莫比乌斯反演小记
写在前面
这是蒟蒻第一次写这么长的博文
\(gyh\ nb\),\(\text{oi-wiki}\ nb\)
如果觉得写得凑合就点个支持吧\(qwq\)
前置知识
、、(有空慢慢补)
mobius函数
定义
莫比乌斯函数\(\mu(n)\)定义为:
其中\(p_1p_2…,p_s\)是不同素数。
可以看出,当\(n\)没有平方因子时,\(\mu(n)\)非零。
\(\mu\)也是积性函数。
性质
莫比乌斯函数具有如下的性质:
使用狄利克雷卷积来表示,即
证明:
\(n=1\)时显然成立。
若\(n>1\),设\(n\)有\(s\)个不同的素因子,由于\(\mu(d)\neq0\)当且仅当\(d\)无平方因子,故\(d\)中每个素因子的指数只能为\(0\)或\(1\)。
若\(n\)的某个因子\(d\),有\(\mu(d)=(-1)^i\),则它由\(i\)个本质不同的质因子构成。因为质因子的总数为\(s\),所以满足上式的因子数有\(c_s^i\)个。
所以我们就可以对于原式,转化为枚举\(\mu(d)\)的值,同时运用二项式定理,故有
该式在\(s=0\)即\(n=1\)时为1,否则为\(0\)。
求莫比乌斯函数
因为莫比乌斯函数是积性函数,所以可以用线性筛
int n, cnt, p[a], mu[a]; bool vis[a]; void getmu() { mu[1] = 1; for (int i = 2; i <= n; i++) { if (!vis[i]) mu[i] = -1, p[++cnt] = i; for (int j = 1; j <= cnt && i * p[j] <= n; j++) { vis[i * p[j]] = 1; if (i % p[j] == 0) { mu[i * p[j]] = 0; break; } mu[i * p[j]] -= mu[i]; } } }
莫比乌斯反演公式
设\(f(n)\),\(g(n)\)为两个数论函数。
如果有
则有
证明
-
法一:对原式做数论变换
-
\(\sum\limits_{d|n}g(d)\)替换\(f(n)\),即
\[\sum\limits_{d|n}\mu(d)f(\frac{n}{d})=\sum\limits_{d|n}\mu(d)\sum\limits_{k|\frac nd}g(k) \] -
变换求和顺序得
\[\sum\limits_{k|n}g(k)\sum\limits_{d|\frac n k}\mu(\frac nk) \] -
因为\(\sum\limits_{d|n}\mu(d)=[n=1]\),所以只有在\(\frac{n}{k}=1\)即\(n=k\)时才会成立,所以上式等价于
\[\sum\limits_{d|n}[n=k]g(k)=g(n) \]
得证
-
-
法二:利用卷积
原问题为:已知\(f=g*1\),证明\(g=f*\mu\)
转化:\(f*\mu=g*1*\mu=g*\varepsilon=g\)(\(1*\mu=\varepsilon\))
再次得证= =
小性质
\([\gcd(i,j)=1]\leftrightarrow\sum\limits_{d|\gcd(i,j)}\mu(d)\)
证明
-
法一:
设\(n=\gcd(i,j)\),那么等式右边\(=\sum\limits_{d|n}\mu(d)=[n=1]=[\gcd(i,j)=1]=\)等式左边
-
法二:
利用单位函数暴力拆开:\([\gcd(i,j)=1]=\varepsilon(\gcd(i,j))=\sum\limits_{d|\gcd(i,j)}\mu(d)\)
做题思路&&应用
利用狄利克雷卷积可以解决一系列求和问题。常见做法是使用一个狄利克雷卷积替换求和式中的一部分,然后调换求和顺序,最终降低时间复杂度。
经常利用的卷积有\(\mu*1=\epsilon\)和\(\text{id}=\varphi*1\)。
题
还是以题为主吧,但是做的题也会单独写题解,毕竟要多水几篇博客的嘛/huaji
洛谷 p2522 [haoi2011]problem b
有\(n\)组询问,每次给出\(a,b,c,d,k\),求\(\sum\limits_{x=a}^{b}\sum\limits_{y=c}^{d}[\gcd(x,y)=k]\)
设\(f(n,m)=\sum\limits_{i=1}^{n}\sum\limits_{j= 1}^{m}[\gcd(i,j)=k]\)
那么根据容斥原理,题目中的式子就转化成了\(f(b,d)-f(b, c - 1) - f(a - 1,d) + f(a - 1, c - 1)\)
所以我们接下来的问题就转化为了如何求\(f\)的值
现在来化简\(f\)的值
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容易得出原式等价于$$\sum\limits_{i = 1}^{\lfloor\frac{n}{k}\rfloor}\sum\limits_{j = 1}^{\lfloor\frac{m}{k}\rfloor}[\gcd(i,j) = 1]$$
-
因为\(\epsilon(n) =\sum\limits_{d|n}\mu(d)=[n=1]\),由此可将原式化为
\[\sum\limits_{i=1}^{\lfloor\frac{n}{k}\rfloor}\sum\limits_{j=1}^{\lfloor\frac{m}{k}\rfloor}\sum\limits_{d|gcd(i,j)}\mu(d) \] -
改变枚举对象并改变枚举顺序,先枚举\(d\),得
\[\sum\limits_{d=1}^{\min(n,m)}\mu(d)\sum\limits_{i=1}^{\lfloor\frac{n}{k}\rfloor}[d|i]\sum\limits_{j=1}^{\lfloor\frac{m}{k}\rfloor}[d|j] \]也就是说当\(d|i\)且\(d|j\)时,\(d|\gcd(i,j)\)
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易得\(1\sim \lfloor\frac{n}{k}\rfloor\)中一共有\(\lfloor\frac{n}{dk}\rfloor\)个\(d\)的倍数,同理\(1\sim \lfloor\frac{m}{k}\rfloor\)中一共有\(\lfloor\frac{m}{dk}\rfloor\)个\(d\)的倍数,于是原式化为$$\sum\limits_{d=1}^{\min(n,m)}\mu(d)\lfloor\frac{n}{dk}\rfloor\lfloor\frac{m}{dk}\rfloor$$
