Talk:Weird number

message: How can the Schnirelmann density of the set of weird numbers be positive?  The defintion given of Schnirelmann density is such that if the set does not contain 1 its density is zero.  Molinari 01:44, 15 Apr 2005 (UTC)
replies: text: I think it is an error. UPINT 2nd ed (Guy 1994) says only "Erdös showed that their density is positive", which I think must be referring to Natural density. Hv 16:39, 13 July 2005 (UTC)
thread_title: Density

message: The placement of the trivia in no way detracts from the mathematical content of the page, so why remove it?  Other major pages have trivia sections, so I doubt there's an actual precedent against it.
:I second this.  It's definitely not "subtrivial" especially due to the heavy symbolism, especially numerical, that they put into their music. Thavron 04:04, 10 August 2006 (UTC)
replies: text: On the other hand this material is unsourced.  I see no reason to keep it without reliable sources.  Deltahedron (talk) 21:25, 8 March 2014 (UTC)
thread_title: Trivia

message: What's the point in the quoted result from the 1976 paper concerning very large weird numbers, given that if N is weird, Np is weird for all p>sigma(N) ?
So taking p=M#44 (the largest known prime), we get a much larger weird number using any other weird number
(provided its divisors' sum does not exceed M#44). — MFH:Talk 17:47, 4 April 2008 (UTC)
replies: text: I haven't seen the paper, but I would guess the point was to construct a large primitive weird, ie one which has no weird factor (and therefore no abundant factor). Hv (talk) 08:00, 30 June 2012 (UTC)
replies: text:  Sorry, haven't seen that reply earlier. Yes of course, it was about primitive weirds. Kravitz considered only weirds of the form 2^n p q, which are always primitive. (Back in 2008 this must have been the first time I met weird numbers. Since then I know (and published ;-) a bit more about the subject. Better late than never, thanks for your answer (which was only 4 years late). Let's meet again here in 8 years! — MFH:Talk 23:07, 10 November 2018 (UTC)
thread_title:  1976 paper 

message: I removed this disputed content:
:Alternatively, a larger weird number can be calculated using the formula 
:: W = w \cdot p^{k} 
:where w is a weird number, p is a prime greater than the sum of divisors of w, and k is any positive integer.
:Proof:
:If the set of divisors of w have a sum greater than w, then the set of divisors of w each multiplied by pk have a sum greater than wpk.  Divisors of w each multiplied by pk are all divisors of W, therefore W is abundant.
:Assume a subset of proper divisors of W has a sum of W: 
:: \sum D = d1 + d2 + d3 + ... = W 
:Let S be the set of divisors of W not divisible by pk. Let X be the set of divisors of w.
:: S = \{ x_{1} p^{0} , x_{2} p^{0} , ...\  ,\  x_{1} p^{1} , x_{2} p^{1} , ...\ ,\  x_{1} p^{k-1} , x_{2} p^{k-1}, ...\  ,\  w p^{k-1} \} 
:Factor the sum:
:: \sum S = (x_{1} + x_{2} + x_{3} + ... + w)(p^{0} + p^{1} + ... + p^{k-1}) = \sigma_{1}(w)(p^{0} + p^{1} + ... + p^{k-1}) 
:Knowing that p is greater than the sum of divisors of w:
:: p^{k} > (p^{k}-1) = (p-1)(p^{0} + p^{1} + ... + p^{k-1}) \ge \sigma_{1}(w)(p^{0} + p^{1} + ... + p^{k-1}) 
:Therefore no subset of S has a sum divisible by pk. All other divisors of W are divisible by pk, therefore all elements of D are divisible by pk.  Dividing both sides of the first equation by pk results in a subset of proper divisors of w with a sum of w, which is a contradiction if w is weird. Therefore there is no set of proper divisors of W with a sum of W and W is weird.
:Note: Using this formula with w = 70 whose divisors are {1,2,5,7,10,14,35}, p = 79 > 74 = 1+2+5+7+10+14+35 and k = 1, yields W = 70 \cdot 79 = 5530, which is weird, but is not among the first few weird numbers listed in .
Deltahedron (talk) 20:59, 8 March 2014 (UTC)
replies: text: Response from author of disputed content: 70 is a divisor or 70, therefore p must be greater than 144, such as 149. 70 * 149 = 10430, which is indeed among the first few weird numbers listed. Also, you may check the proof - it is valid.  RainMan002 (talk) 05:58, 17 March 2014 (UTC)
replies: text: Is there a reference to an independent reliable source to verify this content?  Deltahedron (talk) 06:44, 26 March 2014 (UTC)
replies: text: No. I (erroneously?) assumed a simple proof constitutes verifiability. The proof requires only minimal knowledge to verify. RainMan002 (talk) 22:06, 29 March 2014 (UTC)

text: I see that it is already contained in the paper of Benkowski and Erdős.  Deltahedron (talk) 18:06, 28 March 2014 (UTC)
replies: text: I just skimmed the paper, and it seems their version does not include the exponent on p as I have. The exponent allows direct calculation of arbitrarily large weird numbers without having to search for anything. RainMan002 (talk) 22:06, 29 March 2014 (UTC)
replies: text: In which case, I ask the question again.  Is there a reference to an independent reliable source to verify this content?  If not, then it is original research, and we can't use it.  Deltahedron (talk) 06:49, 30 March 2014 (UTC)
thread_title:  Disputed content 

message: There is no precise definition for 'wierd number' given in this article 75.140.144.193 (talk) 12:21, 6 July 2023 (UTC)
replies: text: First line says: "In number theory, a weird number is a natural number that is abundant but not semiperfect.". If you follow the links to Abundant number and Semiperfect number, you will know exactly how to define "weird numbers". You can also follow the link to OEIS (https://oeis.org/A006037). How more precise a definition do you need? Dhrm77 (talk) 14:59, 6 July 2023 (UTC)
replies: text: Perhaps more saliently, the immediately following sentence gives a precise and self-contained definition.  --JBL (talk) 18:02, 6 July 2023 (UTC)
thread_title:  lacks precise definition