Editing Weird number (section)

===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>