Smooth Number Checker

Check whether every prime factor of n is at most an inclusive bound B, treating one as B-smooth; trial division is limited to n≤10¹².

Description

A B-smooth number has no prime factor greater than B.

A B-smooth number has no prime factor greater than B. This tool uses an inclusive upper bound, so a factor equal to B is allowed.^[1]

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

When to use Smooth Number Checker

Use this for the following task: Check whether every prime factor of n is at most an inclusive bound B, treating one as B-smooth; trial division is limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Smooth 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, bound according to the field table, then run the tool.
Read isSmooth. 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
bound	Inclusive prime-factor bound	integer; minimum 1; maximum 9007199254740991	Required

Factor positive n by trial division and compare each distinct prime with the supplied bound. An empty factorization for one passes.

smooth(n,B) ⇔ every prime factor p of n satisfies p≤B

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

Output field	Meaning	Value type
isSmooth	Whether every prime factor is at most the bound	boolean

json

{
  "input": {
    "n": 360,
    "bound": 5
  },
  "output": {
    "isSmooth": true
  }
}

Limitations and assumptions

The factorization input is capped at 10¹². One is smooth for every accepted bound; factor multiplicities do not affect this predicate.

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 Rough Number Checker when you need its operation: Check whether every prime factor of n is at least an inclusive bound B, treating one as B-rough; trial division is limited to n≤10¹². Choose Distinct Prime Factor Count for the complementary operation: Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².

Background reference: OEIS A051037.

References

OEIS A051037 — OEIS Foundation

Similar or alternative tools

Rough Number Checker
Check whether every prime factor of n is at least an inclusive bound B, treating one as B-rough; trial division is limited to n≤10¹².
Distinct Prime Factor Count
Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².