Editing Weird number (section)

== 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>