divisorcount

Divisorcount, often denoted d(n) or tau(n), is a arithmetic function that returns the number of positive divisors of a positive integer n. By convention, divisorcount refers to the positive divisors, although some contexts discuss the total number of divisors including negative ones (which is twice the positive count for n > 1).

The key formula for divisorcount comes from the prime factorization of n. If n has prime factorization n = p1^a1 p2^a2 ... pk^ak, where the pi are distinct primes and the ai are positive integers, then divisorcount(n) = (a1 + 1)(a2 + 1)...(ak + 1). This follows because each divisor corresponds to choosing an exponent for each prime from 0 up to ai, giving (ai + 1) choices for pi and multiplying across primes.

Divisorcount is a multiplicative function: if gcd(m, n) = 1, then divisorcount(mn) = divisorcount(m) · divisorcount(n). It is a fundamental arithmetic function closely related to the divisor-sum function sigma(n), which sums all positive divisors of n.

Computationally, divisorcount can be obtained by factoring n and applying the product formula, or by enumerating divisors up to sqrt(n) for small n. For large-scale computations, sieve methods or prime factorization algorithms are used to determine the exponents ai efficiently.

Examples: divisorcount(1) = 1; divisorcount(6) = 4 (divisors: 1, 2, 3, 6); divisorcount(12) = 6 (divisors: 1, 2, 3, 4, 6, 12).

See also: divisor function, tau function, prime factorization, sigma function.