我自己琢磨的，恳请大家帮忙论证·

裴进兵

这12天时间，我依据我之前计算的10000以内的每个数字的a(n),在n大于等于5的前提条件下，10000以内的每个数都可以分解成数值不大于该数、数量a(n)个任意数之间的四则运算，我相信我的猜想是对的

whbns

最近在优化24点计算器的算法，顺便帮你验证了一下，到a(n)=8都是对的。
1 Total: 1 Count: 0 Time: 3ms Total Time: 3ms
2 Total: 9 Count: 0 Time: 5ms Total Time: 8ms
3 Total: 45 Count: 0 Time: 11ms Total Time: 19ms
4 Total: 165 Count: 0 Time: 16ms Total Time: 36ms
5 Total: 495 Count: 0 Time: 30ms Total Time: 66ms
6 Total: 1287 Count: 0 Time: 72ms Total Time: 139ms
7 Total: 3003 Count: 0 Time: 130ms Total Time: 269ms
8 Total: 6435 Count: 0 Time: 171ms Total Time: 440ms
9 Total: 12870 Count: 0 Time: 314ms Total Time: 754ms
10 Total: 24310 Count: 0 Time: 661ms Total Time: 1415ms
11 Total: 43758 Count: 0 Time: 1379ms Total Time: 2794ms
12 Total: 75582 Count: 0 Time: 2680ms Total Time: 5474ms
13 Total: 125970 Count: 0 Time: 5440ms Total Time: 10916ms
14 Total: 203490 Count: 0 Time: 9382ms Total Time: 20298ms
15 Total: 319770 Count: 0 Time: 17265ms Total Time: 37563ms
16 Total: 490314 Count: 0 Time: 27884ms Total Time: 65447ms
17 Total: 735471 Count: 0 Time: 47013ms Total Time: 112460ms
18 Total: 1081575 Count: 0 Time: 74247ms Total Time: 186707ms
Count: 0 Time: 186707ms

a(n)=9的数字组合太多了，有7千多万，前1千万（最大的数不超过21）的都是对的。
21以上的也只有24和27了，在有一个至少22的数的情况下，24是可以肯定可以算出来的。27的话，其他8个数凑出1-5应该也不会有问题，所以估计9个数也不会有例外了。

裴进兵

留个微信号  pipi0538,加我时，请说在数学研发论坛上看到的我

lsr314

a(9973)=29.考虑29个整数的集合{9972, 9967, 9949, 9941, 9931, 9929, 9923, 9907, 9901, 9887, 9883, 9871, 9859, 9857, 9851, 9839, 9833, 9829, 9817, 9811, 9803, 9791, 9787, 9781, 9769, 9767, 9749, 9743, 9739}，除了9972之外都是小于n且和9973最近的素数，楼主可以将素数9973表示成这29个整数的四则运算吗？

hujunhua

whbns 发表于 2016-7-4 14:15
最近在优化24点计算器的算法，顺便帮你验证了一下，到a(n)=8都是对的。
1 Total: 1 Count: 0 Time: 3ms To ...

可否共享一下你的优化算法？

mathe

我觉得对于充分大的n,改为任意给定[1,2n]范围内的a(n)个数，都可以有表达式利用这a(n)个数计算出n也应该能够成立。
直接构造应该比计算会容易。
n个数全部为1的情况需要特殊处理。
对于数据中有较大和较小的数字的情况，可以先通过贪心法将这部分较大的数字构造出接近n的结果，然后余下用较小的数字构造差值通常不会难。而对于只有大数据的情况，通过除法可以构造出一些1，通过减法可以得出0，所以也会比较容易构造。
而如果

hujunhua

39#关于基点的想法是可取的，但是定义不合适。

适合定义为：n为基点（我愿意改名为谷），当且仅当a(n)≤a(n-1)且a(n)≤a(n+1)。
这样定义的基点，其各种最小分解式的顶级结点必是乘号，即不能从附近数的最小分解式通过加减衍生。
从这个意义上说，它的最小分解式是新创的，是本原的。这正是称其为基点的意义。
如此，就不能将定义中的≤换成＜。（如39#那样）

