In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an aliquot sequence is a sequence of positive integers in which each term is the sum of the
proper divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Definition and overview
The
aliquot
Aliquot ( la, a few, some, not many) may refer to:
Mathematics
* Aliquot part, a proper divisor of an integer
*Aliquot sum, the sum of the aliquot parts of an integer
*Aliquot sequence, a sequence of integers in which each number is the aliquot su ...
sequence starting with a positive integer ''k'' can be defined formally in terms of the
sum-of-divisors function σ
1 or the
aliquot sum
In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself.
That is,
:s(n)=\sum\nolimits_d.
It can be used to characterize the prime ...
function ''s'' in the following way:
: ''s''
0 = ''k''
: ''s''
n = ''s''(''s''
''n''−1) = σ
1(''s''
''n''−1) − ''s''
''n''−1 if ''s''
''n''−1 > 0
: ''s''
n = 0 if ''s''
''n''−1 = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all aliquot sequences would be infinite sequence, and we can conjecture that all aliquot sequences are
convergent, the limit of these sequences are usually 0 or 6)
and ''s''(0) is undefined.
For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:
:σ
1(10) − 10 = 5 + 2 + 1 = 8,
:σ
1(8) − 8 = 4 + 2 + 1 = 7,
:σ
1(7) − 7 = 1,
:σ
1(1) − 1 = 0.
Many aliquot sequences terminate at zero; all such sequences necessarily end with a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See 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
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ...
* An
amicable number
Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive d ...
has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ...
* A
sociable number
In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and ...
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 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 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.
The lengths of the aliquot sequences that start at ''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, ...
The final terms (excluding 1) of the aliquot sequences that start at ''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, ...
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, ...
Numbers whose aliquot sequence known to terminate in a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
, 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, ...
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, ...
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, ...
A number that is never the successor in an aliquot sequence is called an
untouchable number
An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. ...
.
:
2,
5,
52,
88,
96,
120,
124,
146,
162,
188,
206,
210,
216,
238
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Year 238 ( CCXXXVIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Pius and Pontianus (or, less frequently, year 991 ''Ab ...
,
246
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Year 246 ( CCXLVI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar, the 246th Year of the Common Era ( CE) and Anno Domini ( AD) designations, the 246th year of the 1st millennium, th ...
,
248, 262, 268,
276
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Year 276 ( CCLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tacitus and Aemilianus (or, less frequently, year 1029 ...
,
288
Year 288 ( CCLXXXVIII) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Maximian and Ianuarianus (or, less frequently, year 1041 ...
,
290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ...
Catalan–Dickson conjecture
An important
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
due to
Catalan
Catalan may refer to:
Catalonia
From, or related to Catalonia:
* Catalan language, a Romance language
* Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia
Places
* 13178 Catalan, asteroid #1 ...
, 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. 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
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Year 276 ( CCLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tacitus and Aemilianus (or, less frequently, year 1029 ...
, 552, 564, 660, and 966. However, it is worth noting that 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)).
, there were 898 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9190 such integers less than 1,000,000.
Systematically searching for aliquot sequences
The aliquot sequence can be represented as a
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
,
, for a given integer
, where
denotes the sum of the proper divisors of
.
in
represent sociable numbers within the interval