Talk:Colossally abundant number

Aren't these the same as Superior highly composite numbers????Scythe33 14:44, 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 18:31 (UTC)

See my remarks in the discussion page for superior highly composite numbers. They use different divisor functions; Scythe33's change to the definition is wrong (and would be satifsied by all superabundant numbers). DPJ, 16 Aug 2005 6:26 UTC

According to OEIS data the first 15 numbers of both series are the same.

W.r.t. "All colossally abundant numbers are Harshad numbers." is this true only in base 10 (for the Harshad definition)? Rycanada 18:57, 16 February 2006 (UTC)

Relation to the Riemann hypothesis. How are they related - This is actually a question of interest. Please put this relation explicitely.

f the RH is false, a colossally abundant number will be a counterexample (probably the first counterexample; it's 'obvious' that the first counterexample must be a superabundant number). See [1]. CRGreathouse 04:01, 15 July 2006 (UTC)

I don't understand this sentence. To me a counterexample to the Riemann hypothesis is some non-integer complex number s such that z(s) = 0. Surely you must refer to some function f : N -> {0,1} which is constantly 0 if RH holds, and your "first counterexample" must be the first n such that f(n) = 1. What f are you talking about ? (A quick parse of the article your reference yields me no hint of such an f.) --FvdP (talk) 19:56, 5 December 2007 (UTC) Doh. The answer is in the article. --FvdP (talk) 20:03, 5 December 2007 (UTC)

Couldn't the definition just say "ε > 1" instead of "ε > 0"; then the "1 + ε" terms could be reduced to just "ε"? Or does "ε" have some other significance not mentioned? — sjorford++ 11:21, 17 December 2008 (UTC)

Harshad-ness

I removed the claim that all colossally abundant numbers are Harshad as uncited. If anyone has a citation for a proof of this claim, please add it back. Thanks, Doctormatt (talk) 00:05, 19 November 2009 (UTC)

What does this page mean?

I just wanted to know what a colosally abundant number was and this page doesn't help... (thanks) Monkey Tennis (talk) 04:45, 17 April 2010 (UTC)

Ratio of consecutive superabundant numbers isn't prime

The article says "In their 1944 paper, Alaoglu and Erdős managed to show that the ratio of two consecutive superabundant numbers was always a prime number", but looking at the first few superabundant numbers shows this is clearly not true. They showed that the ratio aproached 1 for large n. — Preceding unsigned comment added by 186.178.217.23 (talk) 13:27, 7 June 2013 (UTC)

This is certainly an error, but I believe the result the contributor was thinking of was the proof of Alaoglu and Erdős that the ratio of two consecutive superior highly composite numbers is always a prime number. I will check to make sure it's the right paper and correct the article. Jaycob Coleman (talk) 09:20, 31 October 2013 (UTC)

Actually it seems that Ramanujan proved the result I mentioned, but I'm not sure which paper it's in. Jaycob Coleman (talk) 10:43, 31 October 2013 (UTC)

Here it is. Someone should cite this properly and fix the article. Jaycob Coleman (talk) 10:58, 31 October 2013 (UTC)

The Ramanujan result from 1916 isn't really relevant to this section of the article, which is discussing the 1944 paper by Alaoglu and Erdős. I have removed the claim about the ratio of consecutive superabundant numbers being a prime number - it wasn't essential in that paragraph anyway. Gandalf61 (talk) 11:23, 31 October 2013 (UTC)

An inconsistency definition versus OEIS table

It seems a minor point, but I know that it confuses people.

When we apply the definition of a colossally abundant number to n = 1, we have that n satisfies the definition with, say, ε = 1.

And yet the referenced OEIS list has 2 as the smallest element in its list. Here the OEIS definition agrees with Wikipedia. But OEIS references other lists, by Amiram Eldar and possibly by T.D. Noe, that apparently have used different definition(s).

I would like to correct the list on the Wikipedia page so that it includes 1. But how do I create a proper reference, now that the correct inclusion of 1 in the list seems to disagree with all the accepted sources? Tommy Rene Jensen (talk) 11:09, 14 August 2023 (UTC)

I have used my registration at OEIS to attempt to correct their list. But nothing has changed yet. The question therefore still stands. Tommy Rene Jensen (talk) 15:11, 4 September 2023 (UTC)

Let me expand a little bit on the point that this confuses people.

The statement "2 is the smallest colossally abundant number" formally implies, using the definition of colossally abundant numbers, that for every k > 0 there exists a natural number n for which σ(n) > n^k.

The latter would seem an important theorem in number theory, so why do we not see that statement around much in the literature, as often as we see the equivalent statement that 2 is the smallest CA number? Tommy Rene Jensen (talk) 16:30, 13 October 2023 (UTC)

The definition in OEIS for A004490 says "for all k > 1", which would exclude 1 from being in the sequence. Bubba73 ^{You talkin' to me?} 22:19, 13 October 2023 (UTC)

I looked at the edit history for A004490, and it is up for review (and someone there commented on it). Bubba73 ^{You talkin' to me?} 23:20, 13 October 2023 (UTC)

The list has been corrected on OEIS, it now starts 1,2,4,6,12, etc. Tommy Rene Jensen (talk) 18:15, 21 October 2023 (UTC)

Apologies: the edit has been submitted and is as yet not effective. I was looking at a different page that I had open for some reason. Sorry about the confusion. Tommy Rene Jensen (talk) 22:45, 21 October 2023 (UTC)