元因數

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在數學上，元因數（unitary divisor）是指一種特殊的因數。若一整數a是另一整數b的因數，且a和ba{displaystyle {frac {b}{a}}}{displaystyle {frac {b}{a}}}互質，則整數a為整數b的元因數。


以60為例，5和605=12{displaystyle {frac {60}{5}}=12}{displaystyle {frac {60}{5}}=12}互質，因此5是整數60的元因數，而6和606=10{displaystyle {frac {60}{6}}=10}{displaystyle {frac {60}{6}}=10}不互質，因此6不是整數60的元因數，1是所有數字的元因數。


一整數b的因數a為其元因數的充份必要條件是a的每一個質因數，其乘幂次數都和該質因數在b出現的次數一様。若整數b為无平方数因数的数，其所有因數均為元因數。


一個整數元因數的和表示為σ*(n)。元因數k次方的和表示為σ*k(n):


σk∗(n)=∑d∣n,gcd(d,n/d)=1dk.{displaystyle sigma _{k}^{*}(n)=sum _{dmid n,gcd(d,n/d)=1}d^{k}.}{displaystyle sigma _{k}^{*}(n)=sum _{dmid n,gcd(d,n/d)=1}d^{k}.}

若一個整數真元因數（小於整數的元因數）的和為整數本身，此整數稱為元完全數。



外部連結



  • 埃里克·韦斯坦因. Unitary Divisor. MathWorld. 


  • OEIS sequences: A034444 is σ0(n). A034448 is σ1(n). A034676 up to A034682 are σ2(n) to σ8(n).


A068068 is σ(o)*0(n).
A192066 is σ(o)*1(n).





































Nuvola apps edu mathematics-p.svg  這是與数论相關的小作品。你可以通过编辑或修订扩充其内容。







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