# Talk:Highly composite number

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Field:  Number theory

## "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: 2k − 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)