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.

Description

A powerful number has every prime divisor appearing at least twice.

A powerful number has every prime divisor appearing at least twice. Equivalently, each prime dividing it also has its square dividing it.^[1]

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

When to use Powerful Number Checker

Use this for the following task: Check whether every prime divisor of n occurs with exponent at least two, treating one as powerful; trial division is limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Powerful 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 isPowerful. 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 the positive bounded integer and require all listed exponents to be at least two. One passes the empty-factorization test.

powerful(n) ⇔ every prime-factor exponent≥2

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

Output field	Meaning	Value type
isPowerful	Whether n is powerful	boolean

json

{
  "input": {
    "n": 72
  },
  "output": {
    "isPowerful": true
  }
}

Limitations and assumptions

Inputs are at most 10¹². A powerful number may also be a perfect power; use the Achilles checker to exclude perfect powers.

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 Achilles Number Checker when you need its operation: 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. Choose Prime Factor Count with Multiplicity for the complementary operation: Return Ω(n), the total number of prime factors counting their exponents, with Ω(1)=0 and trial division limited to n≤10¹².

Background reference: OEIS A001694.

References

OEIS A001694 — OEIS Foundation

Similar or alternative tools

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.
Prime Factor Count with Multiplicity
Return Ω(n), the total number of prime factors counting their exponents, with Ω(1)=0 and trial division limited to n≤10¹².