Primelengthn

Primelengthn is a function in number theory defined as the total number of prime factors of n counted with multiplicity. It is typically denoted primelength(n) or Ω(n) in some literature. If n has the prime factorization n = ∏ p_i^{a_i}, then primelength(n) = ∑ a_i. By convention, primelength(1) = 0. This quantity is distinct from ω(n), which counts only distinct prime factors.

Example: for n = 360 = 2^3 · 3^2 · 5, primelength(360) = 3 + 2 + 1 = 6.

Properties and relations: primelength is completely additive, meaning primelength(ab) = primelength(a) + primelength(b) for all positive integers a and b. It is unbounded, and numbers with longer prime factorizations have larger primelength values.

Asymptotics and distribution: the average order of primelength(n) is roughly log log n, up to a small bounded term. The Erdős–Kac theorem further describes the distribution of Ω(n) (primelength) as normal with mean and variance about log log n, illustrating a typical factorization length for large n.

Computation and applications: to compute primelength(n), factor n and sum the exponents in its prime factorization. While factoring can be difficult for large n, primelength serves as a simple statistic of an integer’s factorization and is used in analytic and probabilistic number theory to model factorization properties and related arithmetic functions.

See also: Ω(n), ω(n), prime factorization, Hardy–Ramanujan theorem, Erdős–Kac theorem.