此时已经可以\(o(n)\)求解,但是过不了,因为有很多值相同的区间,所以可以用数论分块来做
先预处理\(\mu\),再用数论分块,复杂度\(o(n+t\sqrt n)\)
我的代码每次得分玄学,看评测机心情,建议自己写
/* author:loceaner */ #include <cmath> #include <cstdio> #include <cstring> #include <iostream> #include <algorithm> using namespace std; const int a = 1e6 + 11; const int b = 1e6 + 11; const int mod = 1e9 + 7; const int inf = 0x3f3f3f3f; inline int read() { char c = getchar(); int x = 0, f = 1; for( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } int n, a, b, c, d, k, cnt, p[a], mu[a], sum[a]; bool vis[a]; void getmu() { int max = 50010; mu[1] = 1; for (int i = 2; i <= max; i++) { if (!vis[i]) mu[i] = -1, p[++cnt] = i; for (int j = 1; j <= cnt && i * p[j] <= max; j++) { vis[i * p[j]] = true; if (i % p[j] == 0) break; mu[i * p[j]] -= mu[i]; } } for (int i = 1; i <= max; i++) sum[i] = sum[i - 1] + mu[i]; } int work(int x, int y) { int ans = 0ll; int max = min(x, y); for (int l = 1, r; l <= max; l = r + 1) { r = min(x / (x / l), y / (y / l)); ans += (1ll * x / (l * k)) * (1ll * y / (l * k)) * 1ll * (sum[r] - sum[l - 1]); } return ans; } void solve() { a = read(), b = read(), c = read(), d = read(), k = read(); cout << work(b, d) - work(a - 1, d) - work(b, c - 1) + work(a - 1, c - 1) << '\n'; } signed main() { getmu(); int t = read(); while (t--) solve(); return 0; }
洛谷 p3455 [poi2007]zap-queries
有\(t\)组询问,每次询问求
因为我不喜欢用\(x、y、a、b、d\),所以一一对应换成\(i、j、n、m、k\)
直接淦式子
现在就可以每次询问\(o(n)\)做这道题了
但是跑不过啊,不过显然可以数论分块,所以我们就可以\(o(\sqrt n)\)回答每次询问了
/* author:loceaner */ #include <cmath> #include <cstdio> #include <cstring> #include <iostream> #include <algorithm> using namespace std; const int a = 5e5 + 11; const int b = 1e6 + 11; const int mod = 1e9 + 7; const int inf = 0x3f3f3f3f; inline int read() { char c = getchar(); int x = 0, f = 1; for( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } int n, m, k, mu[a], p[a], sum[a], cnt; bool vis[a]; void getmu(int n) { mu[1] = 1; for (int i = 2; i <= n; i++) { if (!vis[i]) p[++cnt] = i, mu[i] = -1; for (int j = 1; j <= cnt && i * p[j] <= n; j++) { vis[i * p[j]] = 1; if (i % p[j] == 0) break; mu[i * p[j]] -= mu[i]; } } for (int i = 1; i <= n; i++) sum[i] = sum[i - 1] + mu[i]; } int solve(int n, int m, int k) { int ans = 0, maxn = min(n, m); for (int l = 1, r; l <= maxn; l = r + 1) { r = min(n / (n / l), m / (m / l)); ans += (sum[r] - sum[l - 1]) * (n / (k * l)) * (m / (k * l)); } return ans; } int main() { getmu(50000); int t = read(); while (t--) { n = read(), m = read(), k = read(); cout << solve(n, m, k) << '\n'; } return 0; }
洛谷 p1829 [国家集训队]crash的数字表格 / jzptab
求
容易想到原式等价于
枚举\(i,j\)的最大公约数\(d\),显然\(\gcd(\frac id,\frac jd)=1\),即\(\frac id\)和\(\frac jd\)互质
变换求和顺序
记\(sum(n,m)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}[\gcd(i,j)=1]i*j\)
对其进行化简,用\(\varepsilon(\gcd(i,j))\)替换\([\gcd(i,j)=1]\)
