# 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, $$\omega (n)$$

Note by Evan Robinson
4 years, 2 months ago

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