Talk:Abundant number

Miscellaneous early comments
proper multiple? -- Cimon Avaro on a pogo-stick 13:51, Sep 19, 2003 (UTC)

"open interval" termonology used with closed interval (brackets) The article mentions an "open interval" (and even provides a link with example usage) but uses brackets instead of parenthesis. Unfortunately, I am unsure whether it is supposed to actually be a "closed interval" (which would match the notation), an "open interval" (in which case, I believe the bounds should be marked with parenthesis and not square brackets), or I am simply wrong. ampre 06:17, 1 May 2004 (UTC)

According to this page, Nicomachus had a different definition for abundant numbers: he "only required that &sigma;(n) exceeds n". As &sigma;(n) is the sum of all divisors of n including n itself, &sigma;(n)>n for all positive integers larger than 1. I don't think this was what he meant, so I have removed the alternative definition from the article. Eugene van der Pijll 23:02, 17 Mar 2005 (UTC)

Mathworld / Wikipedia definitions of "abundance"
This article defines an “abundant number” as an integer whose proper divisors add up to twice the number itself. It gives this example for the abundant number 24: “1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − 2 × 24 = 12.

But in one of the external links cited (Mathworld) author Eric W. Weisstein says there that abundant numbers need only be greater than the sum of the proper divisors of that number, although he does not allow the number to be counted as a divisor of itself. Of course, the two defs are equivalent, but it is a bit fiddly. Would suggest Wikipedia fall in line with Matchworld definition, because it is simpler, and also because people are used to factors of 1 and the number itself being excluded when primes are being defined Myles325a 09:50, 6 March 2007 (UTC)


 * Agreed with Myles325a -- Michael Scott Cuthbert (talk) 14:07, 5 May 2011 (UTC)

amicable numbers
Congrats to all on the beautiful layout of these articles, and the cross-referencing on the front page. Would suggest adding "Amicable Numbers" to thio list.Myles325a 09:53, 6 March 2007 (UTC)

Conjecture
The article states that every integer greater than 20161 can be written as the sum of two abundant numbers. Has that been proven?98.170.203.61 (talk) 04:46, 28 January 2009 (UTC)

According to http://projecteuler.net/index.php?section=problems&id=23, it can be mathematically proven that every number greater than 28123 can be expressed as the sum of two abundant numbers. However, people have tested all the numbers under that and found that 20161 is the greatest integer to not have this property. txttran

Proper divisors / proper factors
In this article it seems that the proper divisors are different from the proper factors, but it's not clear for me what proper factors are. The article Divisor makes no difference. Assuming the author didn't make a mistake (as the definition is explicitely given in two different ways) there are two possibility's for the used definition of 'proper factor' that fit this definition. It seems that to fit this definition the proper factors of n are supposed to be the positive and negative factors excluding n and -n or the proper factors are supposed to be the positive factors including n. Which of both is meant I have no idea of. I'm not familiar with the conventions normally used for this. However comparing with Deficient number I would think the author maybe did make a mistake, since 'proper factor' is apparently considered to be the same as 'proper divisor' in that article. Samaritaan (talk) 19:25, 21 October 2011 (UTC)


 * An editor made a string of bad edits October 12. I have reverted to the article version before that. PrimeHunter (talk) 03:50, 22 October 2011 (UTC)

Wrong numbers?
I was doing a project when I discovered that some of the listed numbers are wrong if we had to follow the steps! The only correct ones from 12 to 36 are: 12, 24, and 30. Jiayangchang (talk) 08:48, 13 January 2012 (UTC)Jiayangchang


 * What makes you think 18 is not an adundant number ? The divisors of 18 are 1, 2, 3, 6, 9 and 18, and their sum is 39. This is greater than 2x18, so 18 is an abundant number. Gandalf61 (talk) 12:43, 13 January 2012 (UTC)


 * For a moment, I thought Jiayangchang was thinking of superabundant numbers or highly composite numbers, but 30 is neither. Owen&times; &#9742;  15:35, 15 January 2012 (UTC)

12 and 24
In the article, there is a difference in computing the abundance. 12: "12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16". 24: "For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60". Given the definition in this article, "In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself", the number itself is not seen as a proper divisor, I would think. Should the example 24 be altered? Kind regards, Vinvlugt (talk) 22:18, 19 April 2013 (UTC)


 * What is a "proper" divisor? Does that mean divisor which is smaller than the number itself (as it is said in Finnish wiki)?


