/* Helper program for https://oeis.org/A309095 Find the smallest number of geometric progressions that cover the numbers 1, 2, ... N See end of file for best known results https://math.stackexchange.com/questions/1385991/what-is-known-about-the-minimal-number-fn-of-geometric-progressions-needed-t?lq=1 By Dmitry Kamenetsky, 20th July 2019 */ import java.util.*; public class CoveringGP { static int N; static List<GP> GPs = new ArrayList<GP>(); public static void main(String[] args) { if (args.length!=1) { System.out.println("usage: java CoveringGP N"); System.out.println("Finds the smallest number of geometric progressions that cover the numbers 1, 2, ..., N"); System.out.println("Only prints progressions with 3 or more terms, the remaining numbers can be covered with one progression for each two numbers"); System.exit(0); } N=Integer.parseInt(args[0]); generateGPSrational(); solve(); } public static void generateGPSrational() { //first,second boolean[][] seen=new boolean[N+1][N+1]; System.out.println("progressions with 3 or more terms:"); for (int i=1; i<=N; i++) { for (int k=i+1; k<=N; k++) { if (seen[i][k]) continue; int div=getMinDivisor(i,k); if (div==-1) continue; int mult=k/(i/div); int cur=i; GP gp=new GP(); while(true) { gp.a.add(cur); if (cur%div!=0) break; cur/=div; cur*=mult; if (cur>N) break; } if (gp.a.size()>=3) { for (int a : gp.a) System.out.print(a+" "); System.out.println(); GPs.add(gp); //update seen for (int p=0; p<gp.a.size(); p++) for (int q=p+1; q<gp.a.size(); q++) seen[gp.a.get(p)][gp.a.get(q)]=true; } } } System.out.println("total number of progressions: "+GPs.size()+"\n"); } //b>a public static int getMinDivisor(int a, int b) { for (int i=1; i<=Math.min(a,b); i++) if (a%i==0 && b%i==0 && b%(a/i)==0) return i; return -1; } public static void solve() { int n=GPs.size(); int maxChanges=5; int bestScore=Integer.MAX_VALUE; if (n==0) { System.out.println("progressions needed: "+(N+1)/2); return; } boolean[] bestA=new boolean[n]; for (int i=0; i<n; i++) bestA[i]=(Math.random()<0.5); boolean[] a=new boolean[n]; while(true) { for (int i=0; i<n; i++) a[i]=bestA[i]; int changes=(int)(Math.random()*maxChanges+1); for (int i=0; i<changes; i++) { int pos=(int)(Math.random()*n); a[pos]=!a[pos]; } optimize(a); int score=score(a); if (score<=bestScore) { if (score<bestScore && score<=N/2) { System.out.println("progressions needed: "+score); print(a); System.out.println("progressions needed: "+score); System.out.println(); } bestScore=score; for (int i=0; i<n; i++) bestA[i]=a[i]; } } } public static void optimize(boolean[] a) { int[] counts=new int[N+1]; int sets=0; int unique=0; for (int i=0; i<GPs.size(); i++) if (a[i]) { sets++; for (int k : GPs.get(i).a) { counts[k]++; if (counts[k]==1) unique++; } } int missing=N-unique; int extraSets=(missing+1)/2; int score=sets+extraSets; if (score!=score(a)) System.out.println("aaargh! "+score+" "+score(a)); int G=GPs.size(); int[] ind=new int[G]; for (int i=0; i<ind.length; i++) ind[i]=i; while(true) { boolean changed=false; shuffle(ind); for (int i=0; i<ind.length; i++) { int pos=ind[i]; int newSets=sets; int newUnique=unique; //remove set if (a[pos]) { newSets--; for (int k : GPs.get(pos).a) { counts[k]--; if (counts[k]==0) newUnique--; } a[pos]=false; } //add set else { newSets++; for (int k : GPs.get(pos).a) { counts[k]++; if (counts[k]==1) newUnique++; } a[pos]=true; } missing=N-newUnique; extraSets=(missing+1)/2; int newScore=newSets+extraSets; //if (newScore!