bzoj3758 数数和bzoj3798 特殊的质数
https://darkbzoj.tk/problem/3758
问 \([l,r]\) 区间内，有几个数的各个位可以分成两个集合，且两个集合内的所有数的和相等
\(l\le r\le 10^9\)
https://darkbzoj.tk/problem/3798
问 \([l,r]\) 之间有多少个质数可以表示成两个正整数的平方和
\(l\le r\le 3\cdot 10^8\)
分段打表，首先这两个题暴力做都很简单，第一个是分离每一位然后背包，第二个就先判一个数是不是质数，然后预处理出一些正整数的平方，然后两个指针 \(i,j\) 去枚举
那么考虑预先把所有 \([1,x]\) 内符合要求的数的个数求出来
对于那个数数，分成 \(1000\) 段，每段 \(10^6\)，就对于每一个 \(k\cdot 10^6\) 把它的结果打进表里
特殊的质数我是分成的 \(3000\) 段
然后想求 \([1,n]\) 的符合要求的数时，就先拿出表中第 \(\lfloor\dfrac{n}{blcok\_size}\rfloor\) 个数，然后剩下的 \([block\_size\cdot \lfloor\dfrac{n}{blcok\_size}\rfloor+1,n]\) 就暴力算
3758：
#define block_size 1000000
const int table[1005]={0,376413,832547,1288828,1744956,2196800,2647716,3090920,3526440,3951372,4366880,4823015,5304766,5797144,6290672,6782004,7272530,7758910,8238396,8710536,9182258,9638539,10130918,10610575,11103529,11590745,12080513,12565094,13045216,13523103,13996047,14452175,14945703,15438658,15913605,16403907,16892727,17371943,17858307,18339125,18810719,19262563,19753895,20241111,20731414,21199399,21684697,22170473,22655525,23128982,23601310,24052226,24542752,25032520,25521340,26006639,26480512,26967738,27450344,27929454,28401296,28844500,29330880,29815461,30294677,30780453,31267680,31727883,32207445,32681531,33151141,33586661,34066147,34546269,35032633,35517685,36000291,36479854,36928875,37398605,37865237,38290169,38762309,39240196,39721014,40194471,40673581,41147667,41617398,42050819,42514127,42929635,43401357,43874301,44345895,44818223,45290065,45759675,46226307,46689616,47111407,47567542,48049293,48541671,49035199,49526531,50017057,50503437,50982923,51455063,51926786,52408537,52900916,53398017,53897101,54395976,54894264,55390376,55882154,56370124,56858336,57350715,57847816,58346901,58846682,59346650,59846494,60345546,60842736,61337762,61832643,62326171,62825256,63325037,63822618,64321881,64821477,65319367,65818159,66315614,66811192,67302524,67801399,68301368,68800631,69295186,69792675,70292149,70790949,71286703,71781713,72272239,72770527,73270371,73769968,74267457,74763402,75261351,75759527,76256605,76751555,77237935,77734047,78233099,78730989,79230464,79728413,80220834,80714563,81209373,81704149,82183635,82675413,83172603,83671395,84170195,84668372,85162101,85646750,86134011,86626601,87098741,87586711,88081737,88579192,89074946,89572024,90066835,90554096,91029271,91513942,91985664,92473876,92968757,93464335,93959345,94454295,94949071,95441662,95926333,96400455,96856736,97349115,97828772,98321726,98808942,99298710,99783291,100263413,100741300,101214244,101706623,102203724,102702809,103202590,103702558,104202402,104701454,105198644,105693670,106188552,106668209,107167293,107654509,108153408,108643571,109140447,109631936,110124934,110615086,111104170,111597125,112096906,112595804,113093027,113592810,114091415,114589181,115087418,115585176,116080738,116567954,117067923,117558086,118057868,118549357,119048442,119540443,120037757,120529913,121025141,121514909,122014753,122511630,123010235,123509319,124005151,124504320,124999107,125496723,125991518,126476099,126975151,127466640,127964407,128456408,128955576,129444361,129941988,130432181,130927549,131407671,131904861,132397859,132896096,133393411,133888198,134385824,134870790,135365761,135853348,136331235,136826261,137316413,137814171,138306327,138803944,139294137,139789107,140268613,140761869,141234813,141729694,142218778,142714340,143209568,143704363,144199732,144687319,145180575,145654892,146111020,146604548,147097503,147572450,148062752,148551572,149030788,149