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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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{{TEX|done}}{{MSC|11A25}}
  
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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''starting from $n$''
  
<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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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]]s function
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$$
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s(a) = \sum_{d|a}d - a \ .
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$$
  
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 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]]): aliquot cycles of length greater than 2 are also termed ''sociable 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 $562,220,284,220,\dots$. Longer cycles are known; e.g., a sequence with cycle length $28$, starting at $n=14316$ (ref [b2]).
  
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.
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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 aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]). 
  
 
====References====
 
====References====
<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></table>
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<table>
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<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>
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<TR><TD valign="top">[a2]</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}}</TD></TR>
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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">[b2]</TD> <TD valign="top">  P. Poulet, "Question 4865", ''L'interméd. des Math.'' '''25''' (1918) 100–101</TD></TR>
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<TR><TD valign="top">[b3]</TD> <TD valign="top">  Richard K. Guy, ''Unsolved Problems in Number Theory'' 3rd ed. Springer-Verlag (2004) {{ISBN|0-387-20860-7}} {{ZBL|1058.11001}}</TD></TR>
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</table>

Latest revision as of 14:02, 12 November 2023

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

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 divisors 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): aliquot cycles of length greater than 2 are also termed sociable numbers.

An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Longer cycles are known; e.g., a sequence with cycle length $28$, starting at $n=14316$ (ref [b2]).

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 aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]).

References

[a1] H.J.J. te Riele, "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam (1976)
[a2] 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
[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
[b2] P. Poulet, "Question 4865", L'interméd. des Math. 25 (1918) 100–101
[b3] Richard K. Guy, Unsolved Problems in Number Theory 3rd ed. Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001
How to Cite This Entry:
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=16085
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article