Aliquot sequence
An aliquot sequence is a sequence of positive integers defined by starting with an initial number n > 0 and iteratively applying the aliquot sum function s(k), which computes the sum of the proper divisors of k (all positive divisors excluding k itself), given by s(k) = \sigma(k) - k where \sigma(k) is the sum of all positive divisors of k.[1] Thus, the sequence is a_1 = n, a_{k+1} = s(a_k) for k \geq 1.[1] Aliquot sequences trace their origins to ancient number theory, particularly the Pythagorean study of perfect numbers, where s(n) = n, forming a cycle of length 1, with the smallest example being 6 since s(6) = 1 + 2 + 3 = 6.[2] They also encompass amicable pairs, such as 220 and 284, where s(220) = 284 and s(284) = 220, creating a cycle of length 2, and sociable numbers that form longer cycles, like the 28-term cycle starting at 14316.[2] Many sequences terminate by reaching a prime p (where s(p) = 1), followed by s(1) = 0, effectively ending at 0.[2] The behavior of aliquot sequences remains a central topic in analytic number theory, with possibilities including termination, periodic cycling, or unbounded growth.[2] The Catalan–Dickson conjecture, proposed by Eugène Charles Catalan in 1888 and extended by Leonard Eugene Dickson in 1913, asserts that every aliquot sequence either terminates or enters a cycle, but this is widely doubted following computational evidence of diverging sequences, such as those starting at 276, which grow without bound.[2] In 1975, H. W. Lenstra Jr. proved the existence of strictly increasing aliquot sequences of any finite length k, while Paul Erdős showed in 1976 that the set of numbers n for which the aliquot sequence strictly increases for k steps has asymptotic density zero.[2] Statistical analyses indicate that, on average, terms in sequences starting from even numbers tend to decrease slightly (geometric mean ratio \approx e^{-0.033}), but for multiples of 4, they grow moderately (\approx e^{0.175}), supporting the likelihood of unbounded growth in certain cases.[2] Open questions abound, including whether there are infinitely many aliquot cycles beyond known lengths, the existence of cycles of length 3, and the asymptotic density of sociable numbers, which is conjectured to be zero. As of 2025, the longest known sociable cycle remains the 28-term one, and no cycles of length 3 have been found.[2] Computational efforts, such as the Aliquot Sequences Project, have explored sequences up to enormous terms (e.g., over 100 digits for starting value 3630, which terminates), revealing no new cycles but confirming divergences.[3] These sequences connect to broader themes in divisor theory, including untouchable numbers (those not equal to s(m) for any m), whose density is heuristically around 0.17.[1]Fundamentals
Definition
An aliquot sequence begins with a positive integer n > 0 and is constructed iteratively by applying the aliquot sum function, which computes the sum of the proper divisors of the previous term. The proper divisors of a number are all positive divisors excluding the number itself.[4][5] The foundation of this construction relies on the divisor function \sigma(n), which gives the sum of all positive divisors of n. For example, the divisors of 10 are 1, 2, 5, and 10, so \sigma(10) = 18. The aliquot sum s(n) is then defined as s(n) = \sigma(n) - n, yielding s(10) = 8 in this case.[4] Formally, the aliquot sequence starting from n is the sequence of terms s^k(n) for k = 0, 1, 2, \dots, where s^0(n) = n, s^1(n) = s(n), and s^k(n) = s(s^{k-1}(n)) for k \geq 2. By convention, the sequence terminates at 0 upon reaching 1, since the proper divisors of 1 are none, so s(1) = 0.[4][5]Basic properties
The terms in an aliquot sequence are classified based on their abundance relative to the aliquot sum function s(n): a term n is deficient if s(n) < n, perfect if s(n) = n, or abundant if s(n) > n. This classification influences the sequence's progression, with sequences often exhibiting alternations between these categories depending on the primality or prime factorization of the terms; for instance, prime numbers and prime powers tend to produce deficient terms leading to decreases, while highly composite numbers can yield abundant terms that cause increases.[4][2] Aliquot sequences can be conceptualized as directed paths in an infinite graph whose nodes are the positive integers and whose edges connect each integer n to s(n). This graphical representation highlights the structure of the sequences as trajectories traversing the graph, potentially merging with other paths or entering cycles.