517152,149997970,150469564,150963092,151462177,151961958,152459539,152958802,153458398,153956288,154455080,154952535,155448113,155941068,156440849,156939747,157436970,157936753,158435358,158933124,159431361,159929119,160424682,160899629,161397209,161894432,162373648,162870479,163366196,163847035,164342307,164834856,165316242,165806545,166305808,166805590,167302421,167797533,168296439,168794206,169291181,169787227,170282583,170771403,171271000,171769605,172265321,172764227,173259685,173756939,174255102,174749455,175244787,175724003,176221893,176719660,177200499,177698265,178195519,178676905,179174053,179669962,180151503,180637867,181136659,181634896,182130169,182627144,183125306,183622454,184112942,184610046,185104838,185585656,186083111,186580869,187073418,187569465,188063818,188559726,189056830,189540680,190034891,190506485,191002063,191497625,191979011,192474367,192969700,193451241,193946033,194440244,194913068,195364912,195856244,196343460,196833763,197301748,197787046,198272822,198757874,199231331,199703659,200194991,200693866,201193835,201693098,202187653,202685142,203184616,203683416,204179170,204674180,205161396,205661365,206151528,206651310,207142799,207641884,208133885,208631199,209123355,209618583,210108886,210608149,211107931,211604762,212099874,212598780,213096547,213593522,214089568,214584925,215052910,215547464,216038953,216534065,217007522,217499718,217990901,218484410,218959905,219446727,219932026,220429515,220928599,221427505,221919701,222413231,222911993,223409872,223904583,224393574,224879350,225378825,225870826,226368592,226859775,227358537,227847477,228346119,228838089,229333643,229818695,230317495,230814810,231311785,231805293,232303172,232801814,233291625,233787283,234281777,234755234,235250988,235743144,236239191,236714686,237209396,237701366,238197024,238673284,239165608,239637936,240132946,240628174,241123530,241610353,242099344,242594898,243089392,243581716,244056062,244506978,244997504,245487272,245976092,246461391,246935264,247422490,247905096,248384206,248856048,249346574,249844862,250344706,250844303,251341792,251837737,252335686,252833862,253330940,253825890,254315658,254815502,255312379,255810984,256310068,256805900,257305069,257799856,258297472,258792267,259281087,259780684,260279289,260775005,261273911,261769369,262266623,262764786,263259139,263754471,264239770,264737259,265236343,265735249,266227445,266720975,267219737,267717616,268212327,268701319,269175192,269671136,270166968,270662426,271155956,271637394,272132980,272626656,273118401,273606213,274093440,274591389,275090557,275587811,276086573,276582159,277074753,277572087,278069041,278564272,279046878,279545055,280039842,280538004,281035883,281529559,282026893,282513143,283009664,283502662,283981772,284478850,284976467,285470820,285965530,286457275,286954229,287450750,287934208,288427696,288899538,289394488,289889283,290384616,290873607,291361419,291856650,292349648,292843136,293317366,293760570,294246950,294731531,295210747,295696523,296183750,296643953,297123515,297597601,298067211,298553591,299049703,299548755,300046645,300546120,301044069,301536490,302030219,302525029,303019805,303504386,304003438,304494927,304992694,305484695,305983863,306472648,306970275,307460468,307955836,308435052,308932942,309430709,309911548,310409314,310906568,311387954,311885102,312381011,312862552,313348328,313847803,314339804,314837570,315328753,315827515,316316455,316815097,317307067,317802621,318289848,318787797,319286965,319784219,320282981,