Highly composite number
Prime factorization

Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:

where p 1 < p 2 < ⋯ < p k {\displaystyle p_{1}<p_{2}<\cdots <p_{k}} are prime, and the exponents c i {\displaystyle c_{i}} are positive integers.

Any factor of n must have the same or lesser multiplicity in each prime:

So the number of divisors of n is:

Hence, for a highly composite number n,

Also, except in two special cases n = 4 and n = 36, the last exponent c must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials or, alternatively, the smallest number for its prime signature.

Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 2 × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors.
change_case:english_lowercase