3^7以内的基点序列如下（1274项）：
6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 27, 30, 32, 34, 36, 38, 39, 40, 42, 45, 48, 50, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, \
72, 74, 75, 76, 78, 81, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 117, 120, 122, 123, \
124, 126, 128, 130, 132, 133, 135, 138, 140, 141, 142, 144, 146, 147, 148, 150, 152, 153, 156, 159, 160, 162, 164, 165, 168, 171, 174, 175, \
176, 177, 178, 180, 182, 183, 184, 185, 186, 189, 192, 194, 195, 196, 198, 200, 202, 204, 207, 208, 210, 212, 213, 214, 216, 218, 219, 220, \
222, 224, 225, 228, 230, 231, 232, 234, 236, 237, 238, 240, 243, 246,248, 249, 250, 252, 254, 256, 258, 259, 260, 261, 264, 265, 266, 267, \
268, 270, 272, 273, 275, 276, 279, 280, 282, 284, 285, 286, 288, 290, 291, 292, 294, 296, 297, 300, 302, 304, 306, 308, 309, 310, 312, 315, \
318, 320, 321, 322, 324, 326, 327, 328, 330, 333, 336, 338, 339, 340, 342, 344, 345, 348, 351, 354, 355, 356, 357, 358, 360, 362, 363, 364, \
365, 366, 368, 369, 370, 371, 372, 375, 376, 378, 380, 381, 382, 384, 386, 387, 388, 390, 392, 394, 396, 399, 400, 402, 405, 408, 410, 412, \
414, 416, 418, 420, 423, 424, 426, 428, 429, 430, 432, 434, 435, 436, 438, 440, 441, 444, 446, 448, 450, 452, 453, 454, 456, 459, 462, 464, \
465, 466, 468, 470, 471, 472, 474, 475, 477, 480, 483, 484, 486, 488, 489, 492, 495, 497, 498, 500, 501, 502, 504, 506, 507, 508, 510, 512, \
513, 516, 518, 520, 522, 525, 528, 530, 531, 532, 534, 535, 536, 537, 538, 540, 542, 543, 544, 545, 546, 549, 550, 552, 555, 558, 560, 562, \
564, 567, 570, 572, 573, 574, 576, 578, 579, 580, 582, 584, 585, 588, 590, 592, 594, 596, 597, 598, 600, 602, 603, 604, 605, 606, 608, 610, \
612, 614, 615, 616, 618, 621, 624, 626, 627, 628, 630, 632, 633, 634, 636, 639, 640, 642, 644, 645, 646, 648, 650, 651, 652, 654, 656, 657, \
660, 662, 663, 664, 666, 668, 669, 670, 672, 675, 678, 679, 680, 681, 682, 684, 686, 687, 688, 689, 690, 693, 696, 699, 700, 702, 704, 705, \
708, 710, 711, 712, 714, 715, 716, 717, 718, 720, 722, 723, 724, 726, 729, 732, 735, 736, 738, 740, 741, 742, 744, 747, 749, 750, 752, 753, \
754, 756, 758, 759, 760, 762, 763, 764, 765, 766, 768, 770, 771, 772, 774, 776, 777, 780, 783, 784, 786, 788, 789, 790, 792, 795, 798, 800, \
801, 804, 805, 807, 808, 810, 812, 813, 815, 816, 819, 820, 822, 825, 828, 830, 832, 834, 835, 837, 840, 842, 843, 844, 846, 848, 850, 852, \
