元完全數

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未解決的數學問題：元完全數是否有無限多個？		Question mark2.svg

元完全數（unitary perfect number）是指一整數其元因數的和等於整數的2倍，元因數是一種特殊的因數，一整數n若有元因數d，則d及n/d互質。


有些完全數不是元完全數（例如28），而也有些數是元完全數，但不是完全數（例如60）。


60的元因數有1, 3, 4, 5, 12, 15, 20, 60，元因數和為1 + 3 + 4 + 5 + 12 + 15 + 20 + 60 = 120 = 60 * 2，因此60是元完全數，頭幾個元完全數：


6, 60, 90, 87360, 146361946186458562560000 OEIS A002827


目前還不知道元完全數是否有無限多個。



參考資料



  • Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag. 2004: 84–86. ISBN 0-387-20860-7.  Section B3.


  • Paulo Ribenboim. My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag. 2000: 352. ISBN 0-387-98911-0. 















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