5040 and other Anti-Prime Numbers - Numberphile
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Fun tidbit: You might naively guess that the number of distinct prime divisors cannot go down from one highly composite number to the next. (By number of distinct prime divisors of a number we mean how many it has not counting repeats. So for example, 48 = 2-4 -3 has only two distinct prime divisors) And this is true for a while. But it actually breaks down eventually. 27720 is highly composite with 27720= (2-3(3-2(5(7(11(so 5 distinct prime factors. The next highly composite is 45360 = (2-4(3-4(5(7. There are some even larger examples where the number of distinct prime factors goes down by 2! There are not any known where it goes down by 3, and as of a few years ago, whether there were any such highly composite numbers was open. I think this is still an open problem but could be mistaken.
Date: 2022-04-08
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Comments and reviews: 9
Danish
-A highly composite number, sometimes called an antiprime number, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915. However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. -
this is from wikipedia
and its official this is anti-prime
reply
-A highly composite number, sometimes called an antiprime number, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915. However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. -
this is from wikipedia
and its official this is anti-prime
reply
LegoChickenGuy
Is there a number that has exactly 5 divisors?
Edit: 16 does. Actually it has to be only numbers that are the fourth power of a prime because that'll give 5 combinations.
The number's (n's) prime factorization is p-4, so the possible combinations for divisors are:
1, p, p-2, p-3, p-4 (which = n.
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Is there a number that has exactly 5 divisors?
Edit: 16 does. Actually it has to be only numbers that are the fourth power of a prime because that'll give 5 combinations.
The number's (n's) prime factorization is p-4, so the possible combinations for divisors are:
1, p, p-2, p-3, p-4 (which = n.
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aLex
I just realized why the dozen is an important commercial unit! And why there's 60 minutes in an hour and 24 hours in a day. 360 (also antiprime) is mighty close to the days in a year, I wonder if there's some mysterious reason for that, hum. but maybe that's why they chose 360 degrees to a full circle, Ha!
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I just realized why the dozen is an important commercial unit! And why there's 60 minutes in an hour and 24 hours in a day. 360 (also antiprime) is mighty close to the days in a year, I wonder if there's some mysterious reason for that, hum. but maybe that's why they chose 360 degrees to a full circle, Ha!
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Son
I liked number 72 for almost my whole life thinking is it a highest composite number under 100
But after watching this video it turns out to be 60.
My whole belief was a mistake for the longest time haha.
But 72 is still my favorite number.
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I liked number 72 for almost my whole life thinking is it a highest composite number under 100
But after watching this video it turns out to be 60.
My whole belief was a mistake for the longest time haha.
But 72 is still my favorite number.
reply
Maria's
There is a very interesting recent research book that have miraculously answered almost all the questions concerning Prime numbers, it is available on Amazon by the name of: THE FORMULAS OF NONPRIMES REVEALING ALL THE PRIME NUMBERS
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There is a very interesting recent research book that have miraculously answered almost all the questions concerning Prime numbers, it is available on Amazon by the name of: THE FORMULAS OF NONPRIMES REVEALING ALL THE PRIME NUMBERS
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fredy
So we can build a very big highly composite number depend on the number of consecutive terms.
So we can build 2 series of primes.
The one and his factorisation.
Never use the term division here.
There is no division
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So we can build a very big highly composite number depend on the number of consecutive terms.
So we can build 2 series of primes.
The one and his factorisation.
Never use the term division here.
There is no division
reply
Aidan
Turns out if you list of the number of factors of the champion composite numbers it's also the harmonic numbers,
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48.
not sure what to do with that link but there you go
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Turns out if you list of the number of factors of the champion composite numbers it's also the harmonic numbers,
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48.
not sure what to do with that link but there you go
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Fallen
720 is a far more useful highly composite number than 5040. It is divisible by all the same factors as 5040 except 7, and how often do you need to divide something into 7 equal parts?
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720 is a far more useful highly composite number than 5040. It is divisible by all the same factors as 5040 except 7, and how often do you need to divide something into 7 equal parts?
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Huh. I was experimenting with ways to make near-perfect polygons with nothing but a cup and a ruler, and now I realize I was playing with highly composite numbers the whole time.
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Huh. I was experimenting with ways to make near-perfect polygons with nothing but a cup and a ruler, and now I realize I was playing with highly composite numbers the whole time.
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