Hyperfactorion Checker

Exact decimal digit hyperfactorialH(d)=Πk^k sum checker, empty product at0=1, input≤ 128 digits. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.

Description

A hyperfactorion equals the sum of the hyperfactorials of its decimal digits, with the zero-digit empty product equal to one.

A hyperfactorion equals the sum of the hyperfactorials of its decimal digits, with the zero-digit empty product equal to one.^[1]

This tool checks that exact relation for a canonical integer string without searching the entire family.

When to use Hyperfactorion Checker

This tool checks that exact relation for a canonical integer string without searching the entire family.

How to use Hyperfactorion Checker and how it works

Prepare canonical decimal, using the input definitions and matching units below.
Enter the required fields and run Hyperfactorion Checker.
Read result.exactDigitInvariantSum, result.isNamedDigitInvariant 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
actualCanonicalDecimal	Canonical nonnegative integer string, at most 128 digits, without signs or leading zeros except zero itself.	string	Required; other constraints below

Require a canonical nonnegative decimal string of at most 128 digits. For each digit d, compute superfactorial as product of k! through d, or hyperfactorial as product of k^k through d, according to the tool. Empty products for digit zero equal one. Sum with BigInt and return the exact decimal sum plus equality to the input integer. No floating-point approximation or family-wide enumeration occurs.

superDigit(d)=product(k!,k=1..d); hyperDigit(d)=product(k^k,k=1..d); invariant=(sum(digitContribution)==input)

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.exactDigitInvariantSum	Exact decimal-string sum of the named digit contributions.	string
result.isNamedDigitInvariant	Whether the exact named digit-contribution sum equals the input integer.	boolean

json

{
  "input": {
    "actualCanonicalDecimal": "1"
  },
  "output": {
    "result": {
      "exactDigitInvariantSum": "1",
      "isNamedDigitInvariant": true
    }
  }
}

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 Untouchable Number Checker for a complementary operation: An untouchable integer is absent from the range of the sum-of-proper-divisors function under a specified domain convention.

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.

References

Hyperfactorion Checker — tool.io

Similar or alternative tools

Untouchable Number Checker
An untouchable integer is absent from the range of the sum-of-proper-divisors function under a specified domain convention.
Superfactorion Checker
A superfactorion uses the sum of superfactorials of its decimal digits, with the empty product at zero equal to one.