Faktorfaktor

Faktorfaktor is a term used in number theory to describe a hypothetical metric that links the divisor structure of a positive integer to its prime factorization. It is not part of standard mathematical nomenclature but appears in educational contexts as a simple illustration of how factors grow with exponents and with the number of distinct primes.

Definition and interpretation

For a positive integer n with prime factorization n = p1^a1 p2^a2 ... pk^ak, let d(n) = ∏(ai + 1) be the number of divisors and let ω(n) = k be the number of distinct prime factors. The Faktorfaktor of n, denoted FF(n), is defined as FF(n) = d(n) / ω(n). This value can be interpreted as the average number of divisors associated with each distinct prime factor, reflecting how the divisor count distributes across the prime components.

Examples

- n = 12 = 2^2 · 3^1: d(12) = (2+1)(1+1) = 6; ω(12) = 2; FF(12) = 3.

- n = 30 = 2 · 3 · 5: d(30) = 2^3 = 8; ω(30) = 3; FF(30) ≈ 2.67.

- n = 8 = 2^3: d(8) = 3; ω(8) = 1; FF(8) = 3.

Variants and properties

FF(n) varies with the exponent pattern. For a prime power p^a, FF(n) = a + 1. For a squarefree number with k distinct primes, FF(n) = 2^k / k. The measure is not a standard arithmetic function and can yield non-integer values, highlighting how divisor structure relates to factorization in different ways.

See also

Divisor function, Omega function, prime factorization, multiplicative functions.