855, 856, 858, 860, 861, 862, 864, 866, 867, 868, 870, 872, 873, 876, 878, 880, 882, 884, 885, 886, 888, 891, 893, 894, 896, 898, 900, 902, \
903, 904, 905, 906, 908, 909, 910, 912, 914, 915, 916, 918, 920, 921, 923, 924, 925, 927, 928, 930, 931, 932, 933, 934, 936, 938, 939, 940, \
942, 945, 948, 949, 950, 951, 952, 954, 956, 957, 958, 960, 963, 966, 968, 969, 970, 972, 974, 975, 976, 978, 981, 984, 986, 987, 988, 990, \
992, 993, 994, 996, 999, 1001, 1002, 1004, 1005, 1006, 1008, 1010, 1011, 1012, 1014, 1015, 1016, 1017, 1020, 1022, 1024, 1026, 1028, \
1029, 1032, 1035, 1036, 1038, 1040, 1042, 1044, 1045, 1047, 1048, 1050, 1053, 1056, 1058, 1060, 1062, 1064, 1065, 1066, 1068, 1070, \
1071, 1072, 1074, 1075, 1076, 1077, 1078, 1080, 1082, 1083, 1084, 1085, 1086, 1088, 1089, 1090, 1092, 1095, 1098, 1100, 1102, 1104, \
1107, 1110, 1113, 1116, 1118, 1120, 1122, 1125, 1127, 1128, 1131, 1132, 1134, 1136, 1137, 1140, 1141, 1143, 1144, 1146, 1148, 1149, \
1150, 1152, 1154, 1155, 1156, 1158, 1160, 1161, 1164, 1166, 1168, 1170, 1172, 1173, 1174, 1176, 1178, 1179, 1180, 1182, 1184, 1186, \
1188, 1190, 1191, 1192, 1194, 1195, 1197, 1200, 1202, 1203, 1204, 1206, 1208, 1210, 1212, 1215, 1218, 1220, 1222, 1224, 1226, 1227, \
1228, 1230, 1232, 1233, 1235, 1236, 1239, 1240, 1242, 1244, 1245, 1246, 1248, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1258, \
1260, 1262, 1263, 1264, 1265, 1266, 1267, 1269, 1272, 1274, 1275, 1276, 1278, 1280, 1282, 1284, 1287, 1288, 1290, 1292, 1293, 1294, \
1296, 1298, 1299, 1300, 1302, 1304, 1305, 1308, 1310, 1312, 1314, 1316, 1317, 1318, 1320, 1323, 1325, 1326, 1328, 1329, 1330, 1332, \
1334, 1335, 1336, 1337, 1338, 1340, 1341, 1342, 1344, 1346, 1347, 1348, 1350, 1352, 1353, 1355, 1356, 1358, 1359, 1360, 1362, 1364, \
1365, 1366, 1368, 1370, 1371, 1372, 1374, 1375, 1377, 1380, 1382, 1383, 1384, 1386, 1387, 1389, 1391, 1392, 1394, 1395, 1398, 1400, \
1401, 1402, 1404, 1406, 1407, 1408, 1410, 1413, 1416, 1417, 1420, 1422, 1424, 1425, 1428, 1431, 1434, 1435, 1436, 1437, 1438, 1440, \
1442, 1443, 1444, 1445, 1446, 1449, 1452, 1455, 1456, 1458, 1460, 1461, 1464, 1467, 1470, 1472, 1473, 1474, 1476, 1478, 1479, 1480, \
1482, 1485, 1488, 1491, 1494, 1496, 1498, 1500, 1503, 1504, 1505, 1506, 1508, 1509, 1510, 1512, 1514, 1515, 1516, 1518, 1519, 1520, \
