x^3 + y^3 + z^3 = k^n恰好有n组解的高效算法
A383689a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^n, where 0 < x < y < z has exactly n integer solutions.DATA
36, 188, 54, 144, 90, 63, 66
COMMENTS
a(9)=15.
EXAMPLE
a(5)=90, because 90^5 =95^3 + 321^3 + 1804^3 = 186^3 + 1272^3 + 1566^3=704^3 + 1230^3 + 1546^3 = 720^3 + 810^3 + 1710^3 = 745^3 + 1335^3 + 1460^3 and no integer less than 90 has 5 solutions.
刚发现我提供的a(7)=66是错误的，{{242,7260,17182},{2892,3480,17532},{3672,10284,16284},{4048,6743,17193},{4356,4356,17424},{4565,6028,17259},{8789,12595,14058}} ,存在x=y的情况，所以66^7只有6组解符合要求。已知a(7)>120。 A383879
a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^n, where 0 < x < y < z < w has exactly n integer solutions.
DATA
100, 42, 55, 34, 74
EXAMPLE
a(3)=55, because 55^3 = 7^3 + 24^3 + 38^3 + 46^3 = 7^3 + 12^3 + 34^3 + 50^3 = 17^3 +19^3 +28^3 +51^3 and no integer less than 55 has 3 solutions.
MATHEMATICA
f:=Do, 0<#[]<#[]<#[]<#[]&]; d={n, k, v, Length@v}; If]==n, Return], {k, 100}]; Do], {n, 4}] 计算发现，$x^3 + y^3 + z^3 + w^3 = 2772^3,0<x<y<z<w$ 恰好有600个整数解：
{{1496,1726,1842,1872},{1554,1603,1880,1893},{1455,1677,1881,1899},{1496,1668,1836,1924},{984,1858,1868,1950},{988,1767,1946,1953},{606,1844,1944,1954},{1444,1517,1934,1963},{1268,1621,1952,1963},{158,1897,1900,1967},{1368,1653,1867,1976},{1530,1602,1794,1986},{1407,1729,1764,1988},{54,1794,1970,1990},{940,1742,1938,1992},{642,1788,1950,1992},{289,1859,1907,1993},{468,1748,1992,1996},{1158,1612,1952,2010},{1048,1716,1904,2016},{1488,1501,1856,2019},{1260,1428,2009,2023},{1576,1622,1690,2024},{1112,1780,1814,2026},{260,1811,1912,2029},{527,1677,2007,2029},{489,1698,1983,2040},{1443,1530,1837,2042},{1044,1722,1869,2043},{198,1749,1950,2043},{39,1665,2013,2043},{1408,1650,1760,2046},{1122,1760,1804,2046},{1410,1440,1908,2046},{354,1599,2046,2049},{437,1824,1867,2052},{558,1737,1935,2052},{909,1620,1971,2052},{302,1798,1894,2054},{334,1654,2006,2054},{1302,1596,1848,2058},{1344,1526,1876,2058},{1278,1410,1973,2059},{1018,1590,1956,2060},{711,1497,2061,2067},{1290,1398,1963,2069},{1326,1563,1845,2070},{585,1647,1980,2070},{86,1832,1840,2074},{804,1632,1954,2078},{741,1548,2011,2084},{519,1733,1903,2085},{597,1739,1890,2086},{798,1442,2058,2086},{761,1411,2077,2087},{1194,1398,1980,2088},{1272,1460,1896,2104},{1446,1516,1760,2106},{666,1602,1962,2106},{540,1485,2043,2106},{1197,1547,1869,2107},{322,1400,2093,2107},{990,1746,1780,2108},{1292,1482,1867,2109},{1265,1474,1882,2111},{936,1584,1917,2115},{352,1349,2106,2115},{942,1428,2002,2120},{336,1680,1911,2121},{1026,1422,1980,2124},{766,1362,2060,2124},{702,1314,2088,2124},{988,1605,1875,2126},{452,1672,1906,2126},{1044,1074,2104,2126},{1309,1430,1862,2131},{361,1601,1955,2131},{1308,1432,1860,2132},{627,1560,1962,2133},{1076,1516,1893,2139},{