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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>
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| {{cite journal
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| | last =Benkoski
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| | first =Stan
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| | authorlink =
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| | coauthors =
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| | title =E2308 (in Problems and Solutions)
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| | journal =The American Mathematical Monthly
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| | volume =79
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| | issue =7
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| | page =774
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| | date =Aug.-September 1972
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| | doi =10.2307/2316276
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| | jstor =2316276
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| }}</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>
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| 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.
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| == Examples ==
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| 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.
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| The first few weird numbers are
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| : 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ... {{OEIS|id=A006037}}.
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| == Properties ==
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| {{unsolved|mathematics|Are there any odd weird numbers?}}
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| 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">
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| {{cite journal
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| | last =Benkoski
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| | first =Stan
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| | authorlink =
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| | coauthors =Paul Erdős
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| | title =On Weird and Pseudoperfect Numbers
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| | journal =Mathematics of Computation
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| | volume =28
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| | issue =126
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| | pages =617–623
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| |date=April 1974
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| | doi =10.2307/2005938
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| | zbl=0279.10005 | mr=347726
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| }}
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| </ref>
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| 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>
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| Sidney Kravitz has shown that if ''k'' is a positive integer, ''Q'' is a [[prime number|prime]], and
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| :<math>R=\frac{2^kQ-(Q+1)}{(Q+1)-2^k}</math>; | |
| if ''R'' is prime, then
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| :<math>n=2^{k-1}QR</math> | |
| is a weird number.<ref>
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| {{cite journal
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| | last =Kravitz
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| | first =Sidney
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| | title =A search for large weird numbers
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| | journal =Journal of Recreational Mathematics
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| | volume =9
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| | issue =2
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| | pages =82–85
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| | publisher =Baywood Publishing
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| | location =
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| | year =1976
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| | zbl=0365.10003
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| }}
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| </ref>
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| With this formula, he found a large weird number
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| :<math>n=2^{56}\cdot(2^{61}-1)\cdot153722867280912929\ \approx\ 2\cdot10^{52}</math>.
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| Alternatively, a larger weird number can be calculated using the formula
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| :<math> W = w \cdot p^{k} </math>
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| where ''w'' is a weird number, ''p'' is a prime greater than the sum of divisors of ''w'', and ''k'' is any positive integer.
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| Proof:
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| 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>''.
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| Divisors of ''w'' each multiplied by ''p<sup>k</sup>'' are all divisors of ''W'', therefore ''W'' is abundant.
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| Assume a subset of proper divisors of ''W'' has a sum of ''W'':
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| :<math> \sum D = d1 + d2 + d3 + ... = W </math>
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| 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''.
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| :<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>
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| Factor the sum:
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| :<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>
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| Knowing that ''p'' is greater than the sum of divisors of ''w'':
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| :<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> | |
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| 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>''.
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| 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.
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| ==In popular culture==
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| 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).
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| ==References==
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| {{Reflist}}
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| == External links ==
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| {{portal|Mathematics}}
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| * {{MathWorld |urlname=WeirdNumber |title=Weird number}}
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| {{Divisor classes}}
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| {{DEFAULTSORT:Weird Number}}
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| [[Category:Divisor function]]
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| [[Category:Integer sequences]]
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