divisorcount12

Divisorcount12 refers to the set of positive integers that have exactly twelve positive divisors. In number theory, the divisor-counting function d(n) (also called tau(n)) gives the number of divisors of n. If n has prime factorization n = p1^a1 p2^a2 ... pk^ak with distinct primes, then d(n) = (a1+1)(a2+1)...(ak+1). Imposing d(n) = 12 restricts the exponents to one of four patterns: 11; 5 and 1; 3 and 2; or 2, 1, and 1. Equivalently, n must be of the form p^11, p^5 q, p^3 q^2, or p^2 q r, where p, q, r are distinct primes.

Examples of numbers with twelve divisors include 60 = 2^2·3·5 (exponents 2,1,1); 72 = 2^3·3^2 (3,2); 84 = 2^2·3·7 (2,1,1); 90 = 2·3^2·5; 96 = 2^5·3 (5,1). Other cases such as 108 = 2^2·3^3, 126 = 2·3^2·7, 132 = 2^2·3·11, and 140 = 2^2·5·7 also fit one of the four patterns.

The smallest number with twelve divisors is 60. There are infinitely many such numbers (for example, any prime raised to the 11th power, p^11, has twelve divisors). The study of numbers with a fixed divisor count is a standard topic in multiplicative number theory and relates to prime distribution and factorization patterns.