莫反小练习

\[\sum_i \sum_j \operatorname{lcm}(a_i, a_j) \]

\[= \sum_i \sum_j \sum_d [\gcd(a_i, a_j) = d] \dfrac{a_i a_j}{d} \]

\[c_x = \sum_i [a_i = x] \]

\[= \sum_d \sum_{1 \le i < j \le \lfloor \frac{V}{d} \rfloor} [\gcd(i, j) = 1] c_{id} c_{jd} ijd \]

\[= \sum_d \sum_k dk ^ 2\mu(k) \sum_{1 \le i < j \le \lfloor \frac{V}{dk} \rfloor} c_{idk} c_{jdk} ij \]

\[T = dk \]

\[\sum_{1 \le i < j \le \lfloor \frac{V}{T} \rfloor} c_{iT} c_{jT} ij \]

\[= \dfrac{(\sum_{1 \le i \le \lfloor \frac{V}{T} \rfloor} ic_{iT}) ^ 2 - \sum_{1 \le i \le \lfloor \frac{V}{T} \rfloor} i ^ 2 c_{iT} ^ 2}{2} \]

直接算即可，复杂度 \(\mathcal{O}(V \log V)\)。