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
- Dec. 1991: Steven Ratering: An Interesting Subset of the Highly Composite Numbers (Math. Mag. Vol. 64, no. 5: pp. 343 – 346) www.jstor.org/stable/2690653