求将集合 {2^s+2^t｜0≤s＜t,s,t为整数} 中的数从小到大排列后得到的第 103 个数

luyuanhong

这是台湾网友 YAG 发表在“陆老师的《数学中国》园地”的一个帖子，

欢迎大家一起来想想如何解答：

luyuanhong

a514688934

http://www.mathchina.com/bbs/for ... &extra=page%3D1




占楼问个问题(O_O)？

markfang2050

升序输出
[3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136, 144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2049, 2050, 2052, 2056, 2064, 2080, 2112, 2176, 2304, 2560, 3072, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4224, 4352, 4608, 5120, 6144, 8193, 8194, 8196, 8200, 8208, 8224, 8256, 8320, 8448, 8704, 9216, 10240, 12288, 16385, 16386, 16388, 16392, 16400, 16416, 16448, 16512, 16640, 16896, 17408, 18432, 24576]

降序输出
[24576, 18432, 17408, 16896, 16640, 16512, 16448, 16416, 16400, 16392, 16388, 16386, 16385, 12288, 10240, 9216, 8704, 8448, 8320, 8256, 8224, 8208, 8200, 8196, 8194, 8193, 6144, 5120, 4608, 4352, 4224, 4160, 4128, 4112, 4104, 4100, 4098, 4097, 3072, 2560, 2304, 2176, 2112, 2080, 2064, 2056, 2052, 2050, 2049, 1536, 1280, 1152, 1088, 1056, 1040, 1032, 1028, 1026, 1025, 768, 640, 576, 544, 528, 520, 516, 514, 513, 384, 320, 288, 272, 264, 260, 258, 257, 192, 160, 144, 136, 132, 130, 129, 96, 80, 72, 68, 66, 65, 48, 40, 36, 34, 33, 24, 20, 18, 17, 12, 10, 9, 6, 5, 3]

a103=18432

说明：数据由python程序自动计算统计排序所得。可索引数组中任意位置的值。

markfang2050

理论解释：当t=n时（t-s=1），从小到大排列的数共有n(n+1)/2个，当n=13时，共有91个，所以接下来从2^0+2^(n+1=14)开始排列起，即2^(0+103-91-1)+2^14=2^11+2^14=18432。:lol

王守恩

求将集合 {2^s+2^t｜0≤s＜t,s,t为整数} 中的数从小到大排列后得到的第 103 个数

陆老师！通项公式 a(n) 可以是这样。还能简化吗？

谢谢陆老师！谢谢 markfang2050！

  3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136,
144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384, 513, 514, 516, 520, 528, 544, 576, 640, 768,
1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2049, 2050, 2052, 2056, 2064, 2080,
2112, 2176, 2304, 2560, 3072, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4224, 4352, 4608, 5120,
6144, 8193, 8194, 8196, 8200, 8208, 8224, 8256, 8320, 8448, 8704, 9216, 10240, 12288, 16385,
16386, 16388, 16392, 16400, 16416, 16448, 16512, 16640, 16896, 17408, 18432, 20480, .................