320778567,321271161,321768495,322265449,322760681,323220884,323713304,324202089,324683475,325172415,325665009,326128508,326618472,327104745,327585453,328065016,328558745,329056371,329553519,330052161,330549495,331039459,331524705,332017239,332510934,332985020,333479831,333970024,334465932,334957902,335454856,335941129,336433663,336912903,337405293,337874903,338369679,338865048,339346589,339842143,340337374,340818082,341311777,341804167,342276694,342712214,343191700,343671822,344158186,344643238,345125844,345605407,346054428,346524158,346990790,347470276,347962054,348459244,348958036,349456836,349955013,350448742,350933391,351420652,351913242,352393364,352890554,353383552,353881789,354379104,354873891,355371517,355856483,356351454,356839041,357325405,357824197,358322434,358817707,359314682,359812844,360309992,360800480,361297584,361792376,362277428,362776228,363273543,363770518,364264026,364761905,365260547,365750358,366246016,366740510,367223116,367721293,368216080,368714242,369212121,369705797,370203131,370689381,371185902,371678900,372158463,372652192,373149818,373646966,374145608,374642942,375132906,375618152,376110686,376604382,377053403,377538051,378023017,378513505,379003316,379489566,379974812,380427196,380908305,381388631,381858362,382345623,382840593,383337697,383833355,384329876,384822410,385303519,385778157,386266143,386732775,387225366,387712953,388207745,388702239,389195237,389688932,390169258,390657244,391130374,391555306,392027446,392505333,392986151,393459608,393938718,394412804,394882535,395315956,395779264,396251404,396739374,397234400,397731855,398227609,398724687,399219498,399706759,400181934,400666605,401144492,401639518,402129670,402627428,403119584,403617201,404107394,404602364,405081870,405575126,406055944,406553399,407051157,407543706,408039753,408534106,409030014,409527118,410010968,410505179,410978636,411474390,411966546,412462593,412938088,413432798,413924768,414420426,414896686,415389010,415868120,416365198,416862815,417357168,417851878,418343623,418840577,419337098,419820556,420314044,420788130,421282941,421773134,422269042,422761012,423257966,423744239,424236773,424716013,425208403,425678134,426165395,426660365,427157469,427653127,428149648,428642182,429123291,429597929,430085916,430519337,430994511,431474017,431957867,432434127,432917585,433396825,433871463,434306086,434779290,435242599,435727270,436220526,436714737,437207061,437700549,438192939,438680925,439154129,439626259,440041767,440513489,440986433,441458027,441930355,442402197,442871807,443338439,443801748,444223539,444695261,445183473,445678354,446173932,446668942,447163892,447658668,448151259,448635930,449110052,449582996,450077877,450566961,451062523,451557751,452052546,452547915,453035502,453528758,454003075,454474669,454970247,455465809,455947195,456442551,456937884,457419425,457914217,458408428,458881252,459353580,459848590,460343818,460839174,461325997,461814988,462310542,462805036,463297360,463771706,464243548,464738498,465233293,465728626,466217617,466705429,467200660,467693658,468187146,468661376,469130986,469625762,470121131,470602672,471098226,471593457,472074165,472567860,473060250,473532777,473999409,474492000,474979587,475474379,475968873,476461871,476955566,477435892,477923878,478397008,478860317,479344988,479838244,480332455,480824779,481318267,481810657,482298643,482771847,483243978,483665769,484139891,484614208,485087032,485561378,486035608,486508135,486981265,487453395,487875963};
int a[15],f[105];
