threeprimefactor

Threeprimefactor is a term used in number theory to describe integers that have exactly three prime factors, counted with multiplicity. Formally, if Ω(n) denotes the total number of prime factors of n in its prime factorization, counted with repetition, then a threeprimefactor is an integer n for which Ω(n) = 3. The concept is commonly referred to in the literature as a 3-almost prime, and threeprimefactor is a descriptive variant of this idea.

Examples include 8 (2^3), 12 (2^2·3), 18 (2·3^2), 20 (2^2·5), and 30 (2·3·5). Note that these numbers may differ from those with three distinct prime factors; for instance, 12 has Ω(12) = 3 but ω(12) = 2, since its prime factors are 2 and 3.

Related concepts include the broader family of k-almost primes, which are numbers with exactly k prime factors (counted with multiplicity). A related distinction is between Ω(n) = 3 (threeprimefactor by multiplicity) and ω(n) = 3 (three distinct prime factors).

Properties and counting: There are infinitely many threeprimefactors. If A_3(x) denotes the number of n ≤ x with Ω(n) = 3, then asymptotically A_3(x) ~ (x / log x) · (log log x)^2 / 2. This reflects the general distribution of almost primes and their growth as x increases.

See also: k-almost primes, almost primes, prime factorization, sieve methods.