SP8099 TABLE - Crash´s number table 题解

合集 - 题解(97)

1.BZOJ3364 Distance Queries 距离咨询 题解2023-08-082.UVA12390 Distributing Ballot Boxes 题解2023-08-243.UVA1108 Mining Your Own Business 题解2023-08-244.CF1425F Flamingoes of Mystery 题解2023-09-295.SP16113 SUBTLEBA - Trucks Transportation 题解2023-10-016.UVA12655 Trucks 题解2023-10-017.P5943 [POI2002] 最大的园地 题解2023-10-018.SP9494 ZSUM - Just Add It 题解2023-10-029.CF1878C Vasilije in Cacak 题解2023-10-0210.CF1010C Border 题解2023-10-0611.CF131D Subway 题解2023-10-0612.B3610 [图论与代数结构 801] 无向图的块 题解2023-11-0413.P9801 [NERC2018] King Kog’s Reception2023-11-0514.CF1089K King Kog's Reception 题解2023-11-0515.SP28304 ADATEAMS - Ada and Teams 题解2023-11-1916.CF327C Magic Five 题解2023-12-1617.P1405 苦恼的小明 题解2023-12-1618.SP10050 POWTOW - Power Tower City 题解2023-12-1619.SP21690 POWERUP - Power the Power Up 题解2023-12-1620.P2898 [USACO08JAN] Haybale Guessing G 题解2023-12-3121.AT_arc125_c [ARC125C] LIS to Original Sequence 题解2024-01-1322.CF1433E Two Round Dances 题解2024-01-2723.CF739A Alyona and mex 题解2024-01-2724.AT_abc270_g [ABC270G] Sequence in mod P 题解2024-02-0725.SP277 CTGAME - City Game 题解2024-02-0726.P8670 [蓝桥杯 2018 国 B] 矩阵求和 题解2024-02-1027.UVA12467 Secret Word 题解2024-02-1328.SP34020 ADAPET - Ada and Pet 题解2024-02-1329.AT_joig2021_d 展覧会 2 (Exhibition 2) 题解2024-03-0330.AT_abc184_f [ABC184F] Programming Contest 题解2024-03-0331.AT_abc343_g [ABC343G] Compress Strings 题解2024-03-1332.CF494C Helping People 题解2024-03-2333.CF922E Birds 题解2024-03-2334.CF1628D1 Game on Sum (Easy Version) 题解2024-03-2335.P7137 [THUPC2021 初赛] 切切糕 题解2024-03-2736.CF1392H ZS Shuffles Cards 题解2024-03-2837.luogu P1543 [POI2004] SZP 题解2024-03-3038.SP64 PERMUT1 - Permutations 题解2024-04-0739.AT_s8pc_2_e 部分文字列 题解2024-04-0740.UVA1223 Editor 题解2024-04-0741.UVA10870 Recurrences 题解2024-05-0242.UVA1362 Exploring Pyramids 题解2024-05-0243.CF1141E Superhero Battle 题解2024-05-2644.SP14943 DRTREE - Dynamically-Rooted Tree 题解2024-05-2645.UVA11922 Permutation Transformer 题解2024-05-2646.UVA1674 闪电的能量 Lightning Energy Report 题解2024-05-2647.P10499 开关问题 题解2024-06-0848.P10466 邻值查找 题解2024-06-0849.P10596 BZOJ2839 集合计数 题解2024-06-2450.UVA12369 Cards 题解2024-06-2851.CF696B Puzzles 题解2024-06-2852.P10672 【MX-S1-T1】壁垒 题解2024-06-2953.P10673 【MX-S1-T2】催化剂 题解2024-06-2954.UVA11181 条件概率 Probability|Given 题解2024-06-2955.SP8177 JZPEXT - Beautiful numbers EXTREME 题解2024-07-0256.AT_tdpc_number 数 题解2024-07-0257.AT_dp_y Grid 2 题解2024-07-0358.P7690 [CEOI2002] A decorative fence 题解2024-07-0459.SP15620 POSTERIN - Postering 题解2024-07-0460.SP14887 GOODA - Good Travels 题解2024-07-1461.P2188 小Z的 k 紧凑数 题解2024-07-14

