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[hdu7082]Pty loves lcm

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11月 28, 2021

先将问题差分,即仅考虑上限$R$(和$L-1$)

注意到$f(x,y)$增长是较快的,对其分类讨论:

1.若$yge x+2$,此时满足$f(x,y)le 10^{18}$的$(x,y)$只有约$10^{6}$组,暴力预处理并通过二分+前缀和即可

2.若$y=x+1$,显然$f(x,y)=x(x+1)$且$varphi(f(x,y))=varphi(x)varphi(x+1)$

换言之,问题即求$sum_{xge 1,x(x+1)le R}varphi(x)varphi(x+1)$,通过二分可以得到新范围$1le xle R’$,这个函数并没有很好的性质,但由于$R’le 10^{9}$因此可以分块打表处理

注意到分块打表预处理以及计算,都需要对区间$[l,r]$快速求出所有$varphi(x)$

关于这个事情,预处理出$sqrt{r}$以内的素数,并对$[l,r]$内的数一起质因数分解——即枚举这些素数并将其在$[l,r]$内的倍数全部筛掉,结合素数的分布以及调和级数,复杂度大约是$o(r-l)$的

 1 #include<bits/stdc++.h>
 2 using namespace std;
 3 #define N 1500005
 4 #define K 500000
 5 #define ll long long
 6 #define ui unsigned int
 7 vector<int>P[N];
 8 vector<pair<ll,ui> >v;
 9 int t,p[N],phi[N],vis[N],Phi[N];
10 ll L,R;
11 ui ans,sum[N],Sum[2001]={0,24914787,3137086944,2464224208,203189032,3449159140,4033773776,2287331592,2259387784,4278809768,2728034220,1421284728,1632991496,3084174244,3081444120,3479813036,58736376,1679860800,2115586824,3108754724,3659593832,1296109640,3335098080,2424792664,1379875632,2361458064,593204068,2881402460,722993064,717671696,2076363960,3790889416,906330240,2800607940,1446608324,692549928,1935163072,1067888940,637979988,590904032,1279173044,1659465532,608332152,1535502336,1891041352,1501490756,765930444,3889000112,2277096284,711498472,3367208768,1756119956,2889809848,585283384,657621328,1541824876,2194003252,2732809620,2642784300,1755708812,2262470360,4173074244,1251704608,4253370716,209913808,1138383812,1608028812,3240288996,3650689700,2007676252,1496808560,3503076724,1420478532,2720640580,3524450412,773237840,492475952,2310041824,3021947092,3194563052,1439103056,2235569260,3126580916,1557228684,987491972,3915267496,2933347432,2282099964,2048389688,2347962116,3247285524,3184840264,1929025984,2449493172,1347519280,3301416460,3277771580,4228458972,2160296740,2769289536,1457276776,3646623936,803425648,1588459728,3860233992,279123652,2577570756,194733472,1376547228,11272364,4076525312,873893156,3501417384,3951953020,450723748,1283858020,3684885104,3545732412,1311441992,1026179192,4027168804,2749737324,199522932,109500564,824016316,3671733688,1205589504,1183468360,2614653344,1029757000,403044452,3454841268,154248176,4044459040,141675072,1613670140,2876918072,2057165660,1298766740,2970849876,3982962024,1967543792,3647156576,2453060856,1938543024,328052996,3733957820,635240368,3044371468,3380895464,2348222168,2665539572,2279227032,653000548,483855044,1383848912,2352257516,3952272008,2063810560,1033606372,3843761672,2454382692,718131844,1427002860,1145868900,3167559928,1165202764,479049916,940390660,486809112,408060884,2759675340,2101157400,4039138988,2867268088,1130282956,3505777716,4250935036,3754375228,2825192912,2193014388,1306950908,3370560616,3757506056,4101934508,109723616,2146880764,1954985608,3934088852,2566477628,290302200,537384024,1630405460,498563084,2625694464,2863751888,3122712480,3251001248,4233810104,673057192,4197110100,2127218472,2523836668,3668266904,3354131716,906374388,395699744,1385771244,4287651028,3963574652,2310966080,3145754248,2046127288,2622090780,1688157804,2840318352,2966738756,4125028160,1194524304,2538677204,878146032,265327284,3561690140,1224537120,240956716,2294783340,18142320,3142532948,2162666488,1871589656,4176744280,2342092228,1494503168,1506241804,2491449088,1620816808,1504507920,1073750940,3116333104,2046658844,4210820820,3255059492,2268436584,2943830836,1810023104,4015681448,2812312400,4216995760,1462536020,479575392,1365253644,605440900,4229281556,3989340404,2369672640,1852703124,2022238304,471956092,1784597484,3898517044,2934957884,3101303088,1669786524,994290008,228938676,1391291236,4107701372,2548862968,1371859560,3497425176,470257164,2049596044,3986104872,620083528,3408035732,28673192,1641790100,2321617464,1983236564,2139056692,1413031664,3804949524,2761525824,3857187260,3040427352,3625915964,609134420,1196720408,3859630688,1300273076,2119654752,4010984268,1255556908,1226038560,1296837416,2548926608,2278914280,2460478740,3849297380,324477936,3404914960,1194364140