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
3 years, 8 months ago

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