Powerful Number 筛 PN 筛。有时候确实比 Min_25 筛快。 首先定义 Powerful Number 为形如 a2b3 的数。一个结论是 n 以内的 Powerful Number 是 O(n) 的。证明枚举 a 对b 乱积分就可以。 假如说我们现在要求 S(n)=∑i=1nf(i) 。那么我们构造一个易求前缀和的积性函数 g ,使其在质数处取值和 f...
Powerful Number 筛学习笔记 用途 \(Powerful\ number\) 筛可以用来求出一类积性函数的前缀和,最快可以达到根号复杂度。 实现 \(Powerful\ number\) 的定义是每个质因子次数都 \(\ge 2\) 的数。 有如下的性质: \(1\)、一个 \(Pow
\(\sum_{i=1}^n f(i)=\sum_{x~is~a~powerful~number} g(x)*\sum_{j=1}^{n/x} d(j)\) 可以线性筛\(1-\sqrt n\)的约数个数,这样\(n/x\le sqrt(n)\)时直接查询,否则利用\(\sum_{i=1}^n d(i)=\sum_{i=1}^n n/i\)分块解决。 通过积分可分析出复杂度为\(O(\sqrt n l...
但是powerful number(简称pn)构造方法有所不同。 对于这个题 数据范围,min_25筛显然是过不去的。最多跑到1e11 $f(p^e)=p^k$启发 构造$g(x)=x^k$和相似一些 再构造$h(x)$,使得$f=h*g$这里表示狄利克雷卷积 那么$\sum_i f(i)=\sum_i\sum_{d|i}h(i)\times g(i/d)=\sum_d h(d)\...
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