Special Highly Composite Number/Examples/6

Example of Special Highly Composite Number
$6$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers.

Proof
By inspection of the sequence of highly composite numbers, $6$ is highly composite.

$n > 6$ is a highly composite number which is not divisible by $6$.

We have that $2$ is a special highly composite number.

Therefore $2$ is a divisor of $n$.

As $6$ is not a divisor of $n$, it follows that $3$ is also not a divisor of $n$.

By Prime Decomposition of Highly Composite Number, that means $n = 2^k$ for some $k \ge 3$.

Then:

But this contradicts our deduction that $n = 2^k$ where $k \ge 3$.

The result follows by Proof by Contradiction.