2019ICPC南昌邀请赛网络赛 G.tsy's number （数论）
积性函数+容斥 2019ICPC南昌邀请赛网络赛 G.tsy's number 题意 求$\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\frac{\phi(i)\phi(j^2)\phi(k^3)}{\phi(i)\phi(j)\phi(k)}\phi(gcd(i,j,k))$ 共T组数据，$T\leq 10000$，$1\leq n \leq 10^7$ 题解 枚举gcd(i,j,k) = d，然后容斥一下 $$ \begin{align*} ans &= \sum_{d=1}^n\phi(d)\sum_{i=1}^{[\frac{n}{d}]}\sum_{j=1}^{[\frac{n}{d}]}\sum_{k=1}^{[\frac{n}{d}]} [gcd(i,j,k)==1]\frac{\phi(i d)\phi((j d)^2)\phi((k d)^3)}{\phi(i d)\phi(j d)\phi(k d)}\ &= \sum_{d=1}^n\phi(d)\sum_{s=1}^{[\frac{n}{d}]}\mu(s)\sum_{i=1}^{[\frac{n}{d s}]}\sum_{j=1}^{[\frac{n}{d s}]}\sum_{k=1}^{[\frac{n}{d s}]} \frac{\phi(i d s)\phi((j d s)