1. What Are Prime Factor Counting Functions?

The functions ω(n) and Ω(n) are fundamental in number theory for counting prime factors of an integer. ω(n) counts distinct prime factors, while Ω(n) counts total prime factors including multiplicity.

2. Mathematical Definitions

ω(n) - Distinct Prime Factors:

Ω(n) - Total Prime Factors (with multiplicity):

Examples:

• n = 12 = 2² × 3¹: ω(12) = 2, Ω(12) = 3
• n = 30 = 2¹ × 3¹ × 5¹: ω(30) = 3, Ω(30) = 3
• n = 8 = 2³: ω(8) = 1, Ω(8) = 3

3. Properties and Applications

Key Properties:

• ω(n) ≤ Ω(n) for all n ≥ 2
• ω(n) = Ω(n) if and only if n is square-free
• Both functions are additive but not completely additive
• Used in studying prime distribution and number theory

Applications: These functions are crucial in analytic number theory, cryptography, and understanding the distribution of prime numbers.

4. Using the Calculator

Instructions: Enter any integer n ≥ 2 to calculate both ω(n) and Ω(n), along with the complete prime factorization.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between ω(n) and Ω(n)?
A: ω(n) counts how many different primes divide n, while Ω(n) counts the total number of prime factors (including repeated ones).

Q2: When are ω(n) and Ω(n) equal?
A: They are equal when n is square-free, meaning no prime squared divides n.

Q3: What are the maximum values of these functions?
A: For n ≤ x, both ω(n) and Ω(n) are approximately log log x on average, but can be as large as log x / log log x.

Q4: Are these functions multiplicative?
A: Ω(n) is completely additive (Ω(mn) = Ω(m) + Ω(n)), while ω(n) is additive but not completely additive.

Q5: How are these functions used in cryptography?
A: They help analyze the security of cryptographic systems based on integer factorization, like RSA.