[egin{aligned}
sumlimits_{i=1}^n [i,n]&= nsumlimits_{i=1}^n frac{i}{(i,n)}\
&= nsumlimits_{d|n} sumlimits_{i=1}^{n/d} i[(i,n/d)=1]\
&= nsumlimits_{d|n} sumlimits_{i=1}^d i[(i,d)=1]\
&= nsumlimits_{d|n} sumlimits_{i=1}^{d} i sumlimits_{p|i,d} mu(p)\
&= nsumlimits_{d|n} sumlimits_{p|d} mu(p) imes psumlimits_{i=1}^{d/p}i\
&= nsumlimits_{p|n}mu(p) imes psumlimits_{d|n/p}sum(d)
end{aligned}
]
其中 (sum(i)=1+2+cdots+i)。时间复杂度 (O(nlogn))。