约数个数和 - 图1%20%20%5C%5C%0A%3D%20%26%20%5Csum%7Bi%3D1%7D%5En%20%5Csum%7Bj%3D1%7D%5Em%5Csum%7Bx%7Ci%7D%5Csum%7By%7Ci%7D%20%5B%5Cgcd(x%2Cy)%3D1%5D%20%5C%5C%0A%3D%20%26%20%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5B%5Cgcd(x%2Cy)%3D1%5D%20%5C%5C%0A%3D%20%26%20%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%5Csum%7Bt%3D1%7D%5En%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5Bt%7C%5Cgcd(x%2Cy)%5D%20%5Cmu(t)%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%20%5Bt%7C%5Cgcd(x%2Cy)%5D%20%20%5Cmu(t)%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5En%20%5Bt%7Cx%5D%5Csum%7By%3D1%7D%5Em%20%5Bt%7Cy%5D%20%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5Cmu(t)%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5En%20%5Bt%7Cx%5D%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Csum%7By%3D1%7D%5Em%20%5Bt%7Cy%5D%20%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%5C%20%5Cmu(t)%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cfrac%7Bn%7D%7Bxt%7D%5Crfloor%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cfrac%7Bm%7D%7Byt%7D%5Crfloor%5C%20%5Cmu(t)%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%20%5Cmu(t)%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Cdfrac%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%7Bx%7D%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cdfrac%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%7By%7D%5Crfloor%5C%20%5C%5C%0A%5Cend%7Baligned%7D%0A#card=math&code=%5Clarge%0A%5Cbegin%7Baligned%7D%0A%26%5Csum%7Bi%3D1%7D%5En%20%5Csum%7Bj%3D1%7D%20%5Em%20d%28ij%29%20%20%5C%5C%0A%3D%20%26%20%5Csum%7Bi%3D1%7D%5En%20%5Csum%7Bj%3D1%7D%5Em%5Csum%7Bx%7Ci%7D%5Csum%7By%7Ci%7D%20%5B%5Cgcd%28x%2Cy%29%3D1%5D%20%5C%5C%0A%3D%20%26%20%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5B%5Cgcd%28x%2Cy%29%3D1%5D%20%5C%5C%0A%3D%20%26%20%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%5Csum%7Bt%3D1%7D%5En%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5Bt%7C%5Cgcd%28x%2Cy%29%5D%20%5Cmu%28t%29%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5Csum%7Bx%3D1%7D%5En%20%5Csum%7By%3D1%7D%5Em%20%5Bt%7C%5Cgcd%28x%2Cy%29%5D%20%20%5Cmu%28t%29%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5En%20%5Bt%7Cx%5D%5Csum%7By%3D1%7D%5Em%20%5Bt%7Cy%5D%20%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%20%5Cmu%28t%29%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5En%20%5Bt%7Cx%5D%5Clfloor%5Cfrac%7Bn%7D%7Bx%7D%5Crfloor%5Csum%7By%3D1%7D%5Em%20%5Bt%7Cy%5D%20%5Clfloor%5Cfrac%7Bm%7D%7By%7D%5Crfloor%5C%20%5Cmu%28t%29%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cfrac%7Bn%7D%7Bxt%7D%5Crfloor%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cfrac%7Bm%7D%7Byt%7D%5Crfloor%5C%20%5Cmu%28t%29%5C%5C%0A%3D%20%26%5Csum%7Bt%3D1%7D%5En%20%5Cmu%28t%29%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Cdfrac%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%7Bx%7D%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cdfrac%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%7By%7D%5Crfloor%5C%20%5C%5C%0A%5Cend%7Baligned%7D%0A)

    设 约数个数和 - 图2%20%3D%5Csum%7Bi%3D1%7D%5Ex%5Cdfrac%7Bx%7D%7Bi%7D#card=math&code=f%28x%29%20%3D%5Csum%7Bi%3D1%7D%5Ex%5Cdfrac%7Bx%7D%7Bi%7D) , 整除分块计算即可。

    约数个数和 - 图3%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Cdfrac%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%7Bx%7D%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cdfrac%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%7By%7D%5Crfloor%5C%20%5C%5C%0A%3D%26%5Csum%7Bt%3D1%7D%5En%20%5Cmu(t)%5C%20f(%5Clfloor%5Cdfrac%7Bn%7D%7Bt%7D%5Crfloor)%20f(%5Clfloor%5Cdfrac%7Bm%7D%7Bt%7D%5Crfloor)%0A%5Cend%7Baligned%7D%0A#card=math&code=%5Clarge%0A%5Cbegin%7Baligned%7D%0A%26%5Csum%7Bt%3D1%7D%5En%20%5Cmu%28t%29%5C%20%5Csum%7Bx%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%20%5Cdfrac%7B%5Clfloor%5Cfrac%7Bn%7D%7Bt%7D%5Crfloor%7D%7Bx%7D%5Csum%7By%3D1%7D%5E%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%20%5Clfloor%5Cdfrac%7B%5Clfloor%5Cfrac%7Bm%7D%7Bt%7D%5Crfloor%7D%7By%7D%5Crfloor%5C%20%5C%5C%0A%3D%26%5Csum_%7Bt%3D1%7D%5En%20%5Cmu%28t%29%5C%20f%28%5Clfloor%5Cdfrac%7Bn%7D%7Bt%7D%5Crfloor%29%20f%28%5Clfloor%5Cdfrac%7Bm%7D%7Bt%7D%5Crfloor%29%0A%5Cend%7Baligned%7D%0A)