Almost Primes: Counting, Properties, and Cryptographic Connections

A deep dive into almost prime numbers: integers with exactly k prime factors (counted with multiplicity). We’ll connect the omega function, explore product and factor properties, learn how to count them with pi_k and its asymptotics, and see how Hardy–Ramanujan and Landau fit in. Along the way we discuss why these numbers matter in cryptography—think semiprimes and RSA—and how these ideas illuminate the broader landscape of prime factorization in number theory.

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