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{{Dablink|The term "weird number" also refers to [[Two's complement#The most negative number|a phenomenon in two's complement arithmetic]].}}
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In [[number theory]], a '''weird number''' is a [[natural number]] that is [[abundant number|abundant]] but not [[semiperfect number|semiperfect]].<ref>
{{cite journal
  | last =Benkoski
  | first =Stan
  | authorlink =
  | coauthors =
  | title =E2308 (in Problems and Solutions)
  | journal =The American Mathematical Monthly
  | volume =79
  | issue =7
  | page =774
  | date =Aug.-September 1972
  | doi =10.2307/2316276
  | jstor =2316276
}}</ref><ref>{{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7|oclc=54611248}}  Section B2.</ref>
In other words, the sum of the proper [[divisor]]s (divisors including 1 but not itself) of the number is greater than the number, but no [[subset]] of those divisors sums to the number itself.
 
== Examples ==
 
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but ''not'' weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2+4+6 = 12.
 
The first few weird numbers are  
: 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ... {{OEIS|id=A006037}}.
 
== Properties ==
{{unsolved|mathematics|Are there any odd weird numbers?}}
It has been shown that an infinite number of weird numbers exist; in fact, the sequence of weird numbers has positive [[asymptotic density]].<ref name="benk1">
{{cite journal
  | last =Benkoski
  | first =Stan
  | authorlink =
  | coauthors =Paul Erdős
  | title =On Weird and Pseudoperfect Numbers
  | journal =Mathematics of Computation
  | volume =28
  | issue =126
  | pages =617–623
  |date=April 1974
  | doi =10.2307/2005938
| zbl=0279.10005 | mr=347726
}}
</ref>
 
It is not known if any odd weird numbers exist; if any do, they must be greater than 2<sup>32</sup> ≈ 4{{e|9}}.<ref>CN Friedman, "Sums of Divisors and Egyptian Fractions", ''Journal of Number Theory'' (1993).  The result is attributed to "M. Mossinghoff at University of Texas - Austin".</ref>
 
Sidney Kravitz has shown that if ''k'' is a positive integer, ''Q'' is a [[prime number|prime]], and
:<math>R=\frac{2^kQ-(Q+1)}{(Q+1)-2^k}</math>;
if ''R'' is prime, then
:<math>n=2^{k-1}QR</math>
is a weird number.<ref>
{{cite journal
  | last =Kravitz
  | first =Sidney
  | title =A search for large weird numbers
  | journal =Journal of Recreational Mathematics
  | volume =9
  | issue =2
  | pages =82–85
  | publisher =Baywood Publishing
  | location =
  | year =1976
| zbl=0365.10003
}}
</ref>
With this formula, he found a large weird number
:<math>n=2^{56}\cdot(2^{61}-1)\cdot153722867280912929\ \approx\ 2\cdot10^{52}</math>.
 
Alternatively, a larger weird number can be calculated using the formula
:<math> W = w \cdot p^{k} </math>
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 ''p<sup>k</sup>'' have a sum greater than ''wp<sup>k</sup>''.
Divisors of ''w'' each multiplied by ''p<sup>k</sup>'' are all divisors of ''W'', therefore ''W'' is abundant.
 
Assume a subset of proper divisors of ''W'' has a sum of ''W'':
:<math> \sum D = d1 + d2 + d3 + ... = W </math>
 
Let ''S'' be the set of divisors of ''W'' not divisible by ''p<sup>k</sup>''. Let ''X'' be the set of divisors of ''w''.
:<math> 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} \} </math>
 
Factor the sum:
:<math> \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}) </math>
 
Knowing that ''p'' is greater than the sum of divisors of ''w'':
:<math> 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}) </math>
 
Therefore no subset of ''S'' has a sum divisible by ''p<sup>k</sup>''. All other divisors of ''W'' are divisible by ''p<sup>k</sup>'', therefore all elements of ''D'' are divisible by ''p<sup>k</sup>''.
Dividing both sides of the first equation by ''p<sup>k</sup>'' 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.
 
==In popular culture==
 
The [[Boards of Canada]] album ''[[Geogaddi]]'' contains a song titled "The Smallest Weird Number," a reference to another song, "Sixtyten" (equalling 70), as well as their record label, [[Music70]]. Despite the song having a running time of 1:17, the audio cuts out at exactly 1:10 (70 seconds).
 
==References==
{{Reflist}}
 
== External links ==
{{portal|Mathematics}}
* {{MathWorld |urlname=WeirdNumber |title=Weird number}}
 
 
{{Divisor classes}}
 
{{DEFAULTSORT:Weird Number}}
[[Category:Divisor function]]
[[Category:Integer sequences]]

Latest revision as of 19:12, 12 November 2014

If you have the desire to procedure settings swiftly, loading files rapidly, however the body is logy and torpid, what would you do? If you are a giant "switchboard" that is lack of effective management program and effective housekeeper, what would you do? If you have send the exact commands to the mind, yet the body may not do properly, what would we do? Yes! We want a full-featured repair registry!

When you registry gets cluttered up with a lot of junk we don't employ, your PC might run slower. Therefore it happens to be prudent that you regularly get a registry cleaned.

StreamCI.dll is a file utilized by the default Windows Audio driver to aid process the different audio settings on the system. Although this file is regarded as the most crucial on countless different Windows systems, StreamCI.dll is continually causing a lot of errors that have to be repaired. The good news is the fact that we can fix this error by utilizing several easy to do steps that will resolve all potential difficulties which are causing the error to show on a PC.

Always see to it which we have installed antivirus, anti-spyware and anti-adware programs plus have them updated on a regular basis. This can help stop windows XP running slow.

After which, I furthermore purchased the Regtool tuneup utilities Software, and it further protected my computer having program crashes. All my registry problems are fixed, plus I could work peacefully.

The initially thing you need to do is to reinstall any program which shows the error. It's typical for various computers to have specific programs which require this DLL to show the error when you try plus load it up. If you see a particular system show the error, you need to initially uninstall that system, restart your PC plus then resinstall the system again. This could replace the damaged ac1st16.dll file plus remedy the error.

When the registry is corrupt or full of mistakes, the signs is felt by the computer owner. The slow performance, the frequent system crashes plus the nightmare of all computer owners, the blue screen of death.

By changing the method you employ the internet you are able to have access more of your precious bandwidth. This may ultimately provide we a quicker surfing experience. Here is a link to 3 methods to personalize a PC speed found on the net.

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