=score(a)) System.out.println("aaaargh2 "+newScore+" "+score(a)); if (newScore<=score) { if (newScore<score) changed=true; score=newScore; sets=newSets; unique=newUnique; } //undo else { //remove set if (a[pos]) { for (int k : GPs.get(pos).a) counts[k]--; a[pos]=false; } //add set else { for (int k : GPs.get(pos).a) counts[k]++; a[pos]=true; } } } if (!changed) break; } } //computes number of sets needed to cover all numbers public static int score(boolean[] a) { Set<Integer> covered=new HashSet<Integer>(); //integers that are covered int sets=0; for (int i=0; i<GPs.size(); i++) if (a[i]) { sets++; for (int k : GPs.get(i).a) covered.add(k); } int missing=N-covered.size(); int extraSets=(missing+1)/2; //extra sets can cover up to 2 numbers each return sets+extraSets; } //shuffle the array randomly public static void shuffle(int[] a) { for (int i=0; i<a.length; i++) { int k=(int)(Math.random()*(a.length-i)+i); int temp=a[i]; a[i]=a[k]; a[k]=temp; } } public static void print(boolean[] a) { for (int i=0; i<GPs.size(); i++) { if (a[i]) { GPs.get(i).print(); System.out.print(", "); } } System.out.println(); } static class GP { List<Integer> a; public GP() { a=new ArrayList<Integer>(); } void print() { System.out.print("("); for (int i=0; i<a.size(); i++) { System.out.print(a.get(i)); if (i<a.size()-1) System.out.print(","); } System.out.print(")"); } } } /* RESULTS N=100 progressions needed: 36 (1,2,4,8,16,32,64), (3,6,12,24,48,96), (5,10,20,40,80), (7,14,28,56), (9,18,36,72), (11,22,44,88), (13,26,52), (15,30,60), (17,34,68), (19,38,76), (21,42,84), (23,46,92), (25,35,49), (27,45,75), (44,66,99), (50,70,98), (81,90,100) N=200 progressions needed: 73 (1,2,4,8,16,32,64,128), (3,6,12,24,48,96,192), (5,10,20,40,80,160), (7,14,28,56,112), (9,18,36,72,144), (11,22,44,88,176), (13,26,52,104), (15,30,60,120), (17,34,68,136), (19,38,76,152), (19,57,171), (21,42,84,168), (23,46,92,184), (25,65,169), (27,45,75,125), (29,58,116), (31,62,124), (33,66,132), (35,70,140), (37,74,148), (39,78,156), (41,82,164), (43,86,172), (47,94,188), (49,98,196), (54,90,150), (63,105,175), (68,102,153), (81,99,121), (108,126,147), (162,180,200) N=500 progressions needed: 182 (1,2,4,8,16,32,64,128,256), (3,6,12,24,48,96,192,384), (5,10,20,40,80,160,320), (7,14,28,56,112,224,448), (9,18,36,72,144,288), (11,22,44,88,176,352), (13,26,52,104,208,416), (15,30,60,120,240,480), (17,34,68,136,272), (19,38,76,152,304), (21,42,84,168,336), (23,46,92,184,368), (25,50,100,200,400), (27,45,75,125), (29,58,116,232,464), (31,62,124,248,496), (33,66,132,264), (35,70,140,280), (37,74,148,296), (39,78,156,312), (41,82,164,328), (43,86,172,344), (46,138,414), (47,94,188,376), (49,98,196,392), (51,102,204,408), (53,106,212,424), (54,90,150,250), (55,110,220,440), (57,114,228,456), (59,118,236,472), (61,122,244,488), (63,105,175), (65,130,260), (67,134,268), (71,142,284), (73,146,292), (77,154,308), (79,158,316), (81,135,225,375), (83,166,332), (85,170,340), (87,174,348), (88,132,198,297), (89,178,356), (91,182,364), (93,186,372), (95,190,380), (97,194,388), (99,165,275), (101,202,404), (103,206,412), (104,156,234,351), (107,214,428), (108,180,300,500), (109,218,436), (111,222,444), (113,226,452), (115,230,460), (119,238,476), (121,143,169), (123,246,492), (126,210,350), (136,204,306,459), (147,189,243), (153,255,425), (162,270,450), (171,285,475), (188,282,423), (207,276,368), (216,252,294,343), (220,330,495), (242,286,338), (245,315,405), (289,357,441), (294,378,486), (324,342,361), (325,390,468), (360,420,490), (363,396,432), (441,462,484) N=1000 