For example, 60, it is highly composite because it has 12 divisors and none of the numbers up to 59 has 12 or more divisors.
The first 19 highly composite numbers coincide with the first 19 superabundant numbers, and the first one which is not superabundant is 7560.
All the highly composite numbers greater than 1 have a prime factorization
where
all the primes from 2 to
appear. Moreover, the exponents
are nonincreasing, i.e.,
and
the last one (
) is always 1 except for the
numbers
and
.
The first highly composite numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080 more terms