divisorcountn

Divisorcountn is a mathematical arithmetic function that assigns to each positive integer n the number of its positive divisors. It is commonly denoted d(n) or tau(n) in number theory literature, and it plays a central role in divisor-related problems and multiplicative functions.

If n has prime factorization n = ∏_{i=1}^k p_i^{a_i}, where the p_i are distinct primes and a_i ≥ 1, then divisorcountn(n) = ∏_{i=1}^k (a_i + 1). For example, 12 = 2^2 · 3^1, so divisorcountn(12) = (2+1)(1+1) = 3·2 = 6, since 12 has divisors 1, 2, 3, 4, 6, and 12.

Key properties include multiplicativity: if gcd(m, n) = 1, then divisorcountn(mn) = divisorcountn(m) · divisorcountn(n). The value of divisorcountn(n) is odd if and only if n is a perfect square; otherwise it is even.

Computationally, divisorcountn can be found by factoring n and applying the product formula above. A straightforward approach uses trial division up to sqrt(n) to determine the exponents a_i, while for many values a sieve or precomputation can yield all d(n) for n ≤ N efficiently. For very large integers, advanced factoring methods may be required.

In related contexts, the function is often denoted d(n) or tau(n). The generating function sum_{n≥1} d(n)/n^s equals zeta(s)^2, and the sequence of d(n) values appears in the OEIS as A000005. Divisorcountn is a basic tool in studying divisor structures, divisor sums, and the multiplicative architecture of integers.