Special Highly Composite Number/Examples/2

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Example of Special Highly Composite Number

$2$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers.


Proof

We have that $2$ is highly composite.

Let $n > 2$ be a highly composite number.


From Prime Decomposition of Highly Composite Number, the multiplicity of $2$ in $n$ is at least as high as the multiplicity of any other prime $p$ in $n$.

Thus if $p \divides n$ it follows that $2 \divides n$.


Thus, by definition, $2$ is a special highly composite number.

$\blacksquare$


Sources