Divisorfrequency

Divisorfrequency is a concept used in number theory to describe how often divisors occur among integers. It encompasses two closely related ideas: the divisor function, which counts the divisors of a single integer, and the broader frequency with which a fixed divisor appears among a range of integers. Together, these perspectives illuminate patterns in the distribution of divisors.

The divisor function, usually denoted d(n) or tau(n), gives the number of positive divisors of n. If n has prime factorization n = p1^a1 p2^a2 ... pr^ar, then d(n) = (a1+1)(a2+1)...(ar+1). The function is multiplicative: if m and n are coprime, then d(mn) = d(m)d(n). Variants include the sum-of-divisors function σ(n), which sums all positive divisors of n.

Frequency across ranges: for a fixed integer k, the number of integers n ≤ N divisible by k is floor(N/k), so the asymptotic density of multiples of k is 1/k. This underpins the study of divisor occurrence in large sets of integers. The divisor function also has known average behavior: the average order of d(n) is about log n, more precisely log n + (2γ − 1) + o(1), where γ is Euler’s constant. Analytic methods yield insights into the distribution and variance of d(n).

Related concepts and applications: divisorfrequency relates to topics such as sieve methods, probabilistic number theory, and the distribution of divisors in arithmetic progressions. It connects to related arithmetic functions like σ(n) and to questions about the typical size of the set of divisors of a random integer. In practice, the term is used informally to discuss both per-number divisor counts and the per-divisor frequency across a range.