Difference between revisions of "Aliquot sequence"
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The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule | 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_1=n,\quad a_k=\left(\sum_{d|a_{k-1}}d\right)-a_{k-1}.$$ |
The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a [[Perfect number|perfect number]], while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also [[Amicable numbers|Amicable numbers]]). | The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a [[Perfect number|perfect number]], while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also [[Amicable numbers|Amicable numbers]]). | ||
Revision as of 21:31, 15 November 2014
starting from $n$
The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule
$$a_1=n,\quad a_k=\left(\sum_{d|a_{k-1}}d\right)-a_{k-1}.$$
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) |
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=34544