progressions needed: 362 (1,2,4,8,16,32,64,128,256,512), (3,6,12,24,48,96,192,384,768), (5,10,20,40,80,160,320,640), (7,14,28,56,112,224,448,896), (9,18,36,72,144,288,576), (11,22,44,88,176,352,704), (13,26,52,104,208,416,832), (15,30,60,120,240,480,960), (17,34,68,136,272,544), (19,38,76,152,304,608), (21,42,84,168,336,672), (23,46,92,184,368,736), (25,50,100,200,400,800), (27,54,108,216,432,864), (29,58,116,232,464,928), (31,62,124,248,496,992), (33,66,132,264,528), (35,70,140,280,560), (37,74,148,296,592), (39,78,156,312,624), (41,82,164,328,656), (43,86,172,344,688), (45,90,180,360,720), (47,94,188,376,752), (49,98,196,392,784), (51,102,204,408,816), (53,106,212,424,848), (55,110,220,440,880), (57,114,228,456,912), (59,118,236,472,944), (61,122,244,488,976), (63,126,252,504), (65,130,260,520), (67,134,268,536), (69,138,276,552), (71,142,284,568), (73,146,292,584), (75,150,300,600), (77,154,308,616), (79,158,316,632), (81,135,225,375,625), (83,166,332,664), (85,170,340,680), (87,174,348,696), (89,178,356,712), (91,182,364,728), (93,186,372,744), (95,190,380,760), (97,194,388,776), (99,198,396,792), (101,202,404,808), (101,303,909), (103,206,412,824), (105,210,420,840), (107,214,428,856), (107,321,963), (109,218,436,872), (111,222,444,888), (112,168,252,378,567), (113,226,452,904), (115,230,460,920), (117,234,468,936), (119,238,476,952), (121,242,484,968), (123,246,492,984), (125,175,245,343), (127,254,508), (129,258,516), (131,262,524), (133,266,532), (137,274,548), (139,278,556), (141,282,564), (143,286,572), (145,290,580), (147,294,588), (149,298,596), (151,302,604), (153,255,425), (155,310,620), (157,314,628), (159,318,636), (161,322,644), (162,270,450,750), (163,326,652), (165,330,660), (167,334,668), (169,273,441), (171,285,475), (173,346,692), (177,354,708), (179,358,716), (181,362,724), (183,366,732), (185,370,740), (187,374,748), (189,315,525,875), (191,382,764), (193,386,772), (195,390,780), (197,394,788), (199,398,796), (201,402,804), (203,406,812), (205,410,820), (207,345,575), (209,418,836), (211,422,844), (213,426,852), (215,430,860), (217,434,868), (219,438,876), (221,442,884), (227,454,908), (229,458,916), (231,462,924), (233,466,932), (235,470,940), (237,474,948), (239,478,956), (241,482,964), (243,405,675), (247,494,988), (249,498,996), (250,350,490,686), (261,435,725), (279,465,775), (289,391,529), (297,495,825), (306,510,850), (324,558,961), (325,455,637), (333,555,925), (338,546,882), (342,570,950), (351,585,975), (361,399,441), (363,561,867), (388,582,873), (486,594,726), (500,550,605), (507,702,972), (539,693,891), (540,630,735), (578,646,722), (612,714,833), (648,756,882), (676,754,841), (684,798,931), (700,770,847), (729,810,900,1000), (845,910,980) N=2000 progressions needed: 725 (1,2,4,8,16,32,64,128,256,512,1024), (3,6,12,24,48,96,192,384,768,1536), (5,10,20,40,80,160,320,640,1280), (7,14,28,56,112,224,448,896,1792), (9,18,36,72,144,288,576,1152), (11,22,44,88,176,352,704,1408), (13,26,52,104,208,416,832,1664), (15,30,60,120,240,480,960,1920), (17,34,68,136,272,544,1088), (19,38,76,152,304,608,1216), (21,42,84,168,336,672,1344), (23,46,92,184,368,736,1472), (25,50,100,200,400,800,1600), (27,54,108,216,432,864,1728), (29,58,116,232,464,928,1856), (31,62,124,248,496,992,1984), (33,66,132,264,528,1056), (35,70,140,280,560,1120), (37,74,148,296,592,1184), (39,78,156,312,624,1248), (41,82,164