Weird numbers: Difference between revisions

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In number theory, a [[wp:weird number|weird number]] is a natural number that is abundant but not semiperfect.

In number theory, a [[wp:weird number|weird number]] is a natural number that is [[wp:abundant number|abundant]] but not [[wp:semiperfect number|semiperfect]].

In other words, the sum of the proper divisors (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.

In other words, the sum of the [[wp:Divisor#Further_notions_and_facts|proper divisors]] of the number (divisors including 1 but not itself) is greater than the number itself (the number is ''abundant''), but no subset of those divisors sums to the number itself (the number is not ''semiperfect'').

For example:

~~E.G.~~ '''12''' is ''not'' a weird number. ~~It is abundant; the proper divisors '''1, 2, 3, 4 & 6''' sum to '''16''', but it is semiperfect,~~

* '''12''' is ''not'' a weird number.

** It is abundant; its proper divisors '''1, 2, 3, 4, 6''' sum to '''16''',

'''6 + 4 + 2 == 12'''.

** but it ''is'' semiperfect, '''6 + 4 + 2 == 12'''.

'''70''' ''is'' a weird number. ~~It is abundant; the proper divisors '''1, 2, 5, 7, 10, 14 & 35''' sum to '''74''', but there is no subset~~

* '''70''' ''is'' a weird number.

of proper divisors ~~that~~ sum to '''70'''.

** It is abundant; its proper divisors '''1, 2, 5, 7, 10, 14, 35''' sum to '''74''',

** and there is no subset of proper divisors that sum to '''70'''.