[NKOJ7707]数论

\[\sum\limits_{i=1}^n\sum\limits_{j=1}^m\gcd(i,j)^n\sum\limits_{k=1}^{ij}[i\perp k][j\perp k]k \]

\[\frac12\sum\limits_{i=1}^n\sum\limits_{j=1}^m\gcd(i,j)^n\varphi(ij)ij \]

\[\frac1 2 \sum\limits_{d=1}^n\frac{d^{n+1}}{\varphi(d)}\sum\limits_{i=1}^n\sum\limits_{j=1}^m[\gcd(i,j)=d]\varphi(i)\varphi(j)ij \]

这里不应该把 \(d\) 提出去的，因为后面要凑 \(T=dx\)。

\[\frac1 2 \sum\limits_{d=1}^n\frac{d^{n+3}}{\varphi(d)}\sum\limits_{i=1}^{\frac nd}\sum\limits_{j=1}^{\frac md}[\gcd(i,j)=1]\varphi(id)\varphi(jd)ij \]

\[\frac1 2 \sum\limits_{d=1}^n\frac{d^{n+3}}{\varphi(d)}\sum\limits_{i=1}^{\frac nd}\sum\limits_{j=1}^{\frac md}\varphi(id)\varphi(jd)ij\sum\limits_{x|\gcd(i,j)}\mu(x) \]

\[\frac12\sum\limits_{d=1}^n\frac{d^{n+1}}{\varphi(d)}\sum\limits_{x=1}^n\mu(x)\sum\limits_{i=1}^{\frac n {xd}}\varphi(idx)idx\sum\limits_{j=1}^{\frac m {xd}}\varphi(jdx)jdx \]

\[\frac12\sum\limits_{x=1}^n\mu(x)\sum\limits_{d=1}^n\frac{d^{n+1}}{\varphi(d)}\sum\limits_{i=1}^{\frac n {xd}}\varphi(idx)idx\sum\limits_{j=1}^{\frac m {xd}}\varphi(jdx)jdx \]

\[\frac12\sum\limits_{T=1}^n\sum\limits_{i=1}^{\frac n {T}}\varphi(iT)iT\sum\limits_{j=1}^{\frac m {T}}\varphi(jT)jT\sum\limits_{d|T}\frac{d^{n+1}}{\varphi(d)}\mu(\frac Td) \]

\[\frac12\sum\limits_{T=1}^n\sum\limits_{T|i}\varphi(i)i\sum\limits_{T|j}\varphi(j)j\sum\limits_{d|T}\frac{d^{n+1}}{\varphi(d)}\mu(\frac Td) \]

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\[\begin{aligned} f(n)&=\sum\limits_{i=1}^{n}i[\gcd(n,i)=1]\\ &=\frac 1 2\varphi(n)\times n \end{aligned} \]