on the Catalan–Dickson Conjecture" , ''Mathematics of Computation'' '''27''' No. 121 (1973) 189-192. DOI 10.2307/2005261 URL  www.jstor.org/stable/2005261</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J.J. te Riele,  "A Note on the Catalan–Dickson Conjecture" , ''Mathematics of Computation'' '''27''' No.121 (1973) 189-192. DOI 10.2307/2005261 URL  www.jstor.org/stable/2005261</TD></TR>
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</table>
  
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====Comments====
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Aliquot cycles of length greater than 2 are termed ''sociable numbers''.
  
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The aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]).  The cycle of length 28 starts at $n=14316$ (ref [b2]).
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====References====
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<table>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Benito, Manuel; Varona, Juan L. "Advances in aliquot sequences", ''Mathematics of Computation'' '''68''', No.225 (1999) 389-393. DOI:10.1090/S0025-5718-99-00991-6 Zbl 0957.11060</TD></TR>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  P. Poulet, "Question 4865", ''L'interméd. des Math.' '''25''' (1918) 100–101</TD></TR>
 
</table>
 
</table>
 
 
[[Category:Number theory]]
 
[[Category:Number theory]]

Revision as of 21:58, 15 November 2014

starting from $n$

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule $a_1 = n$, $a_{k+1} = s(a_k)$ where $s(a)$ is the sum of aliquot divisor function $$ s(a) = \sum_{d|a}d - a \ . $$

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 all $n$ 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)
[a1] H.J.J. te Riele, "A Note on the Catalan–Dickson Conjecture" , Mathematics of Computation 27 No.121 (1973) 189-192. DOI 10.2307/2005261 URL www.jstor.org/stable/2005261

Comments

Aliquot cycles of length greater than 2 are termed sociable numbers.

The aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]). The cycle of length 28 starts at $n=14316$ (ref [b2]).

References

[b1] Benito, Manuel; Varona, Juan L. "Advances in aliquot sequences", Mathematics of Computation 68, No.225 (1999) 389-393. DOI:10.1090/S0025-5718-99-00991-6 Zbl 0957.11060
[b1] P. Poulet, "Question 4865", L'interméd. des Math.' 25 (1918) 100–101
How to Cite This Entry:
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=34547
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article