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5040 and other Anti-Prime Numbers - Numberphile

Numberphile1 HN recommenders · 2018–2024
2.4M views · explainer · evergreen · Uploaded 2016-07-06

Numberphile's James Grime discusses highly composite numbers, nicknamed anti-primes, which have more factors than any smaller number, using 5040 as the central example. It touches on Plato and Ramanujan.

HN points to it whenever highly divisible numbers matter, such as sharding, base 12, time units and music.

Pure mathematicsRecreational & visual math

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4 comments from Hacker News

@Balgair2018 · in “The fifth hyperfactorial: 5⁵×4⁴×3³×2²×1¹=86400000 milliseconds is exactly 1 day”

Numbers like 24, 60, and 360 are 'highly composite numbers'. Here is a GREAT video on these 'anti-prime' numbers : [VIDEO] These anti-prime numbers were studied first by Ramanujan (of course) and conditions are that: 1) The prime factors must be consecutive (2^n X 3^m not 2^n X 5^m) 2) The powers of the prime factors must be deceasing as the prime number increases (2^4 X 3^3 not 2^1 X 3^5) 3) The numbers end with the the highest prime to the highest prime (2^4 X 3^1 not 2^3 X 3^2). Except for the numbers 2 and 4. More here: [LINK]

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@infogulch2021 · in “Lessons learned from sharding Postgres at Notion”

Numberphile made a video about "highly composite numbers" / "anti-primes", which are numbers that have more factors than all smaller numbers. [VIDEO] 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. [LINK]

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@nikeee2022 · in “Why 12 notes in Western music?”

These numbers are called "Highly Composite Numbers" [0]. Basically, it is a series of numbers where each number has more factors than the number before it (and is the first number with that number of factors). As you hinted, they are especially useful if your number system does not have fractions or decimal places and you still want to divide things. You may recognize the beginning of the series: 1 2 4 6 12 24 36 48 60 120 180 240 360 720 Numberphile [1] calls them "Anti-Primes". [0]: [LINK] [1]: [VIDEO]

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@willis9362024 · in “Decimal Time”

They're all magical because they are small. 12 is a highly factorable number and 10 is not. Ancient Greece was not ignorant of this fact. [LINK] [VIDEO]

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