Lowest Unused Anti-Prime (Highly Composite Number) Factors

Sunday, February 3, 2019
By: Matthew Doucette

I may have discovered a new number sequence, 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: null
2: null
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
...

 

See Anything Special?

It appears to be dominated by prime numbers starting at five: 2, 3, 5, 7, 11, 13, 17, 19, ... However, not all of them are prime.

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.

 

History:

Interestingly, Jason Doucette is already in the OEIS database: https://oeis.org/search?q=jason+doucette. He is listed in some sequences relating to his world record work in palindrone math theory.

 

More to come!

 

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 $180,000 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.