Aliquot sequence calculator

This calculator performs that iteration deterministically, reports every term it reaches, and stops when it uses the requested step limit, reaches zero, or detects a repeated value. A strict safe bound protects the calculation from unexpectedly large intermediate terms, making the result suitable for experiments, lessons, validation scripts, and reproducible number-theory workflows.

Choose the starting value and understand each transition

Enter a positive whole number as start and choose a step_limit from 1 through 100. The starting value is always the first item in the returned sequence. Each permitted transition then replaces the most recent term with its aliquot sum: the sum of every positive divisor smaller than the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the next term is 16. The proper divisors of 16 are 1, 2, 4, and 8, making the following term 15. Because the limit counts transitions rather than displayed values, a limit of ten can return at most eleven sequence entries. It may return fewer when a natural stopping condition appears. This convention keeps the requested computational effort unambiguous and makes it easy to compare the output with a hand calculation, classroom worksheet, database record, or another mathematical implementation. Aliases such as n, number, or value may supply the start in API integrations, while steps may supply the limit.

Read the termination and cycle information

The result includes the complete sequence produced during the request, the number of transitions actually completed, and a termination label. A label of step_limit means the calculator completed every requested transition without encountering an earlier stopping condition. A label of zero means the sequence reached zero; this commonly follows 1 because 1 has no proper positive divisors. A label of cycle means a newly calculated value was already present. The repeated value is included as the last sequence entry so the cycle is visible, and cycle_start_index identifies its earlier zero-based position. Perfect numbers therefore produce an immediate one-value loop: starting at 6 gives 6, 6 and points back to index zero. For non-cyclic results, cycle_start_index is -1, providing a stable response shape without a nullable field. Cycle detection avoids wasting the remaining step allowance on repetitions while preserving enough information to reconstruct the periodic portion and distinguish it from the nonrepeating lead-in.

Work safely with large terms and reproducible results

Every starting value must be no greater than the published safe bound of one trillion, and every generated term is checked against that same bound before it is accepted. If an intermediate aliquot sum exceeds the bound, the request returns an invalid-input error that names the transition where growth became unsafe. The calculator never truncates, rounds, or silently substitutes a smaller number, because doing so would create a sequence that is mathematically false. Internally, each aliquot sum is found by deterministic integer factorization and the multiplicative divisor-sum formula. There is no network access, random choice, current-time dependency, or stored state, so identical input produces identical output in the browser and through the API. Use a modest step limit when exploring an unfamiliar start, then increase it when the observed terms remain manageable. The API price is $0.002 per request. The item unit describes one requested calculation; the explicit term and step limits bound the work performed within it.