is called highly composite
if it has more divisors than any smaller number, i.e., if it sets a new record in the
number of divisors.
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
All the highly composite numbers greater than 1 have a prime factorization
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
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many highly composite numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p