Sociable numbers checker

It records every term until the sequence reaches zero or repeats a value, then distinguishes a cycle that returns to the original starting number from a path that merely enters some other cycle. The result includes the full observed sequence, the repeated cycle, its length, and the number of iteration steps, making the calculation easy to inspect or reuse in a program.

What the checker means by a sociable cycle

An aliquot sum is the sum of all positive divisors of an integer except the integer itself. Starting from 12496, for example, the operation produces another integer; applying the same operation repeatedly eventually returns to 12496 after visiting several distinct terms. This checker calls that a sociable cycle because the repeated value is the original starting value. The response sets forms_sociable_cycle to true, labels the status as a sociable cycle, and returns the members in cycle order without duplicating the closing value inside the cycle array. The complete sequence does include that final repeated value so you can verify the closure directly. A perfect number therefore appears as a cycle of length one, and an amicable pair appears as a cycle of length two. Longer cycles are handled by exactly the same rule. This explicit convention avoids hiding mathematically related special cases and lets callers decide whether they want to filter for cycles of length three or greater. The reported step count is the number of aliquot transformations, not the number of distinct values stored.

How termination and other cycles are reported

Not every starting value belongs to a sociable cycle. If an iteration reaches 1, its next aliquot sum is 0 because 1 has no positive proper divisors. The checker then returns a terminated status, a false result, and a terminal value of zero. Another trajectory may merge into a cycle whose first member is not the requested starting value. In that case the response says that the path entered another cycle, identifies the index where that cycle begins, and returns only the repeating portion in the cycle field. These distinctions matter when exploring aliquot dynamics: merely reaching a repeated value proves eventual periodicity for the observed path, but it does not prove that the initial integer is a member of that period. The sequence array preserves the entire route from the supplied value through the closing repetition, so it can be plotted, audited, or compared with an independent divisor-sum implementation. Output fields that do not apply to a terminated path are omitted rather than filled with null values.

Safe bounds, deterministic calculation, and API use

Aliquot sequences can grow quickly, and factoring increasingly large terms can consume far more work than a small browser or edge calculation should accept. This capability therefore enforces a safe maximum term of 1,000,000,000 and a maximum of 1,000 transformations. The starting value must already be within the term bound. If any computed aliquot sum crosses it, the request fails with an invalid-input error that names the source term and the bound; it never returns a partial result that could be mistaken for a mathematical conclusion. A sequence that neither terminates nor repeats within the step limit also fails explicitly. Within those limits, the algorithm is deterministic: it enumerates divisor pairs up to the square root of each term, counts a square root only once, and uses no network access, clock, randomness, or stored state. Send the required start field as a positive integer. The same input always produces the same JSON result, whether you use the browser tool or automate checks through the API at $0.002 per request.