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Aren't these the same thing as Colossally abundant numbers???? The articles should be combined if so. Scythe33 14:35, 14 Jun 2005 (UTC)
Further research shows that the definition is wrong. (At 14:27, 14 June 2005 Scythe33 changed the definition; the article was right the first time.) According to the definition supplied by Scythe33, every highly composite number would qualify. DPJ, 16 Aug 2005 6:21 UTC
Why isn't n=1 included as a superior highly composite number? Putting = 1, we get , which suggests that 1 is a superior highly composite number. DRLB 14:56, 19 October 2006 (UTC)
Wow, this is densely packed information. There's a formal definition given, which I'm sure is easy to understand for mathematicians, but I'm no slouch at math myself and it's clear as mud to me. Can someone give an informal definition? Matt Yeager ♫ (Talk?) 22:55, 20 May 2008 (UTC)
There has been controversy over whether 840 should be included in the list: 164.127.191.68 ( talk · contribs · WHOIS) asserts that it should. However, (sequence A002201 in the OEIS) disagrees. Deltahedron ( talk) 18:23, 11 July 2014 (UTC)
Sorry, but i don,t understand the definition of highly composite numners. 94.254.209.29 ( talk) 20:12, 25 July 2014 (UTC)
Plot[{32 - 24*(7/3)^eps, 32 - 48*(1/3)^eps}, {eps, .2, 1}]
This:
In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to the number itself
should be defined. Wlod ( talk) 07:01, 22 October 2014 (UTC)
I obviously did not work this out by hand. :-) I did this for putting in the article, but they quickly became walls of text so enormous that I truncated it at the tenth SHCN (720720) – which already has every integer from 1 to 16 as a divisor!
see: /Factors
A neat illustration of just how superior highly composite these numbers are, I guess, but an impenetrable wall of text for the last few. Double sharp ( talk) 14:44, 26 February 2015 (UTC)
This graph:
...and the graph below it:
File:Highly_composite_numbers.svg
both present good information. But this isn't the best info to present in order to communicate how highly composite a number is. One of the most important math concepts when analyzing data is
normalization. For example, we don't say that China is a much more highly composite country than, say, Luxembourg. You expect there to be a much wider range of diversity of people within a country of more than a billion people than you would with a small country. So the important info jumps forward when you normalize the stats. How diverse is the population with respect to the size of the population? This is why you see "
per capita" used so often. If you fail to normalize your data, important aspects within your dataset can get buried. Ok, now back to the topic of this article...
The key info to present in the graph is not merely how many divisors are within any number. You obviously expect there to be more divisors for larger numbers. So to capture the actual composite quality of any number, it must be normalized to the size of the number. One normalization method is to do a straight division of the number of divisors by the integer. But more meaningful info comes forward when the # of divisors is normalized by the method of dividing by the square of the integer. And a strong argument says that the proper normalization method is to divide the # of divisors by the cube of the integer. The argument for normalizing with a higher power of the integer is that divisors are not used to form a one-dimensional entity. A divisor need to be multiplied by something else in order to get back to the integer, so it can be considered to be an area or a volume-type of parameter (analogous to area/volume). And so the proper normalization is by dividing by the square or the cube of the integer.
