题解 P3911 【最小公倍数之和】

设 \(cnt_i\) 为 \(i\) 的出现次数。

则这题要求的是 \(\sum_{i=1}^{N} \sum_{j=1}^{N} lcm(i, j) \times cnt_i \times cnt_j\)

\(\left( lcm (i,j) = \frac{ij}{\gcd(i,j)}\right)\)

\(\sum_{i=1}^{N} \sum_{j=1}^{N} \frac{ij}{\gcd(i,j)} \times cnt_i \times cnt_j\)

枚举 \(\gcd\)。

\(\sum_{d=1}^{N} \sum_{i=1}^{N/d} \sum_{j=1}^{N/d} [\gcd(i,j) =1]d\times ij\times cnt_{id} \times cnt_{jd}\)

\(\sum_{k|n} \mu_k = [n==1]\)

\(\sum_{d=1}^{N} \sum_{i=1}^{N/d} \sum_{j=1}^{N/d} \sum_{k|\gcd(i,j)}\mu_k \times d\times ij\times cnt_{id} \times cnt_{jd}\)

枚举 \(k\)。

\(\sum_{d=1}^{N} \sum_{k=1}^{N/d}\sum_{i=1}^{n/kd}\sum_{j=1}^{n/kd}\mu_k \times d \times k^2 \times i \times j \times cnt_{idk} \times cnt_{jdk}\)

\(T = dk\)。

\(\sum_{T=1}^{N} T \times \left(\sum_{i=1}^{N/T} i \times cnt_{iT}\right)^2 \sum_{k|T} \mu_k \times k\)

然后就做完了。