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  • 20201201A组T3(过程推导)

    20201201A组T3(过程推导)

    [egin{equation}egin{aligned}&sum_{i=1}^{n}sum_{j=1}^{m}gcd^n(i,j)sum_{k=1}^{ij}[i ot k][j ot k]k\ &=sum_{i=1}^{n}sum_{j=1}^{m}gcd^n(i,j)varphi(ij)ij+[ij=1]\&=frac{1}{2}+frac{1}{2}sum_{i=1}^{n}sum_{j=1}^{m}gcd^n(i,j)frac{varphi(i)varphi(j)gcd(i,j)}{varphi(gcd(i,j))}ij\ &=frac{1}{2}+frac{1}{2}sum_{d=1}^{min(n,m)}frac{d^{n+1}}{varphi(d)}sum_{d|i,ile n}sum_{d|j,jle m}varphi(i)varphi(j)ij[gcd(i,j)=d]\ &这里用到了莫比乌斯反演\ &=frac{1}{2}+frac{1}{2}sum_{d=1}^{min(n,m)}frac{d^{n+1}}{varphi(d)}sum_{d|i,ile n}sum_{d|j,jle m}varphi(i)varphi(j)ijsum_{u|frac{gcd(i,j)}{d}}mu(u)\ &=frac{1}{2}+frac{1}{2}sum_{d=1}^{min(n,m)}frac{d^{n+1}}{varphi(d)}sum_{u=1}^{min(n,m)}mu(u)sum_{du|i,ile n}varphi(i)isum_{du|j,jle m}varphi(j)j\ &=frac12+frac12sum_{T=1}^{n}sum_{T|i}varphi(i)isum_{T|j}varphi(j)jsum_{d|T}frac{d^{n+1}}{varphi(d)}mu(frac{T}{d})end{aligned}end{equation} ]

    注意到(sum_{i|T}varphi(i)i)是可以预处理的

    同时,(sum_{d|T}frac{d^{n+1}}{varphi(d)}mu(frac{T}{d})),可以用线筛搞定

    下载.png

    (来自MHT)

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  • 原文地址:https://www.cnblogs.com/Kelvin2005/p/14071007.html
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