# Talk:Weird number

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Field: Number theory

## Density

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)

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)

## Trivia

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)
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)

## Lower bound for odd weird numbers

I took out the claim "if any [odd weird numbers exist], they must be greater than 10^18 (as noted by Bob Hearn in a July 2005 posting to the SeqFans mailing list)" since it was insufficiently sourced, due to the fact that the result is not published (as far as I can tell). I replaced it with the rather trivial lower bound 10^6 given by Benkoski and Erdos. If anyone knows a better published lower bound, please add it. Thanks. Doctormatt 18:45, 25 August 2007 (UTC)

If you must: the 10^6 result is ridiculously trivial. Frankly I don't think you can expect to find a published result on this, since even the 10^18 bound is too trivial to be accepted as a paper in any mathematical journal. Considering that fact, I think that a reference to Sloane's encyclopedia is more than sufficient -- but as a WP:0RR devotee I will leave it.
Hmm. Perhaps some undergraduate journal would accept such a result?
CRGreathouse (t | c) 03:20, 12 October 2007 (UTC)
The 10^6 bound is not the point of the Erdos paper. An improved bound could be simply part of a paper proving something significant. Thanks for not reverting. Doctormatt 05:59, 12 October 2007 (UTC)
That was a 1974 paper, I believe. Searching JSTOR and a few other databases from 1975 on, I wasn't able to find a single paper mentioning "weird numbers" in this context. (JSTOR had half a dozen ads and 2-3 papers from undergraduate journals calling various transcendentals "weird"; Academic Search Complete pulled up only a newspaper article which was similarly unrelated.)
CRGreathouse (t | c) 13:04, 12 October 2007 (UTC)
Guy lists this paper (in section B2 of Unsolved Problems in NT):
• Sidney Kravitz, A search for large weird numbers, J. Recreational Math., 9 (1976-77) 82-85.
I think that might be worth a look (unfortunately we don't have JRM at my institution). MathSciNet also lists this:
• On primitive weird numbers. A collection of manuscripts related to the Fibonacci sequence, pp. 162--166, Fibonacci Assoc., Santa Clara, Calif., 1980.
which might also be worth a look (this isn't in our library either...) Doctormatt 14:50, 12 October 2007 (UTC)
I'll see if I can locate either. My library, like yours, doesn't have the Journal of Recreational Mathematics, but I should be able to get a copy somehow. I'm not even sure what the other one is but I'll ask after it.
As clarification, I added only the 10^17 bound from the OEIS, not the 10^18 bound from the seqfans list. I still feel the 10^17 result is proper, regardless of the 10^18 bound's propriety. (I didn't know about the 10^18 bound at all until I looked in the history after my addition.)
CRGreathouse (t | c) 16:52, 12 October 2007 (UTC)
I've requested the Kravitz article by interlibrary loan. Doctormatt 17:17, 17 October 2007 (UTC)
I did the same thing earlier today.
I was able to find one reference -- CN Friedman, "Sums of Divisors and Egyptian Fractions",Journal of Number Theory (1993) -- which shows the weak lower bound of 232 ≈ 4×109. The result is attributed to "M. Mossinghoff at University of Texas - Austin".
CRGreathouse (t | c) 03:20, 19 October 2007 (UTC)
Great, I'll check out that paper. I got the Kravitz article: no mention of odd weird numbers, just a method for generating large even ones. I added his results to the article. The "large" weird number he gives is clearly outdated - I think it's high time that someone publish a paper with some modern weird number calculations... Doctormatt 04:32, 19 October 2007 (UTC)
Yes, I did not consider the search to 10^18 worth publishing. But I will investigate the options. If nothing else I can make my code available. Actually there were several tricks required to get to 10^18; even checking a single candidate odd weird (to see whether it is semiperfect) in this range is extremely slow if done naively, and of course the vast majority of the space must be weeded out before explicit checking. Bobhearn 00:11, 31 October 2007 (UTC)
UPINT would be a likely place to get this to be a published result. I know the author posts to seqfan occasionally, but I'm not sure how assiduously he reads it. Hv (talk) 13:06, 20 January 2008 (UTC)

## 1976 paper

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)

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)

## Disputed content

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 (sequence A006037 in OEIS).

Deltahedron (talk) 20:59, 8 March 2014 (UTC)

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)
Is there a reference to an independent reliable source to verify this content? Deltahedron (talk) 06:44, 26 March 2014 (UTC)
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)
I see that it is already contained in the paper of Benkowski and Erdős. Deltahedron (talk) 18:06, 28 March 2014 (UTC)
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)
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)

## Simpler definition for weird numbers studied by Sidney Kravitz

If Q and R are primes and k a positive integer such that (Q+1)(R+1)=2^k(Q+R) and Q≥R>2^k, then 2^(k-1)QR is weird. I realize that Sidney Kravitz defined R in terms of Q and k as is currently done in the article, but it's not obvious without rearranging that Q and R are defined symmetrically. Should this be changed? Jaycob Coleman (talk) 01:32, 19 April 2014 (UTC)