Editing Weird number

{{Short description|Number that is abundant but not semiperfect}}
{{Euler_diagram_numbers_with_many_divisors.svg}}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
  | title =E2308 (in Problems and Solutions)
  | journal =The American Mathematical Monthly
  | volume =79
  | issue =7
  | page =774
  | date =August–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 ([[divisor]]s 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&nbsp;+&nbsp;4&nbsp;+&nbsp;6&nbsp;=&nbsp;12.

The first several weird numbers are 
: [[70 (number)|70]], [[836 (number)|836]], 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... {{OEIS|id=A006037}}.

== Properties ==
{{unsolved|mathematics|Are there any odd weird numbers?}}
Infinitely many weird numbers exist.<ref>{{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=113–114}}</ref> For example, 70''p'' is weird for all [[prime number|primes]] ''p'' ≥ 149. In fact, the [[set (mathematics)|set]] of weird numbers has positive [[asymptotic density]].<ref name="benk1">
{{cite journal
  | last1=Benkoski
  | first1=Stan
  | author2-link=Paul Erdős
  | first2=Paul | last2=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
  | jstor=2005938
 | zbl=0279.10005 | mr=347726
| doi-access=free
  }}
</ref>

It is not known if any [[parity (mathematics)|odd]] weird numbers exist. If so, they must be greater than 10<sup>21</sup>.<ref>{{Cite OEIS|1=A006037|2=Weird numbers: abundant (A005101) but not pseudoperfect (A005835)}} -- comments concerning odd weird numbers</ref>

Sidney Kravitz has shown that for ''k'' a positive [[integer]], ''Q'' a prime exceeding 2<sup>''k''</sup>, and
:<math>R = \frac{2^kQ-(Q+1)}{(Q+1)-2^k}</math>
also prime and greater than 2<sup>''k''</sup>, 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 
  | year=1976
  | zbl=0365.10003 
}}</ref>
With this formula, he found the large weird number
:<math>n=2^{56}\cdot(2^{61}-1)\cdot153722867280912929\ \approx\ 2\cdot10^{52}.</math>

===Primitive weird numbers===
A property of weird numbers is that if ''n'' is weird, and ''p'' is a prime greater than the [[sum of divisors]] σ(''n''), then ''pn'' is also weird.<ref name=benk1/> This leads to the definition of ''primitive weird numbers'': weird numbers that are not a [[multiple (mathematics)|multiple]] of other weird numbers {{OEIS|id=A002975}}. Among the 1765 weird numbers less than one million, there are 24 primitive weird numbers. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form <math>2^k p q</math> are primitive, but the existence of infinitely many ''k'' and ''Q'' which yield a prime ''R'' is not guaranteed. It is [[conjecture]]d that there exist infinitely many primitive weird numbers, and [[Giuseppe Melfi|Melfi]] has shown that the infinitude of primitive weird numbers is a consequence of [[Cramér's conjecture]].<ref>
{{cite journal
  | last =Melfi
  | first =Giuseppe
  | title =On the conditional infiniteness of primitive weird numbers
  | journal =Journal of Number Theory
  | volume =147
  | issue =
  | pages = 508–514
  | publisher =Elsevier
  | year =2015
  | doi= 10.1016/j.jnt.2014.07.024
  | zbl= 
| doi-access =
  }}</ref>
Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.<ref>
{{cite journal
  | last1 =Amato
  | first1 =Gianluca
  | last2 =Hasler
  | first2 =Maximilian
  | last3 =Melfi
  | first3 =Giuseppe
  | last4 =Parton
  | first4 =Maurizio
  | title =Primitive abundant and weird numbers with many prime factors
  | journal =Journal of Number Theory
  | volume =201
  | issue =
  | pages = 436–459
  | publisher =Elsevier
  | year =2019
  | doi= 10.1016/j.jnt.2019.02.027
  | zbl= 
| arxiv =1802.07178
  | s2cid =119136924
 }}</ref>

==See also==
{{Wikifunctions|Z14991|weird number checking}}
* [[Untouchable number]]
{{clear}}

==References==
{{Reflist}}

== External links ==
{{portal|Mathematics}}
* {{MathWorld |urlname=WeirdNumber |title=Weird number}}

{{Divisor classes}}

{{DEFAULTSORT:Weird Number}}
[[Category:Divisor function]]
[[Category:Integer sequences]]