1521, 1524, 1526, 1528, 1530, 1532, 1533, 1534, 1536, 1539, 1542, 1544, 1546, 1548, 1550, 1552, 1554, 1556, 1558, 1560, 1562, 1563, \
1564, 1566, 1568, 1570, 1572, 1573, 1575, 1577, 1578, 1580, 1581, 1582, 1584, 1586, 1587, 1588, 1590, 1593, 1596, 1598, 1600, 1602, \
1605, 1608, 1610, 1611, 1614, 1615, 1616, 1617, 1618, 1620, 1622, 1623, 1624, 1625, 1626, 1629, 1630, 1632, 1635, 1638, 1640, 1642, \
1643, 1644, 1645, 1647, 1650, 1652, 1653, 1654, 1656, 1658, 1659, 1660, 1662, 1664, 1665, 1668, 1670, 1671, 1672, 1674, 1676, 1677, \
1678, 1680, 1682, 1683, 1684, 1685, 1686, 1687, 1688, 1689, 1690, 1692, 1694, 1696, 1698, 1701, 1704, 1706, 1708, 1710, 1712, 1714, \
1716, 1719, 1720, 1722, 1724, 1725, 1726, 1728, 1730, 1731, 1732, 1734, 1736, 1737, 1740, 1742, 1744, 1746, 1748, 1749, 1750, 1752, \
1755, 1757, 1758, 1760, 1762, 1764, 1766, 1767, 1768, 1770, 1771, 1772, 1773, 1774, 1776, 1778, 1779, 1780, 1782, 1784, 1785, 1786, \
1788, 1790, 1792, 1794, 1795, 1796, 1797, 1798, 1800, 1802, 1803, 1804, 1805, 1806, 1808, 1809, 1810, 1812, 1813, 1815, 1816, 1818, \
1819, 1820, 1821, 1822, 1824, 1826, 1827, 1828, 1830, 1832, 1833, 1834, 1836, 1838, 1839, 1840, 1842, 1843, 1845, 1846, 1848, 1850, \
1853, 1854, 1855, 1856, 1859, 1860, 1863, 1866, 1868, 1869, 1870, 1872, 1874, 1875, 1876, 1878, 1880, 1881, 1883, 1884, 1885, 1886, \
1887, 1888, 1890, 1892, 1893, 1895, 1896, 1897, 1898, 1899, 1900, 1902, 1904, 1905, 1906, 1908, 1910, 1911, 1912, 1914, 1915, 1917, \
1920, 1922, 1923, 1924, 1926, 1928, 1929, 1930, 1932, 1935, 1936, 1938, 1940, 1941, 1942, 1944, 1946, 1947, 1948, 1950, 1952, 1953, \
1956, 1958, 1959, 1960, 1962, 1964, 1965, 1966, 1968, 1971, 1974, 1976, 1977, 1978, 1980, 1982, 1983, 1984, 1986, 1988, 1989, 1992, \
1995, 1996, 1998, 2000, 2001, 2002, 2004, 2007, 2008, 2009, 2010, 2012, 2013, 2014, 2016, 2018, 2019, 2020, 2022, 2023, 2025, 2028, \
2030, 2032, 2033, 2034, 2035, 2037, 2040, 2043, 2044, 2046, 2048, 2050, 2052, 2054, 2055, 2056, 2058, 2061, 2064, 2067, 2070, 2071, \
2072, 2074, 2075, 2076, 2079, 2080, 2082, 2084, 2085, 2086, 2088, 2090, 2093, 2095, 2097, 2100, 2103, 2104, 2106, 2108, 2109, 2112, \
2115, 2117, 2119, 2120, 2122, 2124, 2126, 2128, 2130, 2132, 2133, 2136, 2138, 2140, 2142, 2144, 2145, 2148, 2150, 2151, 2152, 2154, \
2155, 2156, 2157, 2158, 2160, 2162, 2163, 2164, 2165, 2166, 2168, 2169, 2170, 2172, 2175, 2176, 2178, 2180, 2181, 2184, 2187

