Superior highly composite number

Divisor function d(n) up to n = 250
Prime-power factors

In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.

For any possible exponent, whichever integer has the greatest ratio is a superior highly composite number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.

The first ten superior highly composite numbers and their factorization are listed.

# prime
factors
SHCN
n
Prime
factorization
Prime
exponents
# divisors
d(n)
Primorial
factorization
1 2 2 1 2 2
2 6 2 ⋅ 3 1,1 4 6
3 12 22 ⋅ 3 2,1 6 2 ⋅ 6
4 60 22 ⋅ 3 ⋅ 5 2,1,1 12 2 ⋅ 30
5 120 23 ⋅ 3 ⋅ 5 3,1,1 16 22 ⋅ 30
6 360 23 ⋅ 32 ⋅ 5 3,2,1 24 2 ⋅ 6 ⋅ 30
7 2520 23 ⋅ 32 ⋅ 5 ⋅ 7 3,2,1,1 48 2 ⋅ 6 ⋅ 210
8 5040 24 ⋅ 32 ⋅ 5 ⋅ 7 4,2,1,1 60 22 ⋅ 6 ⋅ 210
9 55440 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 4,2,1,1,1 120 22 ⋅ 6 ⋅ 2310
10 720720 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 4,2,1,1,1,1 240 22 ⋅ 6 ⋅ 30030
Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.

For a superior highly composite number n there exists a positive real number ε > 0 such that for all natural numbers k > 1 we have where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915).[1]

For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12.

120 is another superior highly composite number because it has the highest ratio of divisors to itself raised to the .4 power.

The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other.

  1. ^ Weisstein, Eric W. "Superior Highly Composite Number". mathworld.wolfram.com. Retrieved 2021-03-05.