[6] A fundamental property of aliquot sequences starting from even integers n > 0 is that they eventually reach an odd integer or enter a cycle (with known cycles being odd). This occurs because s(n) \equiv n \pmod{2} unless n is a square or twice a square, in which case the parity changes; thus, even sequences remain even until such a form is encountered, transitioning to odd. For odd n, s(n) remains odd unless n is an odd square, in which case s(n) is even.[2] For powers of 2, the aliquot sum admits a closed form: s(2^k) = 2^k - 1 for k \geq 1. To derive this, note that the proper divisors of $2^k are $1, 2, 4, \dots, 2^{k-1}, forming a geometric series whose sum is $1 + 2 + \cdots + 2^{k-1} = 2^k - 1. Thus, the sequence starting at $2^k proceeds as $2^k, 2^k - 1, s(2^k - 1), \dots, where $2^k - 1 (a Mersenne number) is odd and deficient, leading to a rapid descent to 1 and then 0, ensuring termination.[4]Examples and Classifications
Terminating sequences
Terminating aliquot sequences are those that eventually reach 1 and then 0, effectively ending the process. This occurs when the sequence encounters a prime number p, for which the sum of proper divisors s(p) = 1, followed by s(1) = 0. All prime numbers and 1 terminate immediately in this manner, as their proper divisor sums are trivially 1 and 0, respectively.[2] A concrete example is the aliquot sequence starting at 10: s(10) = 1 + 2 + 5 = 8, s(8) = 1 + 2 + 4 = 7, s(7) = 1, and s(1) = 0. Here, the sequence descends through composite numbers until hitting the prime 7, after which it terminates rapidly. This illustrates the typical descent in terminating sequences, where each step reduces the value until a prime is reached.[2] Certain numbers, known as untouchable or nonaliquot numbers, cannot appear as terms in any aliquot sequence except possibly as starting points, because no integer has them as its sum of proper divisors. Examples include 2, 5, and 52; despite this property, most starting values, including many untouchables, lead to terminating sequences.[2] Early computations of aliquot sequences, beginning with tabulations by Leonard Eugene Dickson in 1913 and extensive work by Paul Poulet in 1918, revealed that the vast majority of small starting values terminate. For instance, powers of 2 terminate via their Mersenne number counterparts: s(2^k) = 2^k - 1, which either is prime (leading directly to 1) or factors into smaller terms that continue descending to a prime.[7][8] Sequences originating from deficient numbers—those with abundance measure d(n) = s(n)/n < 1—often terminate quickly, as the terms strictly decrease toward smaller values until a prime is encountered. This descent is a key characteristic driving termination in such cases.[7]Cyclic sequences
In an aliquot sequence, a cycle occurs when the sequence returns to a previously encountered term, thereby forming a periodic loop of length k \geq 1. Such cycles represent bounded behaviors distinct from sequences that terminate at 0.[9] Cycles are classified by their length. For k=1, the sequence is fixed at a perfect number n, where the sum of proper divisors s(n) = n. The smallest such number is 6, whose proper divisors are 1, 2, and 3, summing to 6; thus, the sequence is 6 → 6 → ⋯. All known perfect numbers are even and of the form $2^{p-1}(2^p - 1) for prime p where $2^p - 1 is a Mersenne prime; 52 such numbers are known as of 2025.[2][10] For k=2, the cycle consists of an amicable pair (m, n) with m \neq n, s(m) = n, and s(n) = m. The smallest pair is (220, 284), discovered in antiquity and attributed to Iamblichus around 300 CE. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, summing to 284; those of 284 are 1, 2, 4, 71, and 142, summing to 220. Thus, the sequence cycles as 220 → 284 → 220 → ⋯. Over 1.2 billion amicable pairs are known as of 2023, with ongoing searches yielding more.[11][12] For k > 2, the numbers are sociable, forming cycles longer than two. The smallest known sociable cycle, of length 4, was discovered by H. J. J. te Riele in 1970 (initially attributed to Paul Cohen) and starts at 1264460:1264460 → 1547860 → 1727636 → 1305184 → 1264460 → ⋯,
where each term is the sum of proper divisors of the previous. As of July 2025, 5433 sociable cycles of length greater than 2 are known, with 5421 of length 4, one of length 5, five of length 6, four of length 8, one of length 9, and one of length 28 (discovered in 1995). These cycles are predominantly even and consist of a mix of abundant and deficient numbers, though no cycles of length 3 are known.[13][14]