转化为首先枚举约数
设\(i=i'*d,j=j'*d\),则可以进一步转化
前半段可以处理前缀和,后半段可以\(o(1)\)求,设
显然可以\(o(1)\)求解
到现在
可以用数论分块求解
回带到原式中
又可以数论分块求解了
然后就做完啦
/* author:loceaner */ #include <cmath> #include <cstdio> #include <cstring> #include <iostream> #include <algorithm> #define int long long using namespace std; const int a = 1e7 + 11; const int b = 1e6 + 11; const int mod = 20101009; const int inf = 0x3f3f3f3f; inline int read() { char c = getchar(); int x = 0, f = 1; for( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } bool vis[a]; int n, m, mu[a], p[b], sum[a], cnt; void getmu() { mu[1] = 1; int k = min(n, m); for (int i = 2; i <= k; i++) { if (!vis[i]) p[++cnt] = i, mu[i] = -1; for (int j = 1; j <= cnt && i * p[j] <= k; ++j) { vis[i * p[j]] = 1; if (i % p[j] == 0) break; mu[i * p[j]] = -mu[i]; } } for (int i = 1; i <= k; i++) sum[i] = (sum[i - 1] + i * i % mod * mu[i]) % mod; } int sum(int x, int y) { return (x * (x + 1) / 2 % mod) * (y * (y + 1) / 2 % mod) % mod; } int solve2(int x, int y) { int res = 0; for (int i = 1, j; i <= min(x, y); i = j + 1) { j = min(x / (x / i), y / (y / i)); res = (res + 1ll * (sum[j] - sum[i - 1] + mod) * sum(x / i, y / i) % mod) % mod; } return res; } int solve(int x, int y) { int res = 0; for (int i = 1, j; i <= min(x, y); i = j + 1) { j = min(x / (x / i), y / (y / i)); res = (res + 1ll * (j - i + 1) * (i + j) / 2 % mod * solve2(x / i, y / i) % mod) % mod; } return res; } signed main() { n = read(), m = read(); getmu(); cout << solve(n, m) << '\n'; }
洛谷 p2257 yy的gcd
给定\(n,m\),求二元组\((x,y)\)的个数,满足\(1\leq x\leq n,1\leq y\leq m\),且\(gcd(x,y)\)是素数。
\(n,m\leq 10^7\),自带多组数据,至多\(10^{4}\)组数据。
思路与第一题problem b类似,在这里不再赘述,只给出代码= =
#include <cmath> #include <cstdio> #include <cstring> #include <iostream> using namespace std; const int a = 1e7 + 11; inline int read() { char c = getchar(); int x = 0, f = 1; for ( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for ( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } bool vis[a]; long long sum[a]; int prim[a], mu[a], g[a], cnt, n, m; void get_mu(int n) { mu[1] = 1; for (int i = 2; i <= n; i++) { if (!vis[i]) { mu[i] = -1; prim[++cnt] = i; } for (int j = 1; j <= cnt && prim[j] * i <= n; j++) { vis[i * prim[j]] = 1; if (i % prim[j] == 0) break; else mu[prim[j] * i] = - mu[i]; } } for (int j = 1; j <= cnt; j++) for (int i = 1; i * prim[j] <= n; i++) g[i * prim[j]] += mu[i]; for (int i = 1; i <= n; i++) sum[i] = sum[i - 1] + (long long)g[i]; } signed main() { int t = read(); get_mu(10000000); while (t--) { n = read(), m = read(); if (n > m) swap(n, m); long long ans = 0; for (int l = 1, r; l <= n; l = r + 1) { r = min(n / (n / l), m / (m / l)); ans += 1ll * (n / l) * (m / l) * (sum[r] - sum[l - 1]); } cout << ans << '\n'; } return 0; }
洛谷 p3327 [sdoi2015]约数个数和
求
\(d(x)\)为\(x\)的约数个数和
需要用到
证明我也不会
然后自己推导吧,在此不再赘述
