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860:. As much as I like the word, I'm not sure that this term has any use in scholarly contexts, but then again I'm not one to read the latest academic journals. Should it be removed from the article for not being used academically? Or has the term become popular enough among casual mathematicians for that to not matter?
665:
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 3780, 6480, 6720, 7560, 13860, 21840, 31680, 32760, 45360, 73920, 83160, 231840, 317520, 371280, 524160, 786240, 1375920, 1572480, 2489760, 5397840, 6153840, 7469280, 14938560, 39584160, 59376240, 75841920, 79168320, 231547680,
641:
The smallest highly composite number that repeats digits is 2520. The list the same for numbers up to 2520. After that, the next number is 3780, with 48 divisors. From this website, I think the highest number is 7691523840, with 1728 divisors.
896:
In addition to not being scholarly at all... the term "antiprime" isn't even a good term for what a highly composite number is! If you want something that is truly the opposite of a prime number, the best thing for that would be a
947:
901:. No sensible mathematician would ever refer to highly composite numbers as "antiprimes". Until one does (Brady Haran is a journalist, not a mathematician), it does not belong in this article. --
539:. But the number "4" doesn't appear anywhere in that row of the table. Seems to me that rather than listing the prime factors of numbers, or anyhow as well as, we should list the divisors. --
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1 2 3 4 6 8 9 10 12 16 18 20 24 30 32 36 40 48 50 56 64 72 80 84 96 100 112 128 144 150 160 192 224 240 256 288 300 320 384 448 576 600 640 672 768 960 1120 1152 1280 1600 1728
532:
than any smaller positive integer". And there's a big table of "the initial or smallest 38 highly composite numbers". But the table doesn't tells us what the divisors are!
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The term "antiprime" or "anti-prime" has been added and removed as an alternate term several times throughout the article's history. The term has gained popularity since
561:
The largest number in the table is 720720 with 240 divisors. That seems too much to list but I have added a table with divisors of the first 15 highly composite numbers.
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915:
it's really irrelevant who approves of the term, their opinion of it, etc. it's become a popular term, and people who search for it should be able to find this page.
585:
OK, thanks! Yeah, 240 is rather a lot... :-/ But this makes sense to me, we can at least see what divisors look like now (for folks who might be wondering). --
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Now how do we know that they don't run out, and that you don't discard nearly all terms? Simple. Suppose the last term in the HCN sequence is
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is kind of its own thing. Not all recreational math concepts have to be taken up by academic mathematicians to be suitable for
Knowledge...
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How did you compute the full sequence? I knew it would include 83160 because it is the highest highly composite number with distinct digits.
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divisors. Then we can simply discard every term which is larger than a subsequent term. VoilĂ , the list of highly composite numbers.
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Article talk pages are for discussing improvements to the article. If it's not about changing the article then it belongs at
463:
You're right, it's even simpler (I must've been mentally considering only proper divisors). The point stands, though. (^_^)
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If we added it as a new column in the existing table then the cells would become up to 32 lines high on narrow screens.
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Yes, we shouldn't say a mathematician "showed" a completely trivial observation anyone could make. I have removed it.
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What are all the numbers with distinct digits that have more divisors than any smaller number with distinct digits?
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Isn't it a trivial consequence from the fact that there are numbers with arbitrarily large number of divisors?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Factors of 240: 1, 2, 3, 4. 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240.
791:
Here are numbers a, b where all numbers in this sequence greater than a are divisible by b.
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What about 296? Including 1 and 2 it has 8 divisors. Or am I misunderstanding something?
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For the half-asleep (including me when I first read that): there exists a number with
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864:, for example, recognizes antiprime as a synonym for a highly composite number.
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by a now-blocked IP sock. Shall we set a reasonable maximum for this table?
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The asterisk behind 2,4,5,9,10, ... doesn't seem to be explained anywhere.
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r=0;for(n=1,10^10,if(setisset(vecsort(digits(n))),d=numdiv(n);if(d: -->
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It got the job done in a few hours so I didn't bother to optimize it.
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A highly composite number has more divisors than any smaller number.
666:
376215840, 752431680, 1475923680, 2591348760, 2834591760, 7691523840
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It isn't in published sources so it doesn't belong in the article.
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For example, the 5th row of the table is for the number 12. And
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This list was expanded to the first 19 a few months ago by
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divisors.) And since there are such numbers for each
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359:, there must be a smallest one among them for each
232:This article has not yet received a rating on the
1012:Knowledge level-5 vital articles in Mathematics
964:Factors of 296: 1, 2, 4, 8, 37, 74, 148, 296.
742:I was not sure what article to post this on.
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363:. So we can construct a sequence, where the
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646:http://www.worldofnumbers.com/ninedig6.htm
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212:Knowledge:WikiProject Numbers
206:and see a list of open tasks.
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