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