# A question about the omega functions $\Omega (n)$ and $\omega (n)$

Given a random positive number $n$, is there a way to determine the number of prime factors of $n$ without factorizing $n$?

There are several useful primality tests for a given number, such as the Miller-Rabin primality test, but is there a test to determine whether number is a semiprime or a 3-almost prime, etc.? Put another way, how can one find either the total number of prime factors (counted with multiplicity) or the number of distinct prime factors without factoring $n$?

After some digging in Wikipedia, I found that these two properties, the number of prime factors to muliplicity and number of distinct prime factors, could be described, respectively, by the functions $\Omega (n)$ and $\omega (n)$.

Thus, we have another way to phrase the question: "Is there a way to calculate either of the omega functions for $n$ without knowing any of the factors of $n$?"

There wasn't much information on those functions at Wikipedia though, and I couldn't parse the information in the notes for the related OEIS sequences.

Here are some links that may be relevant: Prime factor; Prime power decomposition; A001222, The number of prime factors counted with multiplicity, $\Omega (n)$ ; A001221, The number of distinct primes dividing n, $\omega (n)$

6 years, 8 months ago

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