divisorcountmn

Divisorcountmn is a function used in number theory and computer algebra to quantify the number of divisors associated with a pair of integers. In the common interpretation, divisorcountmn(m, n) equals the number of positive divisors of the greatest common divisor of m and n. This is equivalent to applying the ordinary divisor-counting function d(k) to gcd(m, n).

Formally, divisorcountmn(m, n) = d(gcd(m, n)), where d(k) is the number of positive divisors of k. If gcd(m, n) factors as ∏ p_i^{e_i}, then divisorcountmn(m, n) = ∏ (e_i + 1). The value depends only on the shared prime factors of m and n, and is symmetric in m and n.

Computing divisorcountmn involves first determining gcd(m, n) and then counting its divisors, typically via prime factorization or using a precomputed table of divisor counts. In practice, for large inputs, factoring gcd(m, n) is the main cost.

Example: let m = 36 and n = 48. Their gcd is 12, which has divisors 1, 2, 3, 4, 6, 12. Therefore divisorcountmn(36, 48) = 6.

Variants exist in literature and software. Some sources define divisorcountmn using d(lcm(m, n)) or as the product d(m)·d(n) for unrelated purposes. The chosen definition should be specified in any implementation or discussion.

Related concepts include the divisor function d(n), the gcd and lcm operations, and multiplicative functions. The function is primarily used in exploratory number theory, combinatorial counting, and algorithmic number theory where joint divisor structure of a pair is relevant.