Achilles Number Checker
Check whether n is powerful but not a perfect power by testing the prime exponents and their gcd; trial division is limited to n≤10¹².
Description
An Achilles number is powerful but is not a perfect power.
An Achilles number is powerful but is not a perfect power. Prime-factor exponents identify both conditions without trying every possible root.1
The tool returns isAchilles. The indexing and output conventions below determine how to read these values.
When to use Achilles Number Checker
Use this for the following task: Check whether n is powerful but not a perfect power by testing the prime exponents and their gcd; trial division is limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Achilles Number 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 according to the field table, then run the tool.
- Read isAchilles. 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 | Positive integer (trial-division resource limit 10^12) | integer; minimum 1; maximum 1000000000000 | Required |
Factor n, reject the empty factorization or any exponent below two, and apply the Euclidean algorithm to the exponent list.
Achilles ⇔ nonempty factorization, every exponent≥2, gcd(all exponents)=1 The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| isAchilles | Whether n is an Achilles number | boolean |
{
"input": {
"n": 72
},
"output": {
"isAchilles": true
}
}Limitations and assumptions
The input is at most 10¹². One is explicitly excluded, and the tool returns only the membership flag.
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 Powerful Number Checker when you need its operation: Check whether every prime divisor of n occurs with exponent at least two, treating one as powerful; trial division is limited to n≤10¹². Choose Greatest Common Divisor of a List for the complementary operation: Compute the nonnegative greatest common divisor of 1–1,000 signed decimal integers; an all-zero list has gcd zero.
Background reference: OEIS A052486.
References
-
OEIS A052486 — OEIS Foundation
Similar or alternative tools
- Powerful Number Checker
Check whether every prime divisor of n occurs with exponent at least two, treating one as powerful; trial division is limited to n≤10¹².
- Greatest Common Divisor of a List
Compute the nonnegative greatest common divisor of 1–1,000 signed decimal integers; an all-zero list has gcd zero.