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Numberphile made a video about "highly composite numbers" / "anti-primes", which are numbers that have more factors than all smaller numbers. https://youtu.be/2

Numberphile made a video about "highly composite numbers" / "anti-primes", which are numbers that have more factors than all smaller numbers. https://youtu.be/2JM2oImb9Qg https://youtu.be/2JM2oImb9Qg There is an oeis sequence of them that starts "1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840...", which notably does not include 480. https://oeis.org/A002182 https://oeis.org/A002182

moralestapia 5y ago
This is the kind of engineering that it's almost impossible to screen in interviews (particular for HR-minded pals) and is a big part of what makes a 10x-100x engineer (which definitely DO exist). Obviously not this thing just by itself, but this sort of knowledge, applied to everyday decisions that get compound over time.
mikepurvis 5y ago
But 480 is 2*240, so it's definitely a close contender.
alberth 5y ago
That’s super interesting. It’s counterintuitive but 480 has the same # of factors that 360 has (which is 24 factors) https://www.calculatorsoup.com/calculators/math/factors.php https://www.calculatorsoup.com/calculators/math/factors.php
- _v7gu 5y ago
  so it's weakly highly composite?
  - smolder 5y ago
    I think you can still say it's highly composite, but doesn't qualify as an anti-prime when it isn't the lowest number with that many factors.
    - infogulch 5y ago
      "anti-prime" is just Brady trying to rebrand an old boring name to sound cooler (as he does). "highly composite" has a perfectly well defined meaning already, trying to shift it to shoehorn in some artificial distinction between it and "anti-prime" is counterproductive. If the aim is to produce a more descriptive name for a similar sequence of numbers that includes 480, I think GP's "weakly highly composite" is much better. I might propose just "weakly composite" or "mediocrely composite". Edit: Actually, this was (also) already done by Ramanujan and has a better name than any proposed here: http://oeis.org/A067128 http://oeis.org/A067128 - "largely composite numbers" which does in fact contain 480. Perhaps every sharded system should choose a shard count from this list?
      - smolder 5y ago
        Thanks for correcting me, but I don't think drawing a distinction between "anti-prime" (clearly specific) and "highly composite" (vague) is counterproductive. On the other hand, "highly" and "largely" mean roughly the same thing, so treating "highly composite" and "largely composite" as distinct in this way is linguistically counterintuitive and counterproductive. Likewise, "weakly highly" reads like ambiguous nonsense when you don't go out of your way to specially define it.
jedberg 5y ago
It's true that 480 isn't on the list, but you'll notice there is a big skip in the list between 360 and 720, so if you want something in between, 480 is a good choice since it's still a multiple of 12.
- infogulch 5y ago
  Yeah, to be clear, I think there are many important factors that go into choosing a shard count, and blindly jumping to something just because it's some platonic mathematical ideal seems naive. I just thought this is a neat related fact and is probably a good starting point if you're in the process of choosing a shard count.
karakot 5y ago
there is a factorial sequence in it, makes sense. 24, 120, 720...
- varyherb 5y ago
  Not to mention 1, 2, and 6 :) 7! = 5040 shows up as well in the OEIS sequence. Buuut, 8! = 40320 does not...too many useless 2's in the prime factorization?

moralestapia 5y ago

This is the kind of engineering that it's almost impossible to screen in interviews (particular for HR-minded pals) and is a big part of what makes a 10x-100x engineer (which definitely DO exist). Obviously not this thing just by itself, but this sort of knowledge, applied to everyday decisions that get compound over time.

mikepurvis 5y ago

But 480 is 2*240, so it's definitely a close contender.

alberth 5y ago

That’s super interesting. It’s counterintuitive but 480 has the same # of factors that 360 has (which is 24 factors) https://www.calculatorsoup.com/calculators/math/factors.php https://www.calculatorsoup.com/calculators/math/factors.php

_v7gu 5y ago
so it's weakly highly composite?
- smolder 5y ago
  I think you can still say it's highly composite, but doesn't qualify as an anti-prime when it isn't the lowest number with that many factors.
  - infogulch 5y ago
    "anti-prime" is just Brady trying to rebrand an old boring name to sound cooler (as he does). "highly composite" has a perfectly well defined meaning already, trying to shift it to shoehorn in some artificial distinction between it and "anti-prime" is counterproductive. If the aim is to produce a more descriptive name for a similar sequence of numbers that includes 480, I think GP's "weakly highly composite" is much better. I might propose just "weakly composite" or "mediocrely composite". Edit: Actually, this was (also) already done by Ramanujan and has a better name than any proposed here: http://oeis.org/A067128 http://oeis.org/A067128 - "largely composite numbers" which does in fact contain 480. Perhaps every sharded system should choose a shard count from this list?
    - smolder 5y ago
      Thanks for correcting me, but I don't think drawing a distinction between "anti-prime" (clearly specific) and "highly composite" (vague) is counterproductive. On the other hand, "highly" and "largely" mean roughly the same thing, so treating "highly composite" and "largely composite" as distinct in this way is linguistically counterintuitive and counterproductive. Likewise, "weakly highly" reads like ambiguous nonsense when you don't go out of your way to specially define it.

jedberg 5y ago

It's true that 480 isn't on the list, but you'll notice there is a big skip in the list between 360 and 720, so if you want something in between, 480 is a good choice since it's still a multiple of 12.

infogulch 5y ago
Yeah, to be clear, I think there are many important factors that go into choosing a shard count, and blindly jumping to something just because it's some platonic mathematical ideal seems naive. I just thought this is a neat related fact and is probably a good starting point if you're in the process of choosing a shard count.

karakot 5y ago

there is a factorial sequence in it, makes sense. 24, 120, 720...

varyherb 5y ago
Not to mention 1, 2, and 6 :) 7! = 5040 shows up as well in the OEIS sequence. Buuut, 8! = 40320 does not...too many useless 2's in the prime factorization?