Convex Hull

$$\begin{gathered} gay(i)=\left\{\begin{array}{ccc}0& if& i=k\times x\times x,x>k,k>1\\ i\times i& else& \end{array}\right\} \\ 求\sum _{i=1}^n\sum _{j=1}^{i}gay(j) \\ =\sum _{i=1}^n(n?i+1)gay(i) \\ =\sum _{i=1}^n(n?i+1){\mu }^2(i)i^2 \\ 因為{\mu }^2(n)=\sum _{i^2\mid n}\mu (i)，容斥定理顯然得到 \\ 有原式=\sum _{i=1}^n(n?i+1)i^2\sum _{j^2\mid i}\mu (j) \\ =(n+1)\sum _{i=1}^n\sum _{j^2\mid i}\mu (j)?\sum _{i=1}^n{i}^3\sum _{j^2\mid i}\mu (j) \\ =(n+1)\sum _{j=1}^{\sqrt{n}}\mu (j)\sum _{j^2\mid i}{i}^2?\sum _{j=1}^{\sqrt{n}}\mu (j)\sum _{j^2\mid i}{i}^3 \\ =(n+1)\sum _{j=1}^{\sqrt{n}}\mu (j)\sum _{t=1}^{\frac{n}{j^2}}{t}^2{j}^4?\sum _{j=1}^{\sqrt{n}}\mu (j)\sum _{t=1}^{\frac{n}{j^2}}{t}^3{j}^6 \\ =\sum _{i=1}^{\sqrt{n}}\mu (i)\sum _{j=1}^{\frac{n}{i^2}}(i^4{j}^2(n+1)?i^6{j}^3) \end{gathered}$$

總結

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