inline int check(int x){
a[0]=0;int sum=0;
while(x) a[++a[0]]=x%10,sum+=a[a[0]],x/=10;
if(sum&1) return 0;
sum>>=1;
std::memset(f,0,sizeof f);f[0]=1;
for(reg int i=1;i<=a[0];i++)
for(reg int j=sum-a[i];j>=0;j--) f[j+a[i]]|=f[j];
return f[sum];
}
inline int calc(int x){
if(!x) return 0;
int ans=0;
for(reg int i=x/block_size*block_size+1;i<=x;i++) ans+=check(i);
return ans+table[x/block_size];
}
int main(){
int l=read(),r=read();
printf("%d",calc(r)-calc(l-1));
return 0;
}
3798：
#include<cstdio>
#include<algorithm>
#include<iostream>
#include<cmath>
#include<map>
#include<iomanip>
#include<cstring>
#define reg register
#define EN std::puts("")
#define LL long long
inline int read(){
register int x=0;register int y=1;
register char c=std::getchar();
while(c<'0'||c>'9'){if(c=='-') y=0;c=std::getchar();}
while(c>='0'&&c<='9'){x=x*10+(c^48);c=std::getchar();}
return y?x:-x;
}
#define block_size 100000
const int table[3006]={0,4784,8978,12981,16901,20732,24523,28249,31916,35585,39176,42818,46430,49962,53516,57023,60523,64040,67489,70985,74417,77879,81285,84679,88055,91433,94878,98240,101608,104979,108284,111653,115039,118375,121695,124997,128270,131626,134912,138219,141503,144830,148121,151368,154619,157925,161166,164444,167675,170946,174194,177433,180617,183844,187048,190283,193493,196709,199900,203111,206333,209526,212732,215966,219123,222272,225463,228651,231844,235069,238261,241443,244581,247733,250868,253974,257141,260352,263482,266623,269760,272885,276014,279149,282271,285396,288556,291710,294878,297984,301101,304177,307326,310444,313579,316642,319813,322904,326007,329106,332181,335268,338394,341465,344551,347665,350739,353840,356909,360000,363080,366160,369261,372328,375439,378518,381635,384728,387817,390920,393993,397033,400087,403209,406288,409309,412333,415431,418499,421556,424569,427583,430660,433680,436753,439791,442818,445857,448901,451962,454974,457972,461015,464069,467135,470153,473178,476215,479250,482265,485275,488279,491293,494311,497242,500236,503251,506276,509308,512383,515414,518404,521398,524428,527422,530472,533469,536442,539450,542467,545501,548499,551461,554513,557493,560489,563514,566514,569508,572481,575503,578464,581445,584484,587448,590416,593357,596373,599365,602379,605357,608340,611346,614316,617284,620283,623219,626204,629187,632137,635171,638124,641088,644018,646968,649933,652929,655897,658857,661819,664771,667743,670676,673688,676670,679672,682576,685541,688504,691478,694490,697405,700326,703273,706181,709115,712113,715038,717989,720960,723896,726882,729831,732782,735740,738676,741606,744563,747458,750384,753318,756217,759188,762199,765150,768087,771012,773928,776896,779827,782754,785698,788611,791513,794449,797361,800286,803241,806180,809116,812028,814963,817868,820796,823752,826720,829643,832596,835498,838436,841364,844292,847194,850080,853022,855895,858875,861761,864672,867592,870514,873482,876358,879282,882186,885138,888023,890933,893857,896758,899686,902573,905460,908421,911332,914214,917110,920007,922881,925822,928780,931678,934604,937520,940453,943326,946295,949181,952088,955026,957871,960726,963624,966507,969407,972309,975196,978085,980962,983867,986746,989647,992538,995405,998272,1001141,1004047,1006935,1009828,1012698,1015619,1018497,1021407,1024321,1027185,1030064,1032962,1035803,1038722,1041665,1044521,1047338,1050175,1053074,1055897,1058798,1061676,1064500,1067388,1070268,1073174,1076029,1078883,1081790,1084692,1087554,1090413,1093331,1096221,1099056,1101903,1104801,1107645,1110568