62.SP8099 TABLE - Crash´s number table 题解2024-07-30

63.AT_abl_e Replace Digits 题解2024-08-0564.AT_past202010_m 筆塗り 题解2024-08-0965.P8575 「DTOI-2」星之河 题解2024-08-0966.CF641E Little Artem and Time Machine 题解2024-08-0967.P10633 BZOJ2989 数列/BZOJ4170 极光 题解2024-08-1568.P10238 [yLCPC2024] F. PANDORA PARADOXXX 题解2024-08-1669.P4271 [USACO18FEB] New Barns P 题解2024-08-1670.P8512 [Ynoi Easy Round 2021] TEST_152 题解2024-08-1871.CF1514D Cut and Stick 题解2024-08-2372.P10933 创世纪 题解2024-08-2473.P10957 环路运输 题解2024-08-2474.SP10502 VIDEO - Video game combos 题解2024-08-2575.P10956 金字塔 题解2024-08-2676.CF704B Ant Man 题解2024-09-0577.P6638 「JYLOI Round 1」常规 题解2024-09-0678.SP1825 FTOUR2 - Free tour II 题解2024-09-2379.P10603 BZOJ4372 烁烁的游戏 题解2024-09-2780.P3364 Cool loves touli 题解2024-09-3081.CF946G Almost Increasing Array 题解2024-10-0582.AT_abc283_g [ABC283G] Partial Xor Enumeration 题解2024-10-1083.CF959F Mahmoud and Ehab and yet another xor task 题解2024-10-1084.P3794 签到题IV 题解2024-10-1685.P7984 [USACO21DEC] Tickets P 题解2024-11-0886.P5479 [BJOI2015] 隐身术 题解2024-11-0887.CF2030D QED's Favorite Permutation 题解2024-11-1088.CF633H Fibonacci-ish II 题解2024-11-1489.AT_abc129_f [ABC129F] Takahashi's Basics in Education and Learning 题解2024-12-1890.P8060 [POI2003] Sums 题解2024-12-2191.AT_keyence2019_e Connecting Cities 题解2024-12-2392.CF1534G A New Beginning 题解2024-12-2693.AT_abc237_g [ABC237G] Range Sort Query 题解2024-12-3094.CF601E A Museum Robbery 题解01-0195.P5680 [GZOI2017] 共享单车 题解01-0496.P3081 [USACO13MAR] Hill Walk G 题解01-0697.AT_abc248_h [ABC248Ex] Beautiful Subsequences 题解01-10

收起

题目传送门

前置知识

一般的积性函数 | 数论分块 | 莫比乌斯反演

解法

令 $n\le m$。

考虑莫比乌斯反演，推式子，有 $\begin{array}{rl}& \sum _{i=1}^n\sum _{j=1}^n\mathrm{lcm}(i,j)\\ & =\sum _{i=1}^n\sum _{j=1}^n\frac{ij}{gcd(i,j)}\\ & =\sum _{k=1}^n\frac{1}{k}\sum _{i=1}^n\sum _{j=1}^{n}ij[gcd(i,j)=k]\\ & =\sum _{k=1}^n\frac{1}{k}\sum _{i=1}^{\lfloor \frac{n}{k}\rfloor }\sum _{j=1}^{\lfloor \frac{m}{k}\rfloor }ij{k}^2[gcd(i,j)=1]\\ & =\sum _{k=1}^{n}k\sum _{i=1}^{\lfloor \frac{n}{k}\rfloor }\sum _{j=1}^{\lfloor \frac{m}{k}\rfloor }ij\sum _{d|gcd(i,j)}\mu (d)\\ & =\sum _{k=1}^{n}k\sum _{d=1}^n\mu (d)\sum _{i=1}^{\lfloor \frac{n}{k}\rfloor }\sum _{j=1}^{\lfloor \frac{m}{k}\rfloor }ij[d|gcd(i,j)]\\ & =\sum _{k=1}^{n}k\sum _{d=1}^n\mu (d)d^2\sum _{i=1}^{\lfloor \frac{n}{kd}\rfloor }\sum _{j=1}^{\lfloor \frac{m}{kd}\rfloor }ij\\ & =\sum _{k=1}^{n}k\sum _{d=1}^n\mu (d)d^2\sum _{i=1}^{\lfloor \frac{n}{kd}\rfloor }i\sum _{j=1}^{\lfloor \frac{m}{kd}\rfloor }j\\ & =\sum _{T=1}^{n}T\sum _{d|T}\mu (d)d\sum _{i=1}^{\lfloor \frac{n}{T}\rfloor }i\sum _{j=1}^{\lfloor \frac{m}{T}\rfloor }j\end{array}$。

$f(n)=\sum _{d|n}\mu (d)d$ 显然是积性函数，线筛筛一下即可。

接着维护前缀和后整除分块。

需要注意的是本题仅限 C++98 提交。

代码

#include<bits/stdc++.h>
using namespace std;
#define ll long long 
#define ull unsigned long long
#define sort stable_sort 
#define endl '\n'
const ll p=20101009;
ll prime[10000010],f[10000010],s[10000010],len=0;
bool vis[10000010];
void isprime(ll n)
{
	memset(vis,0,sizeof(vis));
	f[1]=1;
	for(ll i=2;i<=n;i++)
	{
		if(vis[i]==0)
		{
			len++;
			prime[len]=i;
			f[i]=1-i;
		}
		for(ll j=1;j<=len&&i*prime[j]<=n;j++)
		{
			vis[i*prime[j]]=1;
			if(i%prime[j]==0)
			{
				f[i*prime[j]]=f[i];
				break;
			}
			else
			{
				f[i*prime[j]]=f[i]*f[prime[j]];
			}
		}
	}
	for(ll i=1;i<=n;i++)
	{
		s[i]=(s[i-1]+i)%p;
		f[i]=(f[i-1]+f[i]*i%p+p)%p;
	}
}
ll ask(ll n,ll m)
{
	ll ans=0,l,r;
	for(l=1,r;l<=n;l=r+1)
	{
		r=min(n/(n/l),m/(m/l));
		ans=(ans+(s[n/l]*s[m/l]%p)*(f[r]-f[l-1]+p)%p)%p;
	}
	return ans;
}
int main()
{
	ll n,m;
	cin>>n>>m;
	isprime(max(n,m));
	if(n>m)
	{
		swap(n,m);
	}
	cout<<ask(n,m)<<endl;
	return 0;
}

后记

多倍经验：luogu P1829 [国家集训队] Crash的数字表格 / JZPTAB