/* author:loceaner */ #include <cmath> #include <cstdio> #include <cstring> #include <iostream> #include <algorithm> using namespace std; const int a = 5e5 + 11; const int b = 1e6 + 11; const int mod = 1e9 + 7; const int inf = 0x3f3f3f3f; inline int read() { char c = getchar(); int x = 0, f = 1; for( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } bool vis[a]; int n, m, p[a], mu[a], cnt, sum[a]; long long g[a], ans; void getmu(int n) { mu[1] = 1; for (int i = 2; i <= n; i++) { if (!vis[i]) mu[i] = -1, p[++cnt] = i; for (int j = 1; j <= cnt && i * p[j] <= n; j++) { vis[i * p[j]] = 1; if (i % p[j] == 0) break; mu[i * p[j]] -= mu[i]; } } for (int i = 1; i <= n; i++) sum[i] = sum[i - 1] + mu[i]; for (int i = 1; i <= n; i++) { int ans = 0; for (int l = 1, r; l <= i; l = r + 1) { r = (i / (i / l)); ans += 1ll * (r - l + 1) * (i / l); } g[i] = ans; } } signed main() { int t = read(); getmu(50000); while (t--) { n = read(), m = read(); int maxn = min(n, m); ans = 0; for (int l = 1, r; l <= maxn; l = r + 1) { r = min(n / (n / l), m / (m / l)); ans += 1ll * (sum[r] - sum[l - 1]) * 1ll * g[n / l] * 1ll * g[m / l]; } cout << ans << '\n'; } return 0; }
洛谷 p4449 于神之怒加强版
求
还是直接淦式子
令\(p=dx\),则原式等于
显然前面的\(\lfloor\frac n{p}\rfloor\lfloor\frac m{p}\rfloor\)部分可以分块求解。
现在考虑后面的一部分,令
容易得出这个函数是积性函数,所以我们就可以线性筛然后求出其前缀和
然后就做完了
/* author:loceaner 莫比乌斯反演 */ #include <cmath> #include <cstdio> #include <cstring> #include <iostream> #include <algorithm> #define int long long using namespace std; const int a = 5e6 + 11; const int b = 1e6 + 11; const int mod = 1e9 + 7; const int inf = 0x3f3f3f3f; inline int read() { char c = getchar(); int x = 0, f = 1; for( ; !isdigit(c); c = getchar()) if(c == '-') f = -1; for( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48); return x * f; } bool vis[a]; int t, n, m, k, f[a], g[a], p[a], cnt, sum[a]; inline int power(int a, int b) { int res = 1; while (b) { if (b & 1) res = res * a % mod; a = a * a % mod, b >>= 1; } return res; } inline int mo(int x) { if(x > mod) x -= mod; return x; } inline void work() { g[1] = 1; int maxn = 5e6 + 1; for (int i = 2; i <= maxn; i++) { if (!vis[i]) { p[++cnt] = i, f[cnt] = power(i, k), g[i] = mo(f[cnt] - 1 + mod); } for (int j = 1; j <= cnt && i * p[j] <= maxn; j++) { vis[i * p[j]] = 1; if (i % p[j] == 0) { g[i * p[j]] = g[i] * 1ll * f[j] % mod; break; } g[i * p[j]] = g[i] * 1ll * g[p[j]] % mod; } } for (int i = 2; i <= maxn; i++) g[i] = (g[i - 1] + g[i]) % mod; } inline int abss(int x) { while (x < 0) x += mod; return x; } signed main() { t = read(), k = read(); work(); while (t--) { n = read(), m = read(); int maxn = min(n, m), ans = 0; for (int l = 1, r; l <= maxn; l = r + 1) { r = min(n / (n / l), m / (m / l)); (ans += abss(g[r] - g[l - 1]) * 1ll * (n / l) % mod * (m / l) % mod) %= mod; } ans = (ans % mod + mod) % mod; cout << ans << '\n'; } return 0; }