Talk:Superior highly composite number

From Wikipedia, the free encyclopedia

Aren't these the same thing as Colossally abundant numbers???? The articles should be combined if so.Scythe33 14:35, 14 Jun 2005 (UTC)

I'm not an expert, but it appears they are different. In the definition of superior highly composite number, the exponent is required to be > 0, whereas in the definition of colossally abundant number, the exponent is required to be > 1, so every colossally abundant number is superior highly composite, but not conversely. See also the links to the encyclopedia of integer sequences. Revolver 6 July 2005 18:32 (UTC)

They're two completely different things. Superior highly composite numbers are defined in terms of number of divisors (d function); colossally abundant numbers are defined in terms of sum of divisors (sigma function). DPJ, 16 Aug 2005 6:13 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 ε = 1, we get , which suggests that 1 is a superior highly composite number. DRLB 14:56, 19 October 2006 (UTC)

[edit] Umm...

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)

I cannot think of a good less formal definition. An example might be better (not formatted properly for article space). Consider ε=0.6. Rounded to 3 decimals, the value of d(k)/k^0.6 for k = 1, 2, ... is 1.000, 1.320, 1.035, 1.306, 0.761, 1.365, 0.622, 1.149, 0.803, 1.005, ... It can be proved (skipped here) that the value for k=6, d(6)/6^0.6 = 1.365, is the largest value in this infinite sequence. Therefore 6 is a superior highly composite number. The same value of ε can never be used for two distinct superior highly composite numbers, since at most one of them can give the maximal value of d(k)/k^ε. Does this help? PrimeHunter (talk) 00:20, 21 May 2008 (UTC)

On second thought, there may be boundary values of ε where distinct superior highly composite numbers give the same maximal value of d(k)/k^ε. PrimeHunter (talk) 00:26, 21 May 2008 (UTC)

I think I get it. So... if I take a positive integer, and divide its # of divisors by the original number raised to the power of ε (which is just any positive number), and the result I get is the maximum for that function, than the positive integer I used must be a superior highly composite number. If I'm right, than you have explained this very well (and I realize it's really difficult to give any less formal of a definition than what the article states... thank you very much!). Matt Yeager ♫ (Talk?) 09:49, 24 May 2008 (UTC)

You are right. I will add a better formatted example to the article later. PrimeHunter (talk) 12:36, 24 May 2008 (UTC)