,1113425,1116309,1119226,1122066,1124941,1127772,1130652,1133559,1136475,1139334,1142195,1145026,1147867,1150738,1153592,1156490,1159339,1162244,1165140,1168043,1170911,1173746,1176591,1179444,1182370,1185244,1188093,1190925,1193801,1196660,1199545,1202432,1205285,1208198,1211026,1213868,1216688,1219588,1222393,1225259,1228119,1230996,1233830,1236682,1239605,1242422,1245272,1248112,1250996,1253792,1256652,1259537,1262363,1265205,1268047,1270890,1273689,1276506,1279377,1282286,1285127,1287980,1290771,1293580,1296434,1299249,1302081,1304899,1307765,1310565,1313437,1316314,1319179,1321953,1324799,1327680,1330570,1333417,1336222,1339091,1341880,1344730,1347558,1350402,1353252,1356098,1358925,1361746,1364529,1367411,1370266,1373089,1375934,1378787,1381608,1384448,1387240,1390062,1392850,1395709,1398555,1401374,1404179,1407069,1409899,1412745,1415626,1418381,1421271,1424100,1426913,1429775,1432595,1435430,1438258,1441129,1443939,1446779,1449605,1452383,1455248,1458083,1460920,1463729,1466529,1469369,1472195,1475031,1477838,1480667,1483501,1486323,1489166,1491981,1494787,1497648,1500453,1503239,1506081,1508873,1511686,1514527,1517367,1520184,1523011,1525806,1528622,1531402,1534201,1536997,1539894,1542683,1545503,1548241,1551093,1553884,1556702,1559565,1562352,1565117,1567943,1570716,1573536,1576359,1579176,1581960,1584772,1587542,1590356,1593212,1596005,1598783,1601564,1604386,1607165,1609974,1612824,1615617,1618452,1621217,1624007,1626767,1629636,1632421,1635222,1637991,1640857,1643718,1646485,1649297,1652064,1654854,1657619,1660460,1663265,1666081,1668892,1671700,1674594,1677356,1680125,1682935,1685691,1688539,1691341,1694101,1696863,1699639,1702441,1705278,1708110,1710912,1713725,1716510,1719267,1722095,1724880,1727671,1730436,1733217,1736052,1738807,1741642,1744421,1747205,1750007,1752772,1755537,1758337,1761093,1763839,1766647,1769444,1772246,1775046,1777876,1780671,1783464,1786236,1789019,1791841,1794680,1797448,1800274,1803074,1805871,1808647,1811416,1814143,1816933,1819707,1822528,1825244,1828005,1830793,1833548,1836360,1839214,1841996,1844792,1847581,1850327,1853089,1855856,1858663,1861433,1864207,1866991,1869796,1872540,1875345,1878117,1880878,1883646,1886490,1889297,1892075,1894867,1897679,1900521,1903296,1906069,1908797,1911589,1914346,1917129,1919906,1922737,1925489,1928318,1931029,1933873,1936618,1939421,1942195,1944975,1947727,1950518,1953327,1956076,1958882,1961653,1964430,1967231,19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50048,6952622,6955229,6957811,6960447,6963003,6965624,6968229,6970814,6973374,6975908,6978553,6981142,6983701,6986303,6988895,6991452,6994053,6996636,6999239,7001827,7004386,7006943,7009509,7012092,7014637,7017247,7019911,7022468,7025101,7027733,7030312,7032873,7035446,7038006,7040537,7043119,7045744,7048342,7050888,7053483,7056063,7058671,7061274,7063885,7066487,7069099,7071649,7074208,7076794,7079414,7081992,7084585,7087182,7089730,7092253,7094829,7097378,7099975,7102560,7105136,7107734,7110341,7112923,7115505,7118105,7120687,7123329,7125909,7128529,7131104,7133676,7136261,7138851,7141412,7143963,7146532,7149099,7151678,7154306,7156895,7159502,7162172,7164744,7167291,7169862,7172477,7175070,7177687,7180299,7182859,7185447,7188010,7190543,7193111,7195725,7198306,7200837,7203410,7206026,7208623,7211221,7213777,7216339,7218922,7221467,7224118,7226696,7229224,7231835,7234448,7237074,7239661,7242248,7244831,7247438,7250010,7252624,7255179,7257777,7260342,7262872,7265456,7267970