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