Weird numbers: Difference between revisions
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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]] (and |
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]] (and hence unavoidably not [[wp:perfect number|perfect]] either). |
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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''). |
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''). |
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* '''12''' is ''not'' a weird number. |
* '''12''' is ''not'' a weird number. |
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** It is abundant; its proper divisors '''1, 2, 3, 4, 6''' sum to '''16 |
** It is abundant; its proper divisors '''1, 2, 3, 4, 6''' sum to '''16''' (which ''is'' > 12), |
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** but it ''is'' semiperfect, '''6 + 4 + 2 == 12'''. |
** but it ''is'' semiperfect, eg '''6 + 4 + 2 == 12'''. |
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* '''70''' ''is'' a weird number. |
* '''70''' ''is'' a weird number. |
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** It is abundant; its proper divisors '''1, 2, 5, 7, 10, 14, 35''' sum to '''74 |
** It is abundant; its proper divisors '''1, 2, 5, 7, 10, 14, 35''' sum to '''74''' (which ''is'' > 70), |
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** and there is no subset of proper divisors that sum to '''70'''. |
** and there is no subset of proper divisors that sum to '''70'''. |
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