909,1506,1944,2139},{238,1268,2114,2140},{174,1665,1898,2143},{185,1456,2030,2143},{1326,1420,1842,2144},{22,1760,1815,2145},{1256,1422,1872,2146},{1349,1443,1810,2148},{112,1596,1932,2156},{412,1548,1950,2162},{12,1527,1969,2162},{468,1568,1933,2163},{1368,1560,1688,2164},{539,1690,1834,2165},{24,1668,1866,2166},{987,1148,2046,2173},{1500,1514,1609,2175},{828,1566,1872,2178},{723,1557,1896,2178},{1124,1468,1851,2181},{563,1462,1968,2181},{708,1482,1940,2182},{354,1704,1806,2184},{490,1526,1932,2184},{413,1134,2107,2184},{564,700,2180,2184},{700,897,2135,2190},{864,1572,1838,2194},{947,1077,2052,2194},{544,1429,1970,2195},{324,1692,1800,2196},{206,1729,1764,2199},{1364,1452,1716,2200},{528,1716,1760,2200},{495,1563,1885,2201},{597,1371,1986,2202},{579,1368,1989,2202},{390,711,2169,2202},{597,1055,2097,2203},{600,1612,1836,2204},{640,988,2108,2204},{24,492,2188,2204},{1008,1512,1827,2205},{330,819,2154,2205},{754,1316,1988,2206},{1313,1504,1695,2208},{544,1212,2046,2210},{454,785,2144,2215},{212,472,2176,2216},{468,1530,1887,2217},{264,1683,1773,2220},{171,1533,1890,2220},{912,1284,1956,2220},{285,760,2147,2220},{228,570,2166,2220},{1307,1338,1788,2221},{839,1557,1815,2221},{396,946,2112,2222},{450,648,2152,2222},{342,1665,1782,2223},{846,1414,1898,2226},{192,1632,1801,2231},{72,468,2160,2232},{392,1624,1799,2233},{465,1419,1929,2235},{1310,1327,1769,2236},{887,1353,1908,2236},{692,1495,1854,2241},{549,1646,1755,2242},{384,762,2118,2244},{522,876,2092,2246},{1155,1323,1827,2247},{17,1185,2019,2251},{675,1061,2026,2256},{156,458,2134,2256},{626,1612,1751,2257},{117,1530,1836,2259},{750,1140,1989,2259},{336,749,2104,2259},{1170,1566,1628,2260},{827,1386,1869,2260},{688,944,2048,2260},{994,1202,1915,2261},{49,1253,1981,2261},{245,973,2065,2261},{938,1428,1816,2262},{538,776,2088,2262},{216,666,2112,2262},{689,1500,1818,2263},{466,1328,1937,2263},{688,1486,1826,2264},{299,603,2113,2265},{1072,1382,1796,2266},{1110,1172,1884,2266},{506,990,2046,2266},{407,561,2112,2266},{303,1406,1897,2268},{532,1176,1988,2268},{252,966,2058,2268},{829,1475,1796,2272},{164,606,2104,2274},{1152,1368,1755,2277},{891,1287,1881,2277},{363,682,2090,2277},{1122,1422,1728,2280},{656,1196,1948,2284},{801,1143,1944,2286},{1326,1514,1519,2289},{338,726,2070,2290},{152,676,2076,2292},{193,1148,1974,2295},{522,585,2070,2295},{936,1464,1735,2297},{56,444,2086,2298},{596,780,2034,2302},{332,1612,1692,2304},{936,1320,1812,2304},{642,1116,1950,2304},{18,1174,1949,2307},{71,185,2080,2308},{672,1274,1876,2310},{273,1128,1959,2310},{155,1517,1756,2314},{1,221,2071,2315},{602,1414,1800,2316},{696,988,1964,2316},{62,1150,1942,2318},{538,1003,1973,2318},{93,850,2019,2318},{706,1241,1871,2320},{822,1170,1878,2322},{667,804,1997,2322},{1284,1464,1515,2325},{492,726,