,3301021820,1790305284,287053980,2760416144,1386126484,1247822920,652809760,741961584,2128265312,1289343176,2064858540,3580082328,283359668,1576181248,2635144932,828255140,1711311888,4279003628,2455980084,3587038080,2870275428,3428670176,3004981900,3657853020,3835657616,3989420124,3912529996,3754108732,669815272,193507320,2174869520,930940040,2789310816,1882836864,2464233436,691122160,1087391096,2998406304,3358507564,302280220,376019648,2101945764,520386488,1852624192,3967220704,1910237932,3754220024,2422393108,1227618016,3641630452,3041428888,3898197248,2381040524,1854261112,1382770240,1250933912,173717728,3510536320,3369214196,496413520,1747960956,2780502588,3466916252,2006738144,232710980,1651483436,760721656,2699094512,2351640900,524330412,720257472,1953131604,3696431340,2926381936,2880133180,113565260,1500271148,2033464776,3387964668,231261972,2154224852,1963012776,2009177180,3718492284,1737070860,2764419012,307856720,88141364,1838116664,941891068,3785792232,4262443836,1773319148,3518195748,2362353828,204128956,2672024764,457913836,2233916612,1361447804,3350530492,3844095632,2575191456,75190948,1743304268,1822976928,6892612,4032491320,480761124,1407918716,938181676,3973277320,2784324100,4028473044,129560520,2856358024,122863848,2003941896,2107422596,2420617908,3024749124,2249499396,2361611968,3282352208,4254906680,1083719480,3275034496,1736514656,656708868,2053548296,4282660780,325597828,3383851120,4135566144,1303187380,2213438124,885848444,412660540,1882815512,1362126456,1081499140,1013555412,3496193680,2591522432,4012560364,2426675488,437309036,2219967812,2518375244,567483672,3778349364,826456336,3076394448,2202826080,2387870808,2466400672,1905473120,1504530032,1464214492,288969212,2934540612,170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12 ll gcd(ll x,ll y){
13     if (!y)return x;
14     return gcd(y,x%y);
15 }
16 ui get_sum(int l,int r){
17     r++;
18     for(int i=0;i<=r-l;i++)vis[i]=Phi[i]=i+l;
19     for(int i=1;p[i]*p[i]<=r;i++)
20         for(int j=(l+p[i]-1)/p[i]*p[i];j<=r;j+=p[i]){
21             Phi[j-l]=Phi[j-l]/p[i]*(p[i]-1);
22             while (vis[j-l]%p[i]==0)vis[j-l]/=p[i];
23         }
24     for(int i=0;i<=r-l;i++)
25         if (vis[i]>1)Phi[i]=Phi[i]/vis[i]*(vis[i]-1);
26     ui ans=0;
27     for(int i=0;i<r-l;i++)ans+=(ui)Phi[i]*Phi[i+1];
28     return ans;
29 }
30 ui calc(ll n){
31     int l=0,r=1e9;
32     while (l<r){
33         int mid=(l+r+1>>1);
34         if ((ll)mid*(mid+1)<=n)l=mid;
35         else r=mid-1;
36     }
37     return Sum[l/K]+get_sum(l/K*K+1,l);
38 }
39 int main(){
40     phi[1]=1;
41     for(int i=2;i<N;i++){
42         if (!vis[i]){
43             p[++p[0]]=i;
44             phi[i]=i-1;
45             P[i].push_back(i);
46         }
47         for(int j=1;(j<=p[0])&&(i*p[j]<N);j++){
48             vis[i*p[j]]=1;
49             if (i%p[j]){
50                 P[i*p[j]].push_back(p[j]);
51                 phi[i*p[j]]=phi[i]*phi[p[j]];
52             }
53             else phi[i*p[j]]=phi[i]*p[j];
54             for(int t=0;t<P[i].size();t++)P[i*p[j]].push_back(P[i][t]);
55             if (i%p[j]==0)break;
56         }
57     }
58     for(int i=0;i<2000;i++)Sum[i+1]+=Sum[i];
59     R=1e18;
60     for(int i=1;;i++){
61         ll s=(ll)i*(i+1),Phi=(ll)phi[i]*phi[i+1];
62         if (s*(i+2)>(R<<1))break;
63         for(int j=i+2;;j++){
64             int g=j/gcd(s,j);
65             if (s>R/g)break;
66             Phi*=g;
67             for(int k=0;k<P[g].size();k++)
68                 if (s%P[g][k])Phi=Phi/P[g][k]*(P[g][k]-1);
69             s*=g;
70             v.push_back(make_pair(s,(ui)Phi)); 
71         }
72     }
73     sort(v.begin(),v.end());
74     for(int i=v.size();i;i--)sum[i-1]=sum[i]+v[i-1].second;
75     scanf("%d",&t);
76     while (t--){
77         scanf("%lld%lld",&L,&R);
78         ans=sum[lower_bound(v.begin(),v.end(),make_pair(L,(ui)0))-v.begin()];
79         ans-=sum[lower_bound(v.begin(),v.end(),make_pair(R+1,(ui)0))-v.begin()];
80         printf("%u
",ans+calc(R)-calc(L-1));
81     }
82     return 0;
83 }

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