Talk:Highly composite number
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|WikiProject Mathematics||(Rated Start-class, Mid-priority)|
"The term was coined by Ramanujan (1915), who showed that there are infinitely many such numbers"
- Yes, we shouldn't say a mathematician "showed" a completely trivial observation anyone could make. I have removed it. 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)