Lowest Unused Anti-Prime (Highly Composite Number) Factors

Sunday, February 3, 2019
By: Matthew Doucette

I discovered a new number sequence, A332271, while exploring Highly Composite Numbers (Anti-Primes):

What Are Highly Composite Numbers?

These are numbers that have lots of factors, specifically more factors than any number lower than them. That's how there were coined as "anti-primes" by Brady Haran of Numberphile. While exploring them, I thought of something interesting:

HCNs Seem To Regularly Skip Some Factors:

In the factors that make up highly composite numbers, which by definition there are a lot of them, what about the smallest unused factor? Is there a pattern? Let's see:

Highly Composite Number : Lowest Unused Factor:

1: 2 (or null if you cap at HCN)
2: 3 (or null if you cap at HCN)
4: 3
6: 4
12: 5
24: 5
36: 5
48: 5
60: 7
120: 7
180: 7
240: 7
360: 7
720: 7
840: 9
1260: 8
1680: 9
2520: 11
5040: 11
7560: 11
10080: 11
15120: 11
20160: 11
25200: 11
27720: 13
45360: 11
50400: 11
55440: 13
83160: 13
110880: 13
166320: 13
221760: 13
277200: 13
332640: 13
498960: 13
554400: 13
665280: 13
720720: 17
1081080: 16
1441440: 17
2162160: 17
2882880: 17
3603600: 17
4324320: 17
6486480: 17
7207200: 17
8648640: 17
10810800: 17
14414400: 17
17297280: 17
21621600: 17
32432400: 17
36756720: 19
43243200: 17
61261200: 19
73513440: 19
110270160: 19
122522400: 19
147026880: 19
183783600: 19
245044800: 19
294053760: 19
367567200: 19
551350800: 19
698377680: 23
735134400: 19
...

Here it is as a number sequence:

2, 3, 3, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 8, 9, 11, 11, 11, 11, 11, 11, 11, 13, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 19, ...

...or if you cap at highly composite number...

null, null, 3, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 8, 9, 11, 11, 11, 11, 11, 11, 11, 13, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 19, ...

See Anything Special?

It appears to be dominated by prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, ... However, not all of them are prime: There is a 4, 8, and 16, for as far as I have calculated.

The number sequence appears to be missing from The On-Line Encyclopedia of Integer Sequences, so I am going to submit it, and investigate it further.

2020-AUG-07 Update:

This number sequence was approved by the On-Line Encyclopedia of Integer Sequences on August 7th, 2020: oeis.org/A332271.

History:

Interestingly, Jason Doucette is already in the OEIS database: oeis.org/search?q=jason+doucette. He is listed in some sequences relating to his world record work in palindrone math theory. 2020-AUG-07 Update: Same search for me, now that I am in the OEIS databas: oeis.org/search?q=matthew+doucette.

That is all.

About the Author: I am Matthew Doucette of Xona Games, an award-winning indie game studio that I founded with my twin brother. We make intensified arcade-style retro games. Our business, our games, our technology, and we as competitive gamers have won prestigious awards and received worldwide press. Our business has won \$190,000 in contests. Our games have ranked from #1 in Canada to #1 in Japan, have become #1 best sellers in multiple countries, have won game contests, and have held 3 of the top 5 rated spots in Japan of all Xbox LIVE indie games. Our game engines have been awarded for technical excellence. And we, the developers, have placed #1 in competitive gaming competitions -- relating to the games we make. Read about our story, our awards, our games, and view our blog.