Talk:Highly composite number

"The term was coined by Ramanujan (1915), who showed that there are infinitely many such numbers"

Isn't it a trivial consequence from the fact that there are numbers with arbitrarily large number of divisors? 132.65.251.182 (talk) 08:18, 23 June 2015 (UTC)

Yes, we shouldn't say a mathematician "showed" a completely trivial observation anyone could make. I have removed it.[1] PrimeHunter (talk) 23:12, 23 June 2015 (UTC)

For the half-asleep (including me when I first read that): there exists a number with k divisors, for any nonnegative integer k. (Example: 2^{k − 1} will have k divisors.) And since there are such numbers for each k, there must be a smallest one among them for each k. So we can construct a sequence, where the kth term is the smallest number that has k divisors. Then we can simply discard every term which is larger than a subsequent term. Voilà, the list of highly composite numbers.

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 n. Then find an odd prime p that doesn't divide n (which is obviously always possible). pn > n (obviously). Now pn has all the factors of n, and then some (p of course, and the products of p and the factors of n). So it has more factors, so n cannot be the last term. Double sharp (talk) 09:53, 18 September 2015 (UTC)

Right. kn for any k > 1 would work, since kn is a divisor of itself but never of n. PrimeHunter (talk) 10:14, 18 September 2015 (UTC)

You're right, it's even simpler (I must've been mentally considering only proper divisors). The point stands, though. (^_^) Double sharp (talk) 13:01, 18 September 2015 (UTC)

What are the * for in the table?

The asterisk behind 2,4,5,9,10, ... doesn't seem to be explained anywhere. Maxiantor (talk) 14:47, 28 July 2016 (UTC)

They link to Superior highly composite number but it's probably too subtle. PrimeHunter (talk) 19:53, 28 July 2016 (UTC)

Table Doesn't Show the Divisors!

The article is about a number "with more divisors 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!

For example, the 5th row of the table is for the number 12. And 4 is certainly a divisor of 12. 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. --johantheghost (talk) 08:16, 16 February 2020 (UTC)

@Johantheghost: 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.[2] 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. PrimeHunter (talk) 23:21, 16 February 2020 (UTC)

@PrimeHunter: 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). --johantheghost (talk) 16:46, 22 February 2020 (UTC)

What are all the numbers with distinct digits that have more divisors than any smaller number with distinct digits?

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.

http://www.worldofnumbers.com/ninedig6.htm 98.0.38.216 (talk) 03:17, 24 November 2022 (UTC)

I have computed the full sequence:

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, 376215840, 752431680, 1475923680, 2591348760, 2834591760, 7691523840

It isn't in published sources so it doesn't belong in the article. PrimeHunter (talk) 19:26, 24 November 2022 (UTC)

Sounds like it's eligible for OEIS. -- AnonMoos (talk) 22:02, 24 November 2022 (UTC)

How did you compute the full sequence? I knew it would include 83160 because it is the highest highly composite number with distinct digits. 98.0.38.216 (talk) 22:36, 24 November 2022 (UTC)

I used a highly inefficient PARI/GP line:

r=0;for(n=1,10^10,if(setisset(vecsort(digits(n))),d=numdiv(n);if(d>r,print1(n", ");r=d)))

It got the job done in a few hours so I didn't bother to optimize it. PrimeHunter (talk) 23:05, 24 November 2022 (UTC)

I was not sure what article to post this on. 98.0.38.216 (talk) 23:49, 24 November 2022 (UTC)

Article talk pages are for discussing improvements to the article. If it's not about changing the article then it belongs at Wikipedia:Reference desk/Mathematics. PrimeHunter (talk) 00:30, 25 November 2022 (UTC)

The number of divisors of each number:

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 66.181.118.116 (talk) 21:13, 31 December 2022 (UTC)

Here are numbers a, b where all numbers in this sequence greater than a are divisible by b.

1 2

4 6

6 12

48 60

13860 120

31680 840

371280 2520

231547680 27720

1475923680 360360

2591348760 16576560

2834591760 7691523840 66.181.118.116 (talk) 03:18, 5 January 2023 (UTC)

Antiprime?

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 Numberphile's video on the topic. 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? MathWorld, for example, recognizes antiprime as a synonym for a highly composite number. Saucy^{[talk – contribs]} 09:17, 19 October 2023 (UTC)

I think that recreational mathematics is kind of its own thing. Not all recreational math concepts have to be taken up by academic mathematicians to be suitable for Wikipedia... AnonMoos (talk) 21:04, 22 October 2023 (UTC)

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 superior highly composite number. 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. --172.88.48.178 (talk) 10:17, 24 December 2023 (UTC)