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.

Description

The arithmetic function Ω(n) counts prime factors with multiplicity.

The arithmetic function Ω(n) counts prime factors with multiplicity. For example, each power in the prime factorization contributes its full exponent.^[1]

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

When to use Prime Factor Count with Multiplicity

Use this for the following task: Return Ω(n), the total number of prime factors counting their exponents, 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 Prime Factor Count with Multiplicity 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 sum the exponents. One returns zero because it has no prime factors.

Ω(n)=Σ prime-factor exponents

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

Output field	Meaning	Value type
count	Prime factors counted with multiplicity	integer

json

{
  "input": {
    "n": 72
  },
  "output": {
    "count": 5
  }
}

Limitations and assumptions

The input is capped at 10¹². The output is a count, not a factor list, and differs from the number of distinct prime divisors.

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 Distinct Prime Factor Count when you need its operation: Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹². Choose Liouville Lambda Function for the complementary operation: Return λ(n)=(−1)^Ω(n), counting prime factors with multiplicity and using λ(1)=1; 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

Distinct Prime Factor Count
Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².
Liouville Lambda Function
Return λ(n)=(−1)^Ω(n), counting prime factors with multiplicity and using λ(1)=1; trial division is limited to n≤10¹².