数学题单做题日志

2025-05-27

1. \(\color{blue}{蓝题}\)

   给定 \(n\) , 求

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

   枚举 \(n\) 的因数 \(d\) ,

   \[\sum_{d,d|n}d\times \varphi (\frac nd)\tag 1 \]

   又因为

   \[\varphi (t) = t \prod _{p\in \mathbb{P},p|t} \frac{1-p}{p} \]
   \[\begin{array}{ll} (1)&=\sum_{d,d|n}\frac{n}{d}\times \varphi (d)\newline &=\sum_{d,d|n}\frac{n}{d}d\prod _{p\in \mathbb{P},p|t}\frac{1-p}{p}\newline &=n\sum_{d,d|n}\prod _{p\in \mathbb{P},p|t}\frac{1-p}{p} \end{array} \]

   令 \(n=\prod_{p_k\in \mathbb{P},p_k|t} p_k^{b_k},\)

   \[\prod _{p\in \mathbb{P},p|t}\frac{1-p}{p}=(1+b_0(\frac{1-p_0}{p_0}))\cdot \prod _{p\in \mathbb{P-\{p_0\}},p|t}\frac{1-p}{p} \]

   就可以得到 \(O(\sqrt{n})\) 的做法