hujunhua

39#所定义的基点，不仅自身是本原的、新创的，不能从附近加减衍生，而且还必须可以衍生附近数的最小分解式。
这个定义意义不大，但是因为严格，所以“珍稀”一些。3^7以内的共有395项（只有#50的三分之一弱）：
12, 18, 24, 27, 30, 32, 36, 42, 45, 48, 54, 60, 72, 78, 81, 84, 90, \
96, 102, 108, 114, 117, 120, 126, 128, 135, 138, 144, 150, 156, 162, \
168, 171, 180, 189, 192, 198, 200, 204, 210, 216, 222, 228, 234, 240, \
243, 246, 252, 256, 270, 282, 288, 294, 300, 304, 306, 312, 315, 318, \
324, 330, 333, 336, 342, 348, 351, 360, 378, 384, 390, 392, 396, 402, \
405, 408, 410, 414, 416, 420, 426, 432, 438, 444, 448, 450, 456, 459, \
462, 468, 477, 480, 486, 492, 495, 504, 516, 518, 520, 522, 525, 528, \
540, 552, 555, 558, 560, 564, 567, 570, 576, 582, 588, 592, 594, 600, \
608, 612, 618, 621, 624, 630, 636, 642, 648, 654, 660, 666, 672, 675, \
684, 693, 696, 702, 708, 720, 726, 729, 732, 738, 744, 747, 756, 768, \
774, 780, 792, 795, 798, 810, 825, 828, 832, 837, 840, 846, 848, 852, \
858, 864, 870, 876, 880, 882, 888, 891, 896, 900, 912, 918, 936, 942, \
945, 954, 960, 963, 966, 972, 978, 981, 984, 990, 996, 999, 1008, \
1020, 1024, 1026, 1032, 1040, 1050, 1053, 1056, 1060, 1062, 1068, \
1080, 1092, 1095, 1098, 1100, 1104, 1107, 1110, 1113, 1116, 1120, \
1125, 1134, 1146, 1152, 1158, 1164, 1168, 1170, 1176, 1184, 1188, \
1197, 1200, 1206, 1210, 1212, 1215, 1218, 1220, 1224, 1230, 1242, \
1248, 1260, 1269, 1272, 1278, 1280, 1284, 1290, 1296, 1302, 1308, \
1312, 1314, 1320, 1323, 1332, 1344, 1350, 1362, 1368, 1377, 1380, \
1398, 1404, 1410, 1413, 1420, 1422, 1428, 1431, 1440, 1449, 1452, \
1458, 1464, 1467, 1470, 1476, 1482, 1485, 1488, 1491, 1494, 1498, \
1500, 1512, 1524, 1526, 1528, 1530, 1536, 1539, 1542, 1544, 1548, \
1552, 1554, 1558, 1560, 1566, 1568, 1575, 1584, 1590, 1593, 1596, \
1600, 1602, 1605, 1608, 1620, 1632, 1635, 1638, 1640, 1647, 1650, \
1656, 1668, 1674, 1680, 1692, 1694, 1696, 1698, 1701, 1704, 1708, \
1710, 1712, 1716, 1722, 1728, 1734, 1740, 1744, 1746, 1752, 1755, \
1760, 1764, 1776, 1782, 1788, 1792, 1800, 1824, 1830, 1836, 1848, \
1850, 1863, 1866, 1872, 1878, 1890, 1902, 1908, 1917, 1920, 1926, \
1932, 1938, 1944, 1950, 1956, 1962, 1968, 1971, 1974, 1980, 1986, \
1992, 1998, 2004, 2016, 2025, 2028, 2030, 2037, 2040, 2048, 2052, \
2058, 2061, 2064, 2067, 2088, 2090, 2093, 2097, 2100, 2106, 2112, \
2115, 2124, 2128, 2130, 2136, 2140, 2142, 2148, 2160, 2172, 2178, \
2184, 2187

裴进兵

非常感谢hujunhua老师的慷慨阐述，我真的是醍醐灌顶。这些数，我仔细看了，还非常有兴致的研究了一番，对我，可以说是，倍加相助，在此，深表诚挚的谢意！

裴进兵

lsr314 发表于 2017-3-29 10:18
a(9973)=29.考虑29个整数的集合{9972, 9967, 9949, 9941, 9931, 9929, 9923, 9907, 9901, 9887, 9883, 9871 ...

9973=9972+（9931-9929）/（9859-9857）+【（9949-9941）-（9811-9803）】*（9967+9923+9907+9901+9887+9883+9871+9851+9839+9833+9829+9817+9791+9787+9781+9769+9767+9749+9743+9739）