\[\sum\limits_{i=1}^N\sum\limits_{j=1}^M\frac{ij}{\gcd(i,j)}\]\[\sum\limits_{d=1}^N\frac1d\sum\limits_{i=1}^N\sum\limits_{j=1}^Mij[\gcd(i,j)=d]\]\[\sum\limits_{d=1}^Nd\sum\limits_{i=1}^{\lfloor\fracNd\rfloor}\sum\limits_

[国家集训队]Crash的数字表格/JZPTAB这个题可以低于线性，然后也可以杜教筛到\(O(n^{2/3})\)这个样子。首先暴力推：\[\begin{aligned}&\sum_{i=1}^{n}\sum_{j=1}^{

求\[\sum^{n}_{i=1}\sum^{m}_{j=1}lcm(i,j)\]即\[\sum^{n}_{i=1}\sum^{m}_{j=1}\dfrac{ij}{\gcd(i,j)}\]即\[\sum^{\min(n,m)}_{k=1}\sum^{n}_{i=1}\s

莫比乌斯反演\(\color{red}{f(n)=\sum\limits_{d|n}g(d)\Leftrightarrowg(n)=\sum\limits_{d|n}\mu(d)f(\dfrac{n}{d})}\)\(f(n),g(n)\)均为积性函数。\(f(n)\)称为