# Complete List of Special Highly Composite Numbers

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## Theorem

There are exactly $6$ special highly composite numbers:

- $1, 2, 6, 12, 60, 2520$

This sequence is A106037 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

We have the following:

By inspection of the sequence of highly composite numbers:

- $1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, \ldots$

it can be seen that there are no more special highly composite numbers less than $2520$.

Let $n > 18$.

From Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2, the $n$th highly composite number does not divide the $n+1$th.

Hence the $n$th highly composite number is not a special highly composite number.

The result follows.

$\blacksquare$

## 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