2019,2325},{540,675,2022,2325},{52,920,1992,2328},{18,525,2040,2331},{702,794,1978,2334},{314,937,1976,2335},{706,1404,1758,2336},{1167,1318,1670,2337},{1132,1290,1703,2337},{993,1369,1709,2337},{981,1227,1791,2337},{111,723,2013,2337},{472,1250,1861,2339},{106,628,2021,2339},{244,1608,1628,2340},{1210,1272,1670,2340},{759,1346,1777,2340},{423,1305,1836,2340},{300,772,2000,2340},{744,1476,1690,2342},{117,1453,1751,2343},{12,1380,1796,2344},{708,1130,1878,2344},{128,970,1958,2344},{658,767,1972,2345},{1078,1344,1676,2346},{272,1528,1685,2347},{629,910,1946,2347},{892,1164,1824,2348},{284,1112,1906,2350},{629,844,1955,2350},{66,312,2024,2350},{465,1086,1906,2351},{644,1386,1750,2352},{392,1176,1876,2352},{28,756,1988,2352},{252,693,1995,2352},{383,437,2011,2353},{206,1396,1768,2354},{979,1157,1790,2356},{1059,1128,1774,2357},{382,1505,1676,2359},{602,1289,1798,2359},{358,1312,1802,2360},{705,1131,1851,2361},{178,1120,1880,2366},{66,532,1992,2366},{546,1512,1641,2367},{321,624,1980,2367},{468,944,1920,2368},{324,1228,1824,2372},{20,988,1912,2372},{100,104,1996,2372},{147,1038,1896,2373},{537,1563,1569,2379},{365,1344,1750,2379},{399,1274,1785,2380},{654,874,1898,2382},{111,256,1976,2385},{728,1288,1728,2388},{160,468,1964,2388},{294,1554,1570,2390},{207,676,1938,2393},{564,1494,1596,2394},{852,1092,1782,2394},{297,960,1879,2396},{387,1474,1622,2397},{754,947,1842,2397},{594,957,1859,2398},{936,1320,1631,2401},{461,1016,1843,2404},{1098,1346,1534,2406},{1001,1024,1743,2406},{796,1404,1596,2408},{472,1134,1794,2408},{678,1239,1722,2409},{792,1111,1760,2409},{362,741,1899,2410},{200,516,1924,2412},{232,1115,1802,2413},{264,988,1842,2414},{435,741,1886,2416},{167,654,1905,2416},{401,1119,1786,2418},{456,1038,1812,2418},{352,620,1902,2418},{1074,1339,1514,2421},{754,947,1800,2421},{43,857,1861,2423},{156,852,1860,2424},{623,1477,1519,2429},{644,1372,1603,2429},{45,171,1908,2430},{1067,1397,1441,2431},{264,638,1881,2431},{530,1072,1767,2433},{333,1216,1707,2438},{1016,1329,1495,2442},{242,1441,1551,2442},{414,1395,1581,2442},{198,1188,1716,2442},{150,1143,1737,2442},{774,892,1772,2442},{292,806,1836,2442},{164,358,1884,2442},{210,1029,1778,2443},{740,1210,1654,2444},{27,498,1872,2445},{855,1442,1449,2446},{487,1347,1599,2447},{540,774,1818,2448},{754,1199,1646,2449},{758,902,1759,2449},{678,1422,1494,2454},{291,717,1830,2454},{281,605,1843,2455},{288,532,1848,2456},{1134,1215,1476,2457},{72,1368,1575,2457},{1174,1248,1418,2460},{920,1094,1627,2461},{211,689,1823,2461},{100,1298,1612,2462},{214,1070,1726,2462},{784,1190,1610,2464},{860,1272,1534,2466},{804,1248,1566,2466},{717,1383,1479,2469},{564,1208,1627,2469},{282,896,1764,2470},{309,698,1803,2470},{461,547,1814,2470},{93,355,1832,2472},{1053,1116,1530,2475},{108,1332,1557,2475