When these graphs are properly normalized, then the special status of the numbers 12 and 60 become evident. But as the graphs stand today, all it really communicates is that "there are more people in China". Yes, it does an adequate job of showing that 360 is a special number. But these graphs have buried the reality of how much more special the composite qualities of 12 and 60 are. -- Tdadamemd19 ( talk) 14:54, 5 April 2019 (UTC)
I tweaked the opening statement... In mathematics, a superior highly composite number or convergence point is a natural number which has more divisors than any other number scaled relative to some positive power of the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. 50.245.89.150 ( talk) 01:38, 18 July 2019 (UTC)
I believe it needs to be reworded in a way that's more intuitive than how it is now. I made the (now reverted) change because I believed "when adjusted for magnitude" seemed much easier to understand than "scaled relative to some positive power of the number itself". For example, when I first read this, I thought "the number itself" was referring to the natural number in question, not just "any" number scaled relative to the same power of "any" number. What about a middle ground between my change and how it is now? Avengingbandit 20:14, 18 August 2021 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Aren't these the same thing as Colossally abundant numbers???? The articles should be combined if so. Scythe33 14:35, 14 Jun 2005 (UTC)
Further research shows that the definition is wrong. (At 14:27, 14 June 2005 Scythe33 changed the definition; the article was right the first time.) According to the definition supplied by Scythe33, every highly composite number would qualify. DPJ, 16 Aug 2005 6:21 UTC
Why isn't n=1 included as a superior highly composite number? Putting = 1, we get , which suggests that 1 is a superior highly composite number. DRLB 14:56, 19 October 2006 (UTC)
Wow, this is densely packed information. There's a formal definition given, which I'm sure is easy to understand for mathematicians, but I'm no slouch at math myself and it's clear as mud to me. Can someone give an informal definition? Matt Yeager ♫ (Talk?) 22:55, 20 May 2008 (UTC)
There has been controversy over whether 840 should be included in the list: 164.127.191.68 ( talk · contribs · WHOIS) asserts that it should. However, (sequence A002201 in the OEIS) disagrees. Deltahedron ( talk) 18:23, 11 July 2014 (UTC)
Sorry, but i don,t understand the definition of highly composite numners. 94.254.209.29 ( talk) 20:12, 25 July 2014 (UTC)
Plot[{32 - 24*(7/3)^eps, 32 - 48*(1/3)^eps}, {eps, .2, 1}]
This:
In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to the number itself
should be defined. Wlod ( talk) 07:01, 22 October 2014 (UTC)
I obviously did not work this out by hand. :-) I did this for putting in the article, but they quickly became walls of text so enormous that I truncated it at the tenth SHCN (720720) – which already has every integer from 1 to 16 as a divisor!
see: /Factors
A neat illustration of just how superior highly composite these numbers are, I guess, but an impenetrable wall of text for the last few. Double sharp ( talk) 14:44, 26 February 2015 (UTC)
This graph:
...and the graph below it:
File:Highly_composite_numbers.svg
both present good information. But this isn't the best info to present in order to communicate how highly composite a number is. One of the most important math concepts when analyzing data is
normalization. For example, we don't say that China is a much more highly composite country than, say, Luxembourg. You expect there to be a much wider range of diversity of people within a country of more than a billion people than you would with a small country. So the important info jumps forward when you normalize the stats. How diverse is the population with respect to the size of the population? This is why you see "
per capita" used so often. If you fail to normalize your data, important aspects within your dataset can get buried. Ok, now back to the topic of this article...
The key info to present in the graph is not merely how many divisors are within any number. You obviously expect there to be more divisors for larger numbers. So to capture the actual composite quality of any number, it must be normalized to the size of the number. One normalization method is to do a straight division of the number of divisors by the integer. But more meaningful info comes forward when the # of divisors is normalized by the method of dividing by the square of the integer. And a strong argument says that the proper normalization method is to divide the # of divisors by the cube of the integer. The argument for normalizing with a higher power of the integer is that divisors are not used to form a one-dimensional entity. A divisor need to be multiplied by something else in order to get back to the integer, so it can be considered to be an area or a volume-type of parameter (analogous to area/volume). And so the proper normalization is by dividing by the square or the cube of the integer.
When these graphs are properly normalized, then the special status of the numbers 12 and 60 become evident. But as the graphs stand today, all it really communicates is that "there are more people in China". Yes, it does an adequate job of showing that 360 is a special number. But these graphs have buried the reality of how much more special the composite qualities of 12 and 60 are. -- Tdadamemd19 ( talk) 14:54, 5 April 2019 (UTC)
I tweaked the opening statement... In mathematics, a superior highly composite number or convergence point is a natural number which has more divisors than any other number scaled relative to some positive power of the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. 50.245.89.150 ( talk) 01:38, 18 July 2019 (UTC)
I believe it needs to be reworded in a way that's more intuitive than how it is now. I made the (now reverted) change because I believed "when adjusted for magnitude" seemed much easier to understand than "scaled relative to some positive power of the number itself". For example, when I first read this, I thought "the number itself" was referring to the natural number in question, not just "any" number scaled relative to the same power of "any" number. What about a middle ground between my change and how it is now? Avengingbandit 20:14, 18 August 2021 (UTC)