数学之推式子题合集

P2303 [SDOI2012] Longge 的问题

$$\sum_{i=1}^{n} \gcd(i,n)$$

$$= \sum_{d | n} \sum_{i=1}^{n} d [gcd(i,n) == d]$$

$$= \sum_{d | n} d \sum_{i=1}^{n} [gcd(i,n) == d]$$

$$= \sum_{d | n} d \sum_{i=1}^{\frac{n}{d}} [gcd(i,n) == 1]$$

$$= \sum_{d | n} d \; \varphi (\frac{n}{d})$$

P1891 疯狂 LCM

$$\sum_{i = 1}^n \operatorname{lcm}(i, n)$$

$$=\sum_{i = 1}^n \frac{i\times n}{\gcd(i,n)}$$

$$=n \sum_{i = 1}^n \frac{i}{\gcd(i,n)}$$

$$=n \frac{\sum_{i=1}^n i}{\sum_{i=1}^n \gcd(i,n)}$$

考虑分母

$$\sum_{i=1}^n \gcd(i,n) =$$

$$= \sum_{d | n} \sum_{i=1}^{n} d [gcd(i,n) == d]$$

$$= \sum_{d | n} d \sum_{i=1}^{n} [gcd(i,n) == d]$$

$$= \sum_{d | n} d \sum_{i=1}^{\frac{n}{d}} [gcd(i,n) == 1]$$

$$= \sum_{d | n} d \; \varphi (\frac{n}{d})$$