【min-max容斥】【莫比乌斯反演】【二项式定理】【欧拉函数】欧拉欧拉

题意

一个序列的权值定义为$\varphi (lcm(a_1,a_2,...,a_k))$
求所有长度为$k$的，每个值$a_i\le n$的序列的权值积

分析

首先使用min-max容斥把lcm改写成gcd
ϕ(S)=∏T⊂Sϕ(gcd(T)(−1)∣T∣−1)ans=∏a1=1n∏a2=1n..∏ak=1n∏T⊂{a1,a2,...,ak}ϕ(gcd(T)(−1)∣T∣−1) \phi(S)=\prod_{T \subset S}\phi(gcd(T)^{(-1)^{|T|}-1})\\ ans=\prod_{a_1=1}^n\prod_{a_2=1}^n..\prod_{a_k=1}^n\prod_{T \subset \{a_1,a_2,...,a_k\}}\phi(gcd(T)^{(-1)^{|T|}-1})\\ ϕ(S)=T⊂S∏​ϕ(gcd(T)(−1)∣T∣−1)ans=a1​=1∏n​a2​=1∏n​..ak​=1∏n​T⊂{a1​,a2​,...,ak​}∏​ϕ(gcd(T)(−1)∣T∣−1)
把T拿出来枚举
$$ans=\prod _{w=1}^k\prod _{a_1=1}^n\prod _{a_2=1}^n..\prod _{a_w=1}^n\varphi (gcd(a_1,a_2,...,a_w){)}^{(-1{)}^{w-1}{C}_k^w{n}^{k-w}}$$
先计算
$$res=\prod _{a_1=1}^n\prod _{a_2=1}^n..\prod _{a_w=1}^n\varphi (gcd(a_1,a_2,...,a_w))$$