洛谷P3455 [POI2007]ZAP-Queries

题目大意:

给定(n,m,k,)

[sumlimits_{x=1}^nsumlimits_{y=1}^m[gcd(x,y)==k] ]

莫比乌斯反演入门题,先进行一步转化,将每个(x,y)除以(k),则答案变为

[sumlimits_{x=1}^{lfloorfrac{n}{k} floor} sumlimits_{y=1}^{lfloorfrac{m}{k} floor} [gcd(x,y)==1] ]

发现最右边的条件可以莫比乌斯反演

[sumlimits_{x=1}^{lfloorfrac{n}{k} floor} sumlimits_{y=1}^{lfloorfrac{m}{k} floor} sumlimits_{d|n}mu(d) ]

我们将枚举因子提到前面,改成枚举约数

[sumlimits_{d=1}^{min(frac{n}{k},frac{m}{k})}mu(d) sumlimits_{x=1}^{lfloorfrac{n}{k} floor} sumlimits_{y=1}^{lfloorfrac{m}{k} floor} ]

易证:(1-n)中k个倍数个数为(lfloorfrac{n}{k} floor)$
所以原式等价于

[sumlimits_{d=1}^{min(frac{n}{k},frac{m}{k})}mu(d)lfloorfrac{n}{kd} floor lfloorfrac{m}{kd} floor ]

用除法分块可以做到(O(sqrt{n}))

原文地址:https://www.cnblogs.com/knife-rose/p/12017061.html