Untouchable Number Checker

Exact bounded untouchable integer0–1000underpreimagem≥1 convention;0touchedbym1,1bym2. Forn≥2 primepreimages cannot occur; everycompositemhaslargestproperdivisor≥√m, so anypreimagesatisfiesm≤n². Sieve complete bound, zero witness means absent.

Description

An untouchable integer is absent from the range of the sum-of-proper-divisors function under a specified domain convention.

An untouchable integer is absent from the range of the sum-of-proper-divisors function under a specified domain convention.^[1]

This tool performs a complete bounded preimage search and returns a smallest witness when one exists.

When to use Untouchable Number Checker

This tool performs a complete bounded preimage search and returns a smallest witness when one exists.

How to use Untouchable Number Checker and how it works

Prepare integer, using the input definitions and matching units below.
Enter the required fields and run Untouchable Number Checker.
Read result.isUntouchable, result.smallestProperDivisorSumWitness using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualInteger	Nonnegative integer whose proper-divisor-sum preimages are checked, from 0 to 1000.	number	Required; integer 0–1000; finite; other constraints below

Require n as an integer 0–1000. Under proper-divisor preimages m>=1, n=0 has witness 1 and n=1 has witness 2. For n>=2, sieve all proper-divisor sums through n^2, a complete preimage bound since a composite m has a proper divisor at least sqrt(m). Return the first matching m in ascending order, or isUntouchable=true with witness sentinel 0. Finite completeness relies on this domain and bound, not an arbitrary search cutoff.

s(m)=sum(d where d divides m and 1<=d<m); witness=min(m>=1:s(m)=n), or 0

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.isUntouchable	True when the complete domain-specific bounded preimage search has no witness.	boolean
result.smallestProperDivisorSumWitness	Smallest positive preimage; zero is the no-witness sentinel.	number

json

{
  "input": {
    "actualInteger": 5
  },
  "output": {
    "result": {
      "isUntouchable": true,
      "smallestProperDivisorSumWitness": 0
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Superfactorion Checker for a complementary operation: A superfactorion uses the sum of superfactorials of its decimal digits, with the empty product at zero equal to one.

Use Hyperfactorion Checker for a complementary operation: A hyperfactorion equals the sum of the hyperfactorials of its decimal digits, with the zero-digit empty product equal to one.

References

Untouchable Number Checker — tool.io

Similar or alternative tools

Superfactorion Checker
A superfactorion uses the sum of superfactorials of its decimal digits, with the empty product at zero equal to one.
Hyperfactorion Checker
A hyperfactorion equals the sum of the hyperfactorials of its decimal digits, with the zero-digit empty product equal to one.