Distinct Prime Factor Count
Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².
Description
The arithmetic function ω(n) counts distinct prime divisors.
The arithmetic function ω(n) counts distinct prime divisors. Repeated occurrences of the same prime contribute only one to this count.1
The tool returns count. The indexing and output conventions below determine how to read these values.
When to use Distinct Prime Factor Count
Use this for the following task: Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Distinct Prime Factor Count 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 count. Use the stated units, indexing and interpretation of the numerical result.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Positive integer (trial-division resource limit 10^12) | integer; minimum 1; maximum 1000000000000 | Required |
Factor positive n by trial division and return the length of the distinct-prime factor list. One has an empty factorization and returns zero.
ω(n)=number of distinct primes in the factorization of n The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| count | Number of distinct prime factors | integer |
{
"input": {
"n": 72
},
"output": {
"count": 2
}
}Limitations and assumptions
The candidate is capped at 10¹². Use the multiplicity-counting Ω tool when exponents should contribute separately.
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 Prime Factor Count with Multiplicity when you need its operation: Return Ω(n), the total number of prime factors counting their exponents, with Ω(1)=0 and trial division limited to n≤10¹². Choose Sphenic Number Checker for the complementary operation: Check whether n is the product of exactly three distinct primes, each with exponent one; trial division is limited to n≤10¹².
Background reference: Digital Library of Mathematical Functions, §27.2.
References
-
Digital Library of Mathematical Functions, §27.2 — National Institute of Standards and Technology
Similar or alternative tools
- 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.
- Sphenic Number Checker
Check whether n is the product of exactly three distinct primes, each with exponent one; trial division is limited to n≤10¹².