},{594,957,1716,2475},{357,819,1764,2478},{546,756,1764,2478},{377,1426,1460,2479},{999,1180,1502,2481},{438,1066,1679,2481},{1152,1188,1404,2484},{507,1422,1437,2484},{498,1040,1678,2484},{120,972,1716,2484},{1056,1258,1404,2486},{187,957,1716,2486},{600,1393,1442,2487},{858,1041,1608,2487},{624,1400,1428,2488},{830,988,1634,2488},{455,1036,1672,2489},{1026,1138,1490,2490},{423,834,1729,2492},{306,1320,1515,2493},{601,1383,1424,2496},{480,1140,1608,2496},{626,796,1710,2496},{463,486,1769,2496},{611,625,1739,2497},{819,1281,1449,2499},{849,1209,1491,2499},{924,1092,1533,2499},{334,1010,1664,2500},{714,987,1630,2501},{17,852,1710,2503},{619,1338,1437,2504},{280,1190,1568,2506},{132,1232,1540,2508},{327,1137,1590,2508},{490,1004,1638,2508},{396,990,1650,2508},{580,804,1688,2508},{160,564,1748,2508},{220,396,1760,2508},{590,1268,1483,2509},{468,1378,1404,2510},{52,1144,1586,2510},{507,1053,1612,2510},{492,1209,1530,2511},{260,996,1644,2512},{62,1026,1630,2514},{829,1058,1538,2515},{897,1271,1368,2518},{344,1034,1612,2518},{220,704,1705,2519},{40,1220,1510,2522},{250,1035,1605,2522},{185,310,1735,2522},{975,1207,1367,2523},{111,897,1653,2523},{611,1252,1444,2525},{324,1215,1497,2526},{504,1176,1505,2527},{428,929,1624,2527},{351,990,1602,2529},{630,783,1638,2529},{594,1254,1430,2530},{429,638,1683,2530},{936,960,1501,2531},{19,1061,1573,2531},{900,1272,1305,2535},{451,1233,1443,2537},{804,1248,1348,2540},{968,979,1453,2540},{550,1232,1419,2541},{468,912,1584,2544},{175,273,1688,2544},{442,1166,1462,2546},{72,766,1632,2546},{613,1016,1518,2547},{168,224,1680,2548},{164,1060,1516,2552},{103,114,1670,2553},{190,1022,1525,2555},{252,924,1560,2556},{56,531,1645,2556},{168,1092,1470,2562},{524,1253,1330,2563},{60,1258,1334,2568},{342,832,1548,2570},{284,920,1516,2572},{328,1040,1459,2573},{251,811,1549,2573},{99,990,1485,2574},{396,594,1584,2574},{139,245,1612,2576},{186,1005,1462,2579},{429,976,1464,2579},{992,1096,1224,2580},{788,832,1452,2580},{376,920,1488,2580},{12,582,1578,2580},{354,507,1581,2580},{164,546,1582,2580},{341,1131,1368,2584},{476,700,1532,2584},{314,1215,1297,2586},{446,573,1548,2587},{54,1206,1299,2589},{99,1056,1404,2589},{603,732,1494,2589},{126,1209,1288,2591},{145,1045,1403,2591},{153,630,1540,2591},{468,828,1476,2592},{143,1061,1387,2593},{415,656,1520,2593},{316,1165,1307,2594},{121,1034,1399,2594},{716,1014,1338,2596},{666,806,1440,2596},{180,1047,1377,2598},{483,485,1512,2602},{252,1176,1260,2604},{546,672,1470,2604},{201,420,1527,2604},{507,796,1437,2606},{412,506,1495,2609},{36,864,1422,2610},{336,840,1421,2611},{717,1011,1276,2612},{308,364,1504,2612},{270,1140,1251,2613},{336,474,1491,2613},{292,582,1473,2615},{636,884,1348,2616},{72,468,1488,2616},{18,208,1502,2616},{18,1098,1271,2617},{616,1078