 * And, actually, is makes no sense to have the number itself also. For example, 11's divisor's are 1 and 11, 1+11 = 12, therefore every number would be qualified. 82.141.119.130 (talk) 08:13, 1 May 2013 (UTC)


 * I just noted it is calculated differently, but seems to be right after all. All divisors are counted for 24, and then the sum is required to be at least double the original number. And that is also noted when calculating the abundance. Though, it should probably be explained somehow. 82.141.119.130 (talk) 08:24, 1 May 2013 (UTC)

Is there a term for 'super abundant numbers'
So, what this would be, is any abundant number whose abundance is also greater than the number.

so like for 12, the abundance is 4, because that's what is left over. What is the first number that has an abundance that is higher than the number itself? How would this be defined most succinctly? — Preceding unsigned comment added by 67.176.1.239 (talk) 14:18, 24 April 2013 (UTC)

Also, i don't mean it like the actually existing superabundant number as discussed above. That has a different definition from waht I mean. Maybe calling it 'super abundant' is not a good name, but i imagine one can see the process involved in identifying the numbers. — Preceding unsigned comment added by 67.176.1.239 (talk) 14:21, 24 April 2013 (UTC)


 * The smallest number such that the sum of its proper divisors is twice itself is 120; the proper divisors of 120 are 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60 and the sum of these is 240, so its abundance is 120. The smallest number such that the sum of its proper divisors is greater than twice itself is 180; the sum of the proper divisors of 180 is 366, so its abundance is 186. I don't know whether there is a specific name for such numbers. Gandalf61 (talk) 10:18, 25 April 2013 (UTC)

"Properties" section: misnomer?
The way I understand it, a "property" of a class of objects is something that holds for all objects in the class. For example, a property of all multiples of 4 is that they are also multiples of 2. So "properties" of abundant numbers are statements that hold true for all abundant numbers; they follow from the definition.

With that in mind, it seems to me that most of the statements in the "Properties" section are not really properties. For instance, "The smallest odd abundant number is 945" is not a property of abundant numbers. Hence, I would suggest moving some of these statements to other sections, for instance the "Examples" section. --Jadhachem (talk) 22:21, 16 May 2013 (UTC)

Mistake in Related Concepts section
The (currently) last item of the Related Concepts section claims that "A necessary and sufficient condition for (a list of primes pi to be abundant) is that the product of pi/(pi-1) be at least 2", which cites a postscript file that indeed does claim that (or at least I found the claim that it's sufficient and then stopped reading). However, this claim is false if the sequence of primes consists of just the number 2: this satisfies the criterion (the product then consists of just the factor 2/1 which is indeed at least 2), but no (finite) power of 2 is abundant. The proof sketch in the linked paper (it claims sufficiency of this condition on page 5) would work, I think, if the criterion is replaced by saying "greater than 2" rather than "at least 2", but in the absence of a good source to cite I'd suggest to just remove the whole claim. DutchCanadian (talk) 20:42, 30 April 2019 (UTC)

Reversion of June 20
Should the table of various hemiperfect numbers that was deleted in that reversion go at that article? (I think it should live somewhere--I noticed this deletion exactly because I had been looking at that table a few days ago and was confused not to be able to find it again anywhere.) Spireguy (talk) 17:28, 22 June 2020 (UTC)

Reference to tighten the density bounds
I found a paper which also references another, with tighter bounds on the density of abundant numbers than the older source given in this article. I don't have time to learn how to properly add this to the reference section, but would someone update the bounds and properly reference these two sources? https://digitalcommons.cwu.edu/cgi/viewcontent.cgi?article=1059&context=math https://collections.dartmouth.edu/archive/object/dcdis/dcdis-kobayashim2010

As an aside - from older talk section comments it appears some do not understand why the "sigma - n" definition of abundant is useful - in short, sigma is a multiplicative function which makes calculating its value (and thus sigma - n) easier when the prime factorization is known. Although this is discussed in the article on sum of divisors, perhaps it can be brought here as well to avoid questions as to why both the "sigma > 2N" and "aliquot sum > N" definitions exist. Walt (talk) 11:13, 22 May 2024 (UTC)