Enter a positive integer to detect if it belongs to a sociable, amicable, or perfect cycle
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Result
Classification
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Aliquot Cycle
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Detailed Derivation
Sociable Number Definition
Period 1: Perfect (n = s(n))
Period 2: Amicable (a->b->a)
Period k > 2: Sociable (k-cycle)
Smallest k=5 cycle: 12496 (5 terms)
Sociable numbers form cycles in the aliquot sequence of length greater than 2. They are the rarest form of aliquot cycles, with fewer than 300 known. The study of sociable numbers is an active area of computational number theory.
⚠The tool computes up to 30 terms or until a cycle is detected. Numbers may grow large quickly. Computing large proper divisor sums may be slow.
What Are Sociable Numbers?
Sociable numbers are numbers whose aliquot sequence forms a cycle of period greater than 2. They extend the concepts of perfect numbers (period 1) and amicable pairs (period 2) to longer cycles. They are one of the rarest and most intriguing classes of numbers.
Period 5
12496: 12496->14288->15472->14536->14264->12496. The smallest sociable cycle. Discovered by Poulet in 1918.
Period 28
14316 cycle: the longest known sociable cycle. Starting at 14316, it takes 28 steps to return. Discovered in 1969 by computer search.
Rarity
Only ~250 sociable cycles known. Period 4 cycles are the most common among sociable numbers. Period 3 cycles are extremely rare.
Research
Sociable numbers were first systematically searched by computers in the 1970s. Distributed computing projects continue to discover new cycles.
Number TheoryCycle DetectionComputational MathAliquot ResearchDistributed ComputingEducation
FAQs about Sociable Numbers
What is a sociable number?▼
A number in an aliquot cycle of period >2. Example: 12496 has period 5 (12496 returns to itself after 5 aliquot steps).
What is the difference from amicable?▼
Amicable numbers form 2-cycles (a->b->a). Sociable numbers form longer cycles (k>2). Both are types of aliquot cycles.
How many sociable cycles are known?▼
About 250 cycles as of 2024. Most are period 4 or 6. The longest known has period 28. Period 3 is the rarest.
How are sociable numbers discovered?▼
By computer search of aliquot sequences. Tools iteratively compute proper divisor sums and look for cycles. Distributed computing projects now search numbers up to 10^12.
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