,7270545,7273095,7275708,7278251,7280851,7283410,7285994,7288560,7291115,7293745,7296316,7298848,7301442,7304018,7306655,7309180,7311705,7314293,7316887,7319531,7322112,7324659,7327260,7329881,7332442,7335022,7337604,7340210,7342739,7345280,7347817,7350378,7352945,7355504,7358025,7360615,7363210,7365832,7368429,7371013,7373605,7376145,7378732,7381276,7383811,7386365,7388933,7391502,7394032,7396642,7399245,7401866,7404457,7406982,7409538,7412102,7414691,7417288,7419884,7422441,7424989,7427522,7430101,7432639,7435216,7437740,7440294,7442912,7445516,7448114,7450684,7453261,7455841,7458445,7461027,7463635,7466198,7468787,7471320,7473930,7476552,7479118,7481641,7484264,7486781,7489342,7491952,7494521,7497063,7499688,7502254,7504815,7507392,7509988,7512553,7515087,7517707,7520308,7522824,7525392,7527945,7530543,7533105,7535700,7538260,7540804,7543418,7545943,7548535,7551130,7553729,7556313,7558844,7561435,7563980,7566505,7569080,7571641,7574170,7576783,7579338,7581897,7584422,7587022,7589577,7592151,7594705,7597276,7599794,7602390,7604973,7607619,7610137,7612748,7615269,7617858,7620416,7622960,7625549,7628196,7630753,7633277,7635834,7638354,7640910,7643496,7646121,7648660,7651279,7653833,7656431,7659011,7661580,7664134,7666612,7669193,7671776,7674356,7676915,7679488,7682031,7684595,7687146,7689731,7692293,7694861,7697419,7700031,7702574,7705205,7707783,7710348,7712918,7715474,7718028,7720597,7723207,7725778,7728305,7730889,7733442,7736038,7738641,7741178,7743670,7746199,7748766,7751407,7753967,7756540,7759095,7761686,7764213,7766793,7769379,7771905,7774461,7777035,7779545,7782119,7784671,7787226,7789834,7792385,7795028,7797627,7800194,7802758,7805346,7807903,7810459,7813000,7815577,7818155,7820761,7823285,7825842,7828445,7830988,7833550,7836167,7838714,7841312,7843890,7846422,7848885,7851434,7854000,7856583,7859172,7861776,7864364,7866909,7869478,7872018,7874591,7877131,7879768,7882314,7884851,7887428,7890024,7892632,7895233,7897844,7900429,7902975,7905546,7908066,7910582,7913125,7915728,7918312,7920913,7923470,7926037,7928634,7931211,7933766,7936281,7938828,7941378,7943902,7946432,7949032,7951602,7954217,7956760,7959316,7961868,7964451,7967003,7969567,7972124,7974694,7977293,7979866,7982413,7985018,7987569,7990128,7992722,7995266,7997861,8000394,8002935,8005473,8008047,8010615,8013200,8015753,8018323,8020862,8023413,8025983,8028499,8031030,8033583,8036158,8038758,8041400,8043966,8046482,8049029,8051591,8054132,8056748,8059305,8061849,8064377,8066944,8069508,8072033,8074668,8077206,8079782,8082323,8084914,8087510,8090092,8092568,8095135,8097678,8100189,8102726,8105256,8107767,8110337,8112936,8115467,8118011,8120631,8123161,8125719};
LL sqr[20005];
inline int check(LL x){
int sqrt=std::sqrt(x);
for(reg int i=2;i<=sqrt;i++)
if(!(x%i)) return 0;
reg int i=1,j=sqrt+1;
for(;sqr[i]<x;i++){
while(sqr[i]+sqr[j]>x) j--;
//由于随着 i 的增加，sqr[i] 增加，那么如果对于这个 i，sqr[i]+sqr[j]>x，那么对于更大的 i 肯定也是大于 x
if(sqr[i]+sqr[j]==x) return 1;
}
return 0;
}
inline int calc(int x){
if(!x) return 0;
int ans=0;
for(reg int i=x/block_size*block_size+1;i<=x;i++) ans+=check(i);
return ans+table[x/block_size];
}
int main(){
for(reg int i=1;i<=20000;i++) sqr[i]=i*i;
int l=read(),r=read();
printf("%d",calc(r)-calc(l-1));
return 0;
}
放一个打表程序
for(reg int i=1;i<=3e8;i++){
sum+=check(i);
if(!(i%block_size)){
printf("%d,",sum);
std::fclose(stdout);
std::freopen("CON","w",stdout);
printf("now is %d\n",i);
std::fclose(stdout);
std::freopen("table.txt","a",stdout);
}
}
