Sociable Chain Length 3 Checker

Check exact three-cycle in proper-divisor sum iteration; fixed perfect numbers and amicable two-cycles are not mislabelled length-three sociable cycles; out-of-domain intermediate refused.

Description

Aliquot iteration replaces an integer by the sum of its proper divisors.

Aliquot iteration replaces an integer by the sum of its proper divisors. A sociable cycle of length three requires three distinct states and a return to the starting state.^[1]

Check exact three-cycle in proper-divisor sum iteration; fixed perfect numbers and amicable two-cycles are not mislabelled length-three sociable cycles; out-of-domain intermediate refused.

When to use Sociable Chain Length 3 Checker

Use this tool to check exact three-cycle in proper-divisor sum iteration; fixed perfect numbers and amicable two-cycles are not mislabelled length-three sociable cycles; out-of-domain intermediate refused.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Sociable Chain Length 3 Checker and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter startingInteger using the field table and calculation rules, then run the tool.
Read firstThreeAliquotIterates, isExactLengthThreeSociableCycle. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
startingInteger	Starting integer	number	Required

Enter a starting integer from 1 to one million. Compute three exact divisor-sum iterates; the first and second must remain in the positive domain through one million or input evaluation errors. Return the three iterates and the distinct-state cycle flag.

a=s(n), b=s(a), c=s(b); length-three cycle iff c=n and n,a,b are distinct

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
firstThreeAliquotIterates	First three aliquot iterates	list of number
isExactLengthThreeSociableCycle	Is exact length three sociable cycle	boolean

json

{
  "input": {
    "startingInteger": 6
  },
  "output": {
    "firstThreeAliquotIterates": [
      6,
      6,
      6
    ],
    "isExactLengthThreeSociableCycle": false
  }
}

Limitations and assumptions

Perfect-number fixed points and amicable two-cycles return false. The third reported sum can lie outside the intermediate-input bound.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Weird Number Checker: Check exact weirdness as abundance with no subset of distinct proper divisors summing to n; bounded 0/1 knapsack, not abundance alone or an exhaustive integer classification beyond caller n. For a complementary task, use Pillai Prime Checker: Find exact Pillai prime witnesses k!≡−1 mod p with k not dividing p−1; exhaustive bounded k<p, no inferred classification from one arbitrary k.

References

Sociable Chain Length 3 Checker — tool.io

Similar or alternative tools

Weird Number Checker
Check exact weirdness as abundance with no subset of distinct proper divisors summing to n; bounded 0/1 knapsack, not abundance alone or an exhaustive integer classification beyond caller n.
Pillai Prime Checker
Find exact Pillai prime witnesses k!≡−1 mod p with k not dividing p−1; exhaustive bounded k<p, no inferred classification from one arbitrary k.