,328,656,1312), (43,86,172,344,688,1376), (45,90,180,360,720,1440), (47,94,188,376,752,1504), (49,98,196,392,784,1568), (51,102,204,408,816,1632), (53,106,212,424,848,1696), (55,110,220,440,880,1760), (57,114,228,456,912,1824), (59,118,236,472,944,1888), (61,122,244,488,976,1952), (63,126,252,504,1008), (65,130,260,520,1040), (67,134,268,536,1072), (69,138,276,552,1104), (71,142,284,568,1136), (73,146,292,584,1168), (75,150,300,600,1200), (77,154,308,616,1232), (79,158,316,632,1264), (81,162,324,648,1296), (83,166,332,664,1328), (85,170,340,680,1360), (87,174,348,696,1392), (89,178,356,712,1424), (91,182,364,728,1456), (93,186,372,744,1488), (95,190,380,760,1520), (97,194,388,776,1552), (99,198,396,792,1584), (101,202,404,808,1616), (103,206,412,824,1648), (105,210,420,840,1680), (107,214,428,856,1712), (109,218,436,872,1744), (111,222,444,888,1776), (113,226,452,904,1808), (115,230,460,920,1840), (117,234,468,936,1872), (119,238,476,952,1904), (121,242,484,968,1936), (123,246,492,984,1968), (125,250,500,1000,2000), (127,254,508,1016), (129,258,516,1032), (131,262,524,1048), (133,266,532,1064), (135,270,540,1080), (137,274,548,1096), (139,278,556,1112), (141,282,564,1128), (143,286,572,1144), (145,290,580,1160), (147,294,588,1176), (149,298,596,1192), (151,302,604,1208), (153,306,612,1224), (155,310,620,1240), (157,314,628,1256), (159,318,636,1272), (161,322,644,1288), (163,326,652,1304), (165,330,660,1320), (167,334,668,1336), (169,338,676,1352), (171,342,684,1368), (173,346,692,1384), (175,350,700,1400), (177,354,708,1416), (179,358,716,1432), (179,537,1611), (181,362,724,1448), (181,543,1629), (183,366,732,1464), (185,370,740,1480), (187,374,748,1496), (189,378,756,1512), (191,382,764,1528), (193,386,772,1544), (195,390,780,1560), (197,394,788,1576), (199,398,796,1592), (199,597,1791), (201,402,804,1608), (203,406,812,1624), (203,609,1827), (205,410,820,1640), (207,414,828,1656), (209,418,836,1672), (211,422,844,1688), (213,426,852,1704), (215,430,860,1720), (215,645,1935), (217,434,868,1736), (217,651,1953), (219,438,876,1752), (221,442,884,1768), (221,663,1989), (223,446,892,1784), (225,450,900,1800), (227,454,908,1816), (229,458,916,1832), (231,462,924,1848), (233,466,932,1864), (235,470,940,1880), (237,474,948,1896), (239,478,956,1912), (241,482,964,1928), (243,405,675,1125,1875), (245,490,980,1960), (247,494,988,1976), (249,498,996,1992), (251,502,1004), (253,506,1012), (255,510,1020), (257,514,1028), (259,518,1036), (261,435,725), (263,526,1052), (265,530,1060), (267,534,1068), (269,538,1076), (271,542,1084), (273,546,1092), (275,550,1100), (277,554,1108), (279,558,1116), (281,562,1124), (283,566,1132), (285,570,1140), (287,574,1148), (289,527,961), (291,582,1164), (293,586,1172), (295,590,1180), (297,495,825,1375), (299,598,1196), (301,602,1204), (303,606,1212), (305,610,1220), (307,614,1228), (311,622,1244), (313,626,1252), (315,630,1260), (317,634,1268), (319,638,1276), (321,642,1284), (323,646,1292), (325,650,1300), (327,654,1308), (329,658,1316), (331,662,1324), (333,555,925), (335,670,1340), (337,674,1348), (339,678,1356), (341,682,1364), (343,539,847,1331), (345,690,1380), (347,694,1388), (349,698,1396), (351,585,975,1625), (353,706,1412), (355,710,1420), (357,714,1428), (359,718,1436), (361,551,841), (363,627,1083), (365,730,1460), (367,734,1468), (369,615,1025), (369,738,1476), (371,742,1484), (373,746,1492), (375,525,735,1029), (377,754,1508), (379,758,1516), (381,762,1524), (383,766,1532), (385,770,1540), (389,778,1556), (391,782,1564), (393,786,1572), (395,790,1580), (397,794,1588), (399,798,1596), (401,802,1604), (403,806,1612), (407,814,1628), (409,818,1636), (411,822,1644), (413,826,1652), (415,830,1660), (417,834,1668), (419,838,1676), (421,842,1684), (423,705,1175), (425,595,833), (427,854,1708), (429,858,1716), (431,862,1724), (433,866,1732), (437,874,1748), (439,878,1756), (441,903,1849), (443,886,1772), (445,890,1780), (447,894,1788), (449,898,1796), (451,902,1804), (453,906,1812), (455,910,1820), (457,914,1828), (459,918,1836), (461,922,1844), (463,926,1852), (465,930,1860), (467,934,1868), (469,938,1876), (471,942,1884), (473,946,1892), (475,665,931), (477,795,1325), (479,958,1916), (481,962,1924), (483,966,1932), (485,970,1940), (486,810,1350), (487,974,1948), (488,732,1098,1647), (489,978,1956), (491,982,1964), (493,986,1972), (497,994,1988), (499,998,1996), (507,819,1323), (513,855,1425), (522,870,1450), (529,759,1089), (531,885,1475), (536,804,1206,1809), (549,915,1525), (567,945,1575), (575,805,1127), (578,1054,1922), (584,876,1314,1971), (594,990,1650), (603,1005,1675), (605,1045,1805), (621,1035,1725), (625,875,1225,1715), (637,1001,1573), (639,1065,1775), (657,1095,1825), (666,1110,1850), (668,1002,1503), (686,882,1134,1458), (692,1038,1557), (693,1155,1925), (702,1170,1950), (711,1185,1975), (722,1102,1682), (726,1188,1944), (729,999,1369), (750,1050,1470), (772,1158,1737), (775,1085,1519), (820,1230,1845), (836,1254,1881), (845,1105,1445), (850,1190,1666), (867,1122,1452), (891,1287,1859), (950,1330,1862), (972,1242,1587), (1014,1326,1734), (1044,1218,1421), (1058,1150,1250), (1075,1290,1548), (1078,1386,1782), (1156,1394,1681), (1183,1274,1372), (1210,1430,1690), (1215,1485,1815), (1325,1590,1908), (1332,1554,1813), (1377,1530,1700), (1404,1638,1911), (1444,1482,1521), (1539,1710,1900), (1764,1806,1849) N=10000 progressions needed: 3620 (1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192), (3,6,12,24,48,96,192,384,768,1536,3072,6144), (5,10,20,40,80,160,320,640,1280,2560,5120), (7,14,28,56,112,224,448,896,1792,3584,7168), (9,18,36,72,144,288,576,1152,2304,4608,9216), (11,22,44,88,176,352,704,1408,2816,5632), (13,26,52,104,208,416,832,1664,3328,6656), (15,30,60,120,240,480,960,1920,3840,7680), (17,34,68,136,272,544,1088,2176,4352,8704), (19,38,76,152,304,608,1216,2432,4864,9728), (21,42,84,168,336,672,1344,2688,5376), (23,46,92,184,368,736,1472,2944,5888), (25,50,100,200,400,800,1600,3200,6400), (27,54,108,216,432,864,1728,3456,6912), (29,58,116,232,464,928,1856,3712,7424), (31,62,124,248,496,992,1984,3968,7936), (33,66,132,264,528,1056,2112,4224,8448), (35,70,140,280,560,1120,2240,4480,8960), (37,74,148,296,592,1184,2368,4736,9472), (39,78,156,312,624,1248,2496,4992,9984), (41,82,164,328,656,1312,2624,5248), (43,86,172,344,688,1376,2752,5504), (45,90,180,360,720,1440,2880,5760), (47,94,188,376,752,1504,3008,6016), (49,98,196,392,784,1568,3136,6272), (51,102,204,408,816,1632,3264,6528), (53,106,212,424,848,1696,3392,6784), (55,110,220,440,880,1760,3520,7040), (57,114,228,456,912,1824,3648,7296), (59,118,236,472,944,1888,3776,7552), (61,122,244,488,976,1952,3904,7808), (63,126,252,504,1008,2016,4032,8064), (65,130,260,520,1040,2080,4160,8320), (67,134,268,536,1072,2144,4288,8576), (69,138,276,552,1104,2208,4416,8832), (71,142,284,568,1136,2272,4544,9088), (73,146,292,584,1168,2336,4672,9344), (75,150,300,600,1200,2400,4800,9600), (77,154,308,616,1232,2464,4928,9856), (79,158,316,632,1264,2528,5056), (81,162,324,648,1296,2592,5184), (83,166,332,664,1328,2656,5312), (85,170,340,680,1360,2720,5440), (87,174,348,696,1392,2784,5568), (89,178,356,712,1424,2848,5696), (91,182,364,728,1456,2912,5824), (93,186,372,744,1488,2976,5952), (95,190,380,760,1520,3040,6080), (97,194,388,776,1552,3104,6208), (99,198,396,792,1584,3168,6336), (101,202,404,808,1616,3232,6464), (103,206,412,824,1648,3296,6592), (105,210,420,840,1680,3360,6720), (107,214,428,856,1712,3424,6848), (109,218,436,872,1744,3488,6976), (111,222,444,888,1776,3552,7104), (113,226,452,904,1808,3616,7232), (115,230,460,920,1840,3680,7360), (117,234,468,936,1872,3744,7488), (119,238,476,952,1904,3808,7616), (121,242,484,968,1936,3872,7744), (123,246,492,984,1968,3936,7872), (125,250,500,1000,2000,4000,8000), (127,254,508,1016,2032,4064,8128), (129,258,516,1032,2064,4128,8256), (131,262,524,1048,2096,4192,8384), (133,266,532,1064,2128,4256,8512), (135,270,540,1080,2160,4320,8640), (137,274,548,1096,2192,4384,8768), (139,278,556,1112,2224,4448,8896), (141,282,564,1128,2256,4512,9024), (143,286,572,1144,2288,4576,9152), (145,290,580,1160,2320,4640,9280), (147,294,588,1176,2352,4704,9408), (149,298,596,1192,2384,4768,9536), (151,302,604,1208,2416,4832,9664), (153,306,612,1224,2448,4896,9792), (155,310,620,1240,2480,4960,9920), (157,314,628,1256,2512,5024), (159,318,636,1272,2544,5088), (161,322,644,1288,2576,5152), (163,326,652,1304,2608,5216), (165,330,660,1320,2640,5280), (167,334,668,1336,2672,5344), (169,338,676,1352,2704,5408), (171,342,684,1368,2736,5472), (173,346,692,1384,2768,5536), (175,350,700,1400,2800,5600), (177,354,708,1416,2832,5664), (179,358,716,1432,2864,5728), (181,362,724,1448,2896,5792), (183,366,732,1464,2928,5856), (185,370,740,1480,2960,5920), (187,374,748,1496,2992,5984), (189,378,756,1512,3024,6048), (191,382,764,1528,3056,6112), (193,386,772,1544,3088,6176), (195,390,780,1560,3120,6240), (197,394,788,1576,3152,6304), (199,398,796,1592,3184,6368), (201,402,804,1608,3216,6432), (203,406,812,1624,3248,6496), (205,410,820,1640,3280,6560), (207,414,828,1656,3312,6624), (209,418,836,1672,3344,6688), (211,422,844,1688,3376,6752), (213,426,852,1704,3408,6816), (215,430,860,1720,3440,6880), (217,434,868,1736,3472,6944), (219,438,876,1752,3504,7008), (221,442,884,1768,3536,7072), (223,446,892,1784,3568,7136), (223,669,2007,6021), (225,450,900,1800,3600,7200), (227,454,908,1816,3632,7264), (229,458,916,1832,3664,7328), (231,462,924,1848,3696,7392), (233,466,932,1864,3728,7456), (235,470,940,1880,3760,7520), (237,474,948,1896,3792,7584), (239,478,956,1912,3824,7648), (241,482,964,1928,3856,7712), (243,405,675,1125,1875,3125), (245,490,980,1960,3920,7840), (247,494,988,1976,3952,7904), (249,498,996,1992,3984,7968), (251,502,1004,2008,4016,8032), (253,506,1012,2024,4048,8096), (253,759,2277,6831), (255,510,1020,2040,4080,8160), (257,514,1028,2056,4112,8224), (259,518,1036,2072,4144,8288), (261,522,1044,2088,4176,8352), (263,526,1052,2104,4208,8416), (265,530,1060,2120,4240,8480), (267,534,1068,2136,4272,8544), (269,538,1076,2152,4304,8608), (271,542,1084,2168,4336,8672), (273,546,1092,2184,4368,8736), (275,550,1100,2200,4400,8800), (277,554,1108,2216,4432,8864), (277,831,2493,7479), (279,558,1116,2232,4464,8928), (281,562,1124,2248,4496,8992), (283,566,1132,2264,4528,9056), (285,570,1140,2280,4560,9120), (287,574,1148,2296,4592,9184), (289,578,1156,2312,4624,9248), (291,582,1164,2328,4656,9312), (293,586,1172,