Aliquot sequence#Catalan-Dickson conjecture

The aliquot sequence starting with a positive integer {{mvar|k}} can be defined formally in terms of the sum-of-divisors function {{math|σ₁}} or the aliquot sum function {{mvar|s}} in the following way:{{MathWorld | urlname=AliquotSequence | title=Aliquot Sequence}}

$$\backslash begin\{align\}$$

s_0 &= k \\[4pt]

s_n &= s(s_{n-1}) = \sigma_1(s_{n-1}) - s_{n-1} \quad \text{if} \quad s_{n-1} > 0 \\[4pt]

s_n &= 0 \quad \text{if} \quad s_{n-1} = 0 \\[4pt]

s(0) &= \text{undefined}

\end{align}

If the {{math|1=s{{sub|n-1}} = 0}} condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6.

For example, the aliquot sequence of 10 is {{nowrap|10, 8, 7, 1, 0}} because:

$$\backslash begin\{align\}$$

\sigma_1(10) -10 &= 5 + 2 + 1 = 8, \\[4pt]

\sigma_1(8) - 8 &= 4 + 2 + 1 = 7, \\[4pt]

\sigma_1(7) - 7 &= 1, \\[4pt]

\sigma_1(1) - 1 &= 0.

\end{align}

Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See {{OEIS|id=A080907}} for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:

• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is {{nowrap|6, 6, 6, 6, ...}}
• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is {{nowrap|220, 284, 220, 284, ...}}
• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is {{nowrap|1264460, 1547860, 1727636, 1305184, 1264460, ...}}
• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is {{nowrap|95, 25, 6, 6, 6, 6, ...}} Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers.{{Cite OEIS|1=A063769|2=Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number. }}

The lengths of the aliquot sequences that start at {{mvar|n}} are

:1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... {{OEIS|id=A044050}}

The final terms (excluding 1) of the aliquot sequences that start at {{mvar|n}} are

:1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... {{OEIS|id=A115350}}

Numbers whose aliquot sequence terminates in 1 are

:1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... {{OEIS|id=A080907}}

Numbers whose aliquot sequence known to terminate in a perfect number, other than perfect numbers themselves (6, 28, 496, ...), are

:25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... {{OEIS|id=A063769}}

Numbers whose aliquot sequence terminates in a cycle with length at least 2 are

:220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... {{OEIS|id=A121507}}

Numbers whose aliquot sequence is not known to be finite or eventually periodic are

:276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... {{OEIS|id=A131884}}

A number that is never the successor in an aliquot sequence is called an untouchable number.

:2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... {{OEIS|id=A005114}}

An important conjecture due to Catalan, sometimes called the Catalan–Dickson conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers.{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}} The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the Lehmer five (named after D.H. Lehmer): 276, 552, 564, 660, and 966.{{cite web|url=http://www.aliquot.de/lehmer.htm|title=Lehmer Five|first=Wolfgang|last=Creyaufmüller|date=May 24, 2014|access-date=June 14, 2015}} However, 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1.

Guy and Selfridge believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are unbounded above (i.e., diverge)).A. S. Mosunov, [http://www.cs.uleth.ca/~hadi/2016-09-29-aliquot_sequences.pdf What do we know about aliquot sequences?]

The aliquot sequence can be represented as a directed graph, $G\_\{n,s\}$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$.{{citation|title=Distributed cycle detection in large-scale sparse graphs|first1=Rodrigo Caetano|last1=Rocha|first2=Bhalchandra|last2=Thatte|year=2015|publisher=Simpósio Brasileiro de Pesquisa Operacional (SBPO)|doi=10.13140/RG.2.1.1233.8640}} Cycles in $G\_\{n,s\}$ represent sociable numbers within the interval $[1,n]$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

• Arithmetic dynamics

{{reflist}}

{{refbegin}}

• Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. [https://web.archive.org/web/20041015194432/http://www.expmath.org/expmath/volumes/11/11.2/3630finishes1.pdf Aliquot Sequence 3630 Ends After Reaching 100 Digits]. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201–206.
• W. Creyaufmüller. Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail. Stuttgart 2000 (3rd ed.), 327p.

{{refend}}

• [http://www.rechenkraft.net/aliquot/AllSeq.html Current status of aliquot sequences with start term below 4 million]
• [https://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm Tables of Aliquot Cycles] (J.O.M. Pedersen)
• [http://www.aliquot.de/aliquote.htm Aliquot Page] (Wolfgang Creyaufmüller)
• [http://christophe.clavier.free.fr/Aliquot/site/Aliquot.html Aliquot sequences] (Christophe Clavier)
• [http://www.mersenneforum.org/forumdisplay.php?f=90 Forum on calculating aliquot sequences] (MersenneForum)
• [http://www.rieselprime.de/Others/Aliquot000.htm Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges)] (Karsten Bonath)
• [http://www.aliquotes.com Active research site on aliquot sequences] (Jean-Luc Garambois) {{in lang|fr}}

{{Divisor classes}}

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Category:Arithmetic functions

Category:Divisor function

Category:Arithmetic dynamics