,1232,2618},{550,693,1419,2618},{261,468,1476,2619},{224,468,1477,2619},{45,465,1479,2619},{628,632,1412,2620},{131,627,1443,2623},{122,148,1481,2623},{927,1041,1082,2626},{68,302,1468,2626},{514,741,1380,2627},{114,1020,1278,2628},{32,682,1408,2630},{555,936,1254,2637},{169,249,1433,2637},{572,1012,1188,2640},{426,1014,1212,2640},{468,656,1360,2640},{815,856,1196,2641},{189,1120,1131,2642},{632,716,1300,2644},{840,1008,1050,2646},{702,990,1134,2646},{554,661,1323,2646},{336,420,1386,2646},{320,400,1385,2647},{180,936,1240,2648},{124,468,1380,2648},{36,1088,1108,2652},{291,328,1373,2652},{91,980,1190,2653},{834,912,1070,2656},{620,794,1222,2656},{699,933,1110,2658},{57,303,1356,2658},{176,385,1330,2663},{504,756,1224,2664},{204,633,1287,2664},{143,352,1329,2664},{614,928,1103,2665},{158,814,1223,2665},{52,830,1202,2668},{318,338,1308,2668},{111,711,1235,2671},{91,357,1296,2672},{711,780,1110,2673},{198,396,1287,2673},{822,897,951,2676},{399,910,1097,2676},{11,206,1282,2677},{561,871,1073,2679},{126,777,1170,2679},{62,442,1257,2679},{480,758,1146,2680},{96,660,1208,2680},{344,628,1208,2680},{354,398,1248,2680},{154,975,1032,2681},{294,840,1116,2682},{88,660,1188,2684},{511,513,1172,2688},{90,486,1198,2690},{723,789,981,2691},{549,828,1026,2691},{21,708,1134,2691},{36,432,1196,2692},{324,360,1196,2692},{608,628,1072,2696},{534,663,1074,2697},{264,792,990,2706},{53,195,1116,2710},{102,619,1037,2712},{28,78,1106,2712},{73,221,1097,2713},{354,741,942,2715},{523,804,834,2717},{352,803,880,2717},{468,756,882,2718},{474,741,891,2718},{576,594,936,2718},{312,808,852,2720},{280,728,916,2720},{48,468,1024,2720},{414,735,882,2721},{449,775,810,2724},{369,603,909,2727},{36,369,990,2727},{173,331,993,2727},{480,593,879,2728},{114,206,996,2728},{582,672,768,2730},{198,675,861,2730},{101,547,924,2730},{310,668,845,2731},{380,481,887,2734},{264,522,870,2736},{51,408,909,2736},{116,264,928,2736},{130,638,776,2740},{324,570,787,2741},{91,357,840,2744},{168,341,819,2746},{370,613,662,2747},{114,558,720,2748},{165,411,780,2748},{147,385,749,2751},{135,585,594,2754},{228,396,664,2756},{264,486,522,2760},{272,340,600,2760},{138,191,624,2761},{209,358,559,2762},{72,411,485,2764},{168,392,418,2766},{204,215,370,2769},{65,272,295,2770},{208,212,302,2770}} 目前计算进度：
$x^3 + y^3 + z^3 + w^3 = k^3 , where 0 < x < y < z < w <k<6300$
恰好有 100 个整数解：{854,1184,1185,1311,1450,1454,2189,2225,2357,2423,2579,2693,2801}
恰好有 200 个整数解：{1962,2249,2274,2872,2889,2979,3070,3097,3278,4351,4783,4813,4841}
恰好有 300 个整数解：{2328,2376,3256,3460,3496,4029,4526,5091,5366}
恰好有 400 个整数解：{3614,5385,5756,6141,6152}
恰好有 500 个整数解：{2604,4674,4770,6213}
恰好有 600 个整数解：{2772,4953,5568,5958}
恰好有 700 个整数解：{5889,6060}
恰好有 800 个整数解：{4725}
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