2344,4688,9376), (295,590,1180,2360,4720,9440), (297,594,1188,2376,4752,9504), (299,598,1196,2392,4784,9568), (301,602,1204,2408,4816,9632), (303,606,1212,2424,4848,9696), (305,610,1220,2440,4880,9760), (307,614,1228,2456,4912,9824), (309,618,1236,2472,4944,9888), (311,622,1244,2488,4976,9952), (313,626,1252,2504,5008), (315,630,1260,2520,5040), (317,634,1268,2536,5072), (319,638,1276,2552,5104), (321,642,1284,2568,5136), (323,646,1292,2584,5168), (325,650,1300,2600,5200), (327,654,1308,2616,5232), (329,658,1316,2632,5264), (331,662,1324,2648,5296), (333,666,1332,2664,5328), (335,670,1340,2680,5360), (337,674,1348,2696,5392), (339,678,1356,2712,5424), (341,682,1364,2728,5456), (343,686,1372,2744,5488), (345,690,1380,2760,5520), (347,694,1388,2776,5552), (349,698,1396,2792,5584), (351,702,1404,2808,5616), (353,706,1412,2824,5648), (355,710,1420,2840,5680), (357,714,1428,2856,5712), (359,718,1436,2872,5744), (361,722,1444,2888,5776), (363,726,1452,2904,5808), (365,730,1460,2920,5840), (367,734,1468,2936,5872), (369,738,1476,2952,5904), (371,742,1484,2968,5936), (373,746,1492,2984,5968), (375,750,1500,3000,6000), (377,754,1508,3016,6032), (379,758,1516,3032,6064), (381,762,1524,3048,6096), (383,766,1532,3064,6128), (385,770,1540,3080,6160), (387,774,1548,3096,6192), (389,778,1556,3112,6224), (391,782,1564,3128,6256), (393,786,1572,3144,6288), (395,790,1580,3160,6320), (397,794,1588,3176,6352), (399,798,1596,3192,6384), (401,802,1604,3208,6416), (403,806,1612,3224,6448), (407,814,1628,3256,6512), (409,818,1636,3272,6544), (411,822,1644,3288,6576), (413,826,1652,3304,6608), (415,830,1660,3320,6640), (417,834,1668,3336,6672), (419,838,1676,3352,6704), (421,842,1684,3368,6736), (423,846,1692,3384,6768), (425,850,1700,3400,6800), (427,854,1708,3416,6832), (429,858,1716,3432,6864), (431,862,1724,3448,6896), (433,866,1732,3464,6928), (435,870,1740,3480,6960), (437,874,1748,3496,6992), (439,878,1756,3512,7024), (441,882,1764,3528,7056), (443,886,1772,3544,7088), (445,890,1780,3560,7120), (447,894,1788,3576,7152), (449,898,1796,3592,7184), (451,902,1804,3608,7216), (453,906,1812,3624,7248), (455,910,1820,3640,7280), (457,914,1828,3656,7312), (459,918,1836,3672,7344), (461,922,1844,3688,7376), (463,926,1852,3704,7408), (465,930,1860,3720,7440), (467,934,1868,3736,7472), (469,938,1876,3752,7504), (471,942,1884,3768,7536), (473,946,1892,3784,7568), (475,950,1900,3800,7600), (477,954,1908,3816,7632), (479,958,1916,3832,7664), (481,962,1924,3848,7696), (483,966,1932,3864,7728), (485,970,1940,3880,7760), (486,810,1350,2250,3750,6250), (487,974,1948,3896,7792), (489,978,1956,3912,7824), (491,982,1964,3928,7856), (493,986,1972,3944,7888), (495,990,1980,3960,7920), (497,994,1988,3976,7952), (499,998,1996,3992,7984), (501,1002,2004,4008,8016), (503,1006,2012,4024,8048), (505,1010,2020,4040,8080), (507,1014,2028,4056,8112), (509,1018,2036,4072,8144), (511,1022,2044,4088,8176), (513,1026,2052,4104,8208), (515,1030,2060,4120,8240), (517,1034,2068,4136,8272), (519,1038,2076,4152,8304), (521,1042,2084,4168,8336), (523,1046,2092,4184,8368), (525,1050,2100,4200,8400), (527,1054,2108,4216,8432), (529,1058,2116,4232,8464), (531,1062,2124,4248,8496), (533,1066,2132,4264,8528), (535,1070,2140,4280,8560), (537,1074,2148,4296,8592), (539,1078,2156,4312,8624), (541,1082,2164,4328,8656), (543,1086,2172,4344,8688), (545,1090,2180,4360,8720), (547,1094,2188,4376,8752), (549,1098,2196,4392,8784), (551,1102,2204,4408,8816), (553,1106,2212,4424,8848), (555,1110,2220,4440,8880), (557,1114,2228,4456,8912), (559,1118,2236,4472,8944), (561,1122,2244,4488,8976), (563,1126,2252,4504,9008), (565,1130,2260,4520,9040), (567,945,1575,2625,4375), (569,1138,2276,4552,9104), (571,1142,2284,4568,9136), (573,1146,2292,4584,9168), (575,1150,2300,4600,9200), (577,1154,2308,4616,9232), (579,1158,2316,4632,9264), (581,1162,2324,4648,9296), (583,1166,2332,4664,9328), (585,1170,2340,4680,9360), (587,1174,2348,4696,9392), (589,1178,2356,4712,9424), (591,1182,2364,4728,9456), (593,1186,2372,4744,9488), (595,1190,2380,4760,9520), (597,1194,2388,4776,9552), (599,1198,2396,4792,9584), (601,1202,2404,4808,9616), (603,1206,2412,4824,9648), (605,1210,2420,4840,9680), (607,1214,2428,4856,9712), (609,1218,2436,4872,9744), (611,1222,2444,4888,9776), (613,1226,2452,4904,9808), (615,1230,2460,4920,9840), (617,1234,2468,4936,9872), (619,1238,2476,4952,9904), (621,1035,1725,2875), (623,1246,2492,4984,9968), (625,875,1225,1715,2401), (627,1254,2508,5016), (629,1258,2516,5032), (631,1262,2524,5048), (633,1266,2532,5064), (635,1270,2540,5080), (637,1274,2548,5096), (639,1278,2556,5112), (641,1282,2564,5128), (643,1286,2572,5144), (645,1290,2580,5160), (647,1294,2588,5176), (649,1298,2596,5192), (651,1302,2604,5208), (653,1306,2612,5224), (655,1310,2620,5240), (657,1314,2628,5256), (659,1318,2636,5272), (661,1322,2644,5288), (663,1326,2652,5304), (665,1330,2660,5320), (667,1334,2668,5336), (669,1338,2676,5352), (671,1342,2684,5368), (673,1346,2692,5384), (677,1354,2708,5416), (679,1358,2716,5432), (681,1362,2724,5448), (683,1366,2732,5464), (685,1370,2740,5480), (687,1374,2748,5496), (689,1378,2756,5512), (691,1382,2764,5528), (693,1386,2772,5544), (695,1390,2780,5560), (697,1394,2788,5576), (699,1398,2796,5592), (701,1402,2804,5608), (703,1406,2812,5624), (705,1410,2820,5640), (707,1414,2828,5656), (709,1418,2836,5672), (711,1422,2844,5688), (713,1426,2852,5704), (715,1430,2860,5720), (717,1434,2868,5736), (719,1438,2876,5752), (721,1442,2884,5768), (723,1446,2892,5784), (725,1450,2900,5800), (727,1454,2908,5816), (729,1215,2025,3375,5625,9375), (731,1462,2924,5848), (733,1466,2932,5864), (735,1470,2940,5880), (737,1474,2948,5896), (739,1478,2956,5912), (741,1482,2964,5928), (743,1486,2972,5944), (745,1490,2980,5960), (747,1494,2988,5976), (749,1498,2996,5992), (751,1502,3004,6008), (753,1506,3012,6024), (755,1510,3020,6040), (757,1514,3028,6056), (759,1518,3036,6072), (761,1522,3044,6088), (763,1526,3052,6104), (765,1530,3060,6120), (767,1534,3068,6136), (769,1538,3076,6152), (771,1542,3084,6168), (773,1546,3092,6184), (775,1550,3100,6200), (777,1554,3108,6216), (779,1558,3116,6232), (781,1562,3124,6248), (783,1305,2175,3625), (785,1570,3140,6280), (787,1574,3148,6296), (789,1578,3156,6312), (791,1582,3164,6328), (793,1586,3172,6344), (795,1590,3180,6360), (797,1594,3188,6376), (799,1598,3196,6392), (801,1602,3204,6408), (803,1606,3212,6424), (805,1610,3220,6440), (807,1614,3228,6456), (809,1618,3236,6472), (811,1622,3244,6488), (813,1626,3252,6504), (815,1630,3260,6520), (817,1634,3268,6536), (819,1638,3276,6552), (821,1642,3284,6568), (823,1646,3292,6584), (825,1650,3300,6600), (827,1654,3308,6616), (829,1658,3316,6632), (831,1662,3324,6648), (833,1666,3332,6664), (835,1670,3340,6680), (837,1395,2325,3875), (839,1678,3356,6712), (841,1682,3364,6728), (843,1686,3372,6744), (845,1690,3380,6760), (847,1694,3388,6776), (849,1698,3396,6792), 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