Highly Composite Number/Examples/60

Example of Highly Composite Number

$60$ is a highly composite number, being the smallest positive integer with $12$ divisors or more.

Proof

From $\sigma_0$ of $60$:

$\map {\sigma_0} {60} = 12$

For a positive integer $n$ to have a $\sigma_0$ of $12$, the product of $1$ greater than the multiplicities of all the prime factors must be $12$.

The divisors of $12$ are $1, 2, 3, 4, 6, 12$.

The smallest $n$ with one prime factor such that $\map {\sigma_0} n = 12$ is $2^{11} = 2048$.

The smallest $n$ with two prime factors such that $\map {\sigma_0} n = 12$ are:

$2^3 3^2 = 72$

$2^5 3^1 = 96$

The smallest $n$ with three prime factors such that $\map {\sigma_0} n = 12$ is:

$2^2 3^1 5^1 = 60$

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $60$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $60$