Aliquot sequence

An aliquot sequence begins with a positive integer and repeatedly replaces each term with the sum of its proper divisors. It ends at 0 if a term is prime.

Sum of divisors function: The function that maps an integer to the sum of its positive divisors. Subtracting the integer itself from this total gives the next term.
Amicable numbers: A pair of distinct integers whose proper-divisor sums equal each other’s values. Starting at either number produces a two-term cycle.
Lehmer's problem: The open question of whether every aliquot sequence eventually reaches 0 or a cycle. It states the central unresolved question about the sequence family.
Divisor function: An arithmetic function that assigns values to integers using their divisors. The aliquot rule is built from a standard divisor-based function.
Proper divisor: A positive divisor of an integer that is smaller than the integer. These are exactly the divisors added at every step.
Sociable numbers: A cycle of three or more distinct integers linked by successive proper-divisor sums. These cycles extend the familiar amicable-pair pattern.
Catalan–Dickson conjecture: The conjecture that every aliquot sequence eventually reaches 0 or enters a cycle. This conjecture formalizes the expectation that trajectories do not grow forever.
Multiplicative function: An arithmetic function whose values on coprime products equal the product of its values on each factor. The divisor-sum function’s multiplicativity helps compute aliquot sums from prime factorizations.
Prime number: An integer greater than 1 whose only positive divisors are 1 and itself. A prime has proper-divisor sum 1, so its sequence proceeds to 1 and then 0.
Aliquot sequence of 12496: The sequence beginning 12496, which enters the amicable cycle 14288, 15472, 14536, 14264, 12496. It illustrates a trajectory that reaches a cycle after several distinct terms.

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Aliquot sumBroader topic: This iteration turns one aliquot sum into a study of long-term behavior.