[转载]这样的n有多少个？

Treenewbee

王守恩 发表于 2024-1-24 11:28
楼上的遐想欠火候。

x+y=2024,  x^2024首n位数字 = y^2024首n位数字。n=1, 2, 3, 4, 5, 6, ..., 2024。
...

1. Table[{n,Select[s,Floor[10^(n-1)*10^FractionalPart[2024*Log10[#]]]==Floor[10^(n-1)*10^FractionalPart[2024*Log10[2024-#]]]&]},{n,10}]

复制代码

{{1,{17,20,22,30,33,36,42,44,45,55,58,85,88,96,98,100,101,102,104,120,121,124,126,140,141,143,149,152,165,166,169,172,178,184,189,190,209,212,219,221,232,235,242,243,260,271,283,305,317,324,329,330,334,345,363,376,377,383,389,393,405,420,431,433,438,440,449,451,462,475,480,493,496,502,505,515,522,526,534,538,543,548,553,558,563,565,571,578,588,596,597,609,610,614,628,633,634,635,636,637,641,642,644,645,647,651,652,653,654,655,656,658,661,664,681,682,684,687,695,696,707,708,709,728,729,751,757,758,759,772,777,778,784,790,801,802,806,813,816,817,822,827,831,832,846,847,850,851,854,855,873,886,890,898,903,906,907,911,915,919,923,927,931,935,943,946,947,958,966,970,981,982,989,993,997,1012,1027,1031,1035,1042,1043,1054,1058,1066,1077,1078,1081,1089,1093,1097,1101,1105,1109,1113,1117,1118,1121,1126,1134,1138,1151,1169,1170,1173,1174,1177,1178,1192,1193,1197,1202,1207,1208,1211,1218,1222,1223,1234,1240,1246,1247,1252,1265,1266,1267,1273,1295,1296,1315,1316,1317,1328,1329,1337,1340,1342,1343,1360,1363,1366,1368,1369,1370,1371,1372,1373,1377,1379,1380,1382,1383,1387,1388,1389,1390,1391,1396,1410,1414,1415,1427,1428,1436,1446,1453,1459,1461,1466,1471,1476,1481,1486,1490,1498,1502,1509,1519,1522,1528,1531,1544,1549,1562,1573,1575,1584,1586,1591,1593,1604,1619,1631,1635,1641,1647,1648,1661,1679,1690,1694,1695,1700,1707,1719,1741,1753,1764,1781,1782,1789,1792,1803,1805,1812,1815,1834,1835,1840,1846,1852,1855,1858,1859,1872,1875,1881,1883,1884,1898,1900,1903,1904,1920,1922,1923,1924,1926,1928,1936,1939,1966,1969,1979,1980,1982,1988,1991,1994,2002,2004,2007}},{2,{184,502,610,636,652,682,784,801,846,873,923,927,931,970,1012,1054,1093,1097,1101,1151,1178,1223,1240,1342,1372,1388,1414,1522,1840}},{3,{184,636,1012,1388,1840}},{4,{184,1012,1840}},{5,{184,1012,1840}},{6,{184,1012,1840}},{7,{184,1012,1840}},{8,{184,1012,1840}},{9,{184,1012,1840}},{10,{184,1012,1840}}}

Treenewbee

{184,1012,1840}
184入围是因为184+1840=2024
1012入围是因为1012是2024的一半

王守恩

这样显得工仗一些。

1. Select[Range[2024],Abs[1-2FractionalPart[Log10[2]#]]<1-2Log10[3]&]

复制代码

上面的公式只适合唯一数对:  2^n首位数字 = 5^n首位数字。
其他数对只能用下面的公式:  A^n首位数字 = B^n首位数字。

1. [Select[Range[n],IntegerDigits[A^#][[1]]==IntegerDigits[B^#][[1]] &]

复制代码

所有数对只能用下面的公式:  A^n首k位数字 = B^n首k位数字。k>1

1. Select[Range[n],Take[IntegerDigits[A^#],k]==Take[IntegerDigits[B^#],k]&]

复制代码