Münchhausen Digit Self-Power Checker

Check whether n equals the sum of d^d over its decimal digits under an explicit 0^0 convention of zero or one, using exact integers up to 200 input characters.

Description

A Münchhausen number equals the sum of each decimal digit raised to its own power.

A Münchhausen number equals the sum of each decimal digit raised to its own power. The treatment of a zero digit depends on an explicit convention for 0^0.^[1]

The tool returns isMunchausen. The indexing and output conventions below determine how to read these values.

When to use Münchhausen Digit Self-Power Checker

Use this for the following task: Check whether n equals the sum of d^d over its decimal digits under an explicit 0^0 convention of zero or one, using exact integers up to 200 input characters. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Münchhausen Digit Self-Power Checker and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n, zeroPowerZero according to the field table, then run the tool.
Read isMunchausen. A true flag means the candidate meets the stated property; false means it does not.
If the flag is false, review the candidate and definition. A well-formed nonmember is different from a rejected input.

Input field	Meaning	Value type and bounds	Requirement
n	Nonnegative decimal integer	string (integer/arbitrary); maximum text length 200; pattern ^(0\|[1-9][0-9]*)$	Required
zeroPowerZero	Value assigned to 0^0: zero or one	integer; minimum 0; maximum 1	Required

Parse the nonnegative decimal input, calculate each digit contribution with BigInt and use the supplied integer zero-or-one convention for zero digits.

membership ⇔ n=Σdigit^digit, assigning 0^0=zeroPowerZero

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
isMunchausen	Whether n equals its digit self-power sum	boolean

json

{
  "input": {
    "n": "3435",
    "zeroPowerZero": 1
  },
  "output": {
    "isMunchausen": true
  }
}

Limitations and assumptions

The input text is capped at 200 characters. Leading zeros are rejected, and changing the zero convention can change membership.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Factorion Checker when you need its operation: Check whether a nonnegative decimal integer equals the sum of factorials of its digits, including 0!=1 and an input resource limit of 200 characters. Choose Dudeney Number Checker for the complementary operation: Check whether a nonnegative decimal integer equals the cube of the sum of its decimal digits, using exact arithmetic and a 200-character input resource limit.

Background reference: OEIS A046253.

References

OEIS A046253 — OEIS Foundation

Similar or alternative tools

Factorion Checker
Check whether a nonnegative decimal integer equals the sum of factorials of its digits, including 0!=1 and an input resource limit of 200 characters.
Dudeney Number Checker
Check whether a nonnegative decimal integer equals the cube of the sum of its decimal digits, using exact arithmetic and a 200-character input resource limit.