Question

给定一个从1到N的数字序列，我们必须找到四元组的数量，以使它们的最大公约数（GCD）为1。

约束：N <= 10 ^ 5

示例：N = 5。因此，{1,2,3,4,5}中此类四元组的数量为：

{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}

所以在这种情况下的答案是5。

Answer 1

我们将以相反的顺序解决此问题。如果我们有n个数字，则从n中选择4的方式是nC4。现在，如果我们知道从n中选择4的方式，使得所选择的4个数字的gcd大于1，那么我们很容易就能获得预期的结果。

因此，如果象限数目为m，使得每个象限的gcd> 1，则我们的预期结果将为nc4-m。

那么我们怎么知道象限的数量，使得每个象限的gcd大于1。

因素。

现在将i分解：

让， i的素数是A ^ 3 * B ^ 2 * C ^ 3。所以独特的素因是A，B，C。

现在，我们将逐一考虑A，B，C。

由A划分的总数为X =（i-1）/ A，我们可以通过XC3方式选择3。
用B表示的总数是Y =（i-1）/ A，我们可以用YC3的方式选择3。
由C划分的总数为Z =（i-1）/ A，我们可以通过ZC3方式选择3。

但是问题是一些由A划分的数字可以由B划分。
例如，AB由A和B划分。
因此，通过这种方法，我们将获得重复的象限。

为解决此问题，我们将生成所有不同质数的可能子集，并将使用排除和包含方法。

包含排除公式：

全部象限数量=用A划分的象限数量+用B划分的象限数量 C划分的象限数-AB划分的象限数-BC划分的象限数-CA划分的象限数+ ABC划分的象限数。

常规公式：

通过这种方法，我们将获得每个i的象限总数，以使每个象限的gcd大于1。

所以现在我们有象限数（m），其中每个象限的gcd都大于1。

因此，我们的最终结果将为nC4-m。

复杂度：

找到每个从4到n的素数的素数的复杂度是O（nlogn）

对于每个我，我们最多可获得7个不同的素数。因此，我们将有2 ^ 7个子集。因此总复杂度为（nlogn + n * 2 ^ 7 * 7）。

我已在评论中添加了代码链接。

Answer 2

现在您已经展示了一些自己的代码，我将提供一种算法和Python 3代码。

我们将强调GCD 大于1 的四联体的数量。例如，如果GCD为2，则所有四个数字都可以被2整除。多少个四元组的所有数字都可以被2整除？域N // 2至1中有N个可被2整除的数字，因此有(N // 2) choose 4个四元组。类似地，存在(N // 3) choose 4个四元组，其中所有数字都可以被3整除。我们不需要查看被4整除的那些，因为它们已经被2整除并且我们已经计算了它们。可以将所有四个数相除的最大质数最多为N // 4，因为我们需要组合中的四个不同的数被该最大数整除。

因此，似乎我们看了从2到N // 4的所有素数并计算了组合。但是，6可整除的组合又如何呢？我们对它们进行了双重计算，因为它们可以同时被2和3整除。但是我们可以通过对它们进行计数来取消计数，而不是像我们对2和3那样进行计数。

这是the inclusion-exclusion principle的简要介绍。总而言之，我们采用乘积不大于N // 4的不同素数的所有组合。这样一来，N // product中的数字可能会被该product整除的四元组。我们计算(N // product) choose 4，这是四元组的计数，其中所有这些值可被所有质数整除。如果存在偶数个素数，我们将增加该计数；如果存在奇数个素数，我们将减去该计数。通过查看不包含质数的“空组合”，我们可以获得四个数字的所有组合的全部计数。我们将其相加，然后减去一个质数的组合，添加两个质数的组合，减去三个质数的组合，等等。

这是我的Python 3代码。为了使事情简单，并且由于我经常使用质数编程，因此我存储了一个中等大小的第一个质数列表。在您的问题中，您需要素数最大为25000的素数。如果愿意，可以通过在例程开始时一次计算该列表来做不同的事情，但这是我的列表。

primes_2_25000 = [
      2,     3,     5,     7,    11,    13,    17,    19,    23,    29,
     31,    37,    41,    43,    47,    53,    59,    61,    67,    71,
     73,    79,    83,    89,    97,   101,   103,   107,   109,   113,
    127,   131,   137,   139,   149,   151,   157,   163,   167,   173,
    179,   181,   191,   193,   197,   199,   211,   223,   227,   229,
    233,   239,   241,   251,   257,   263,   269,   271,   277,   281,
    283,   293,   307,   311,   313,   317,   331,   337,   347,   349,
    353,   359,   367,   373,   379,   383,   389,   397,   401,   409,
    419,   421,   431,   433,   439,   443,   449,   457,   461,   463,
    467,   479,   487,   491,   499,   503,   509,   521,   523,   541,
    547,   557,   563,   569,   571,   577,   587,   593,   599,   601,
    607,   613,   617,   619,   631,   641,   643,   647,   653,   659,
    661,   673,   677,   683,   691,   701,   709,   719,   727,   733,
    739,   743,   751,   757,   761,   769,   773,   787,   797,   809,
    811,   821,   823,   827,   829,   839,   853,   857,   859,   863,
    877,   881,   883,   887,   907,   911,   919,   929,   937,   941,
    947,   953,   967,   971,   977,   983,   991,   997,  1009,  1013,
   1019,  1021,  1031,  1033,  1039,  1049,  1051,  1061,  1063,  1069,
   1087,  1091,  1093,  1097,  1103,  1109,  1117,  1123,  1129,  1151,
   1153,  1163,  1171,  1181,  1187,  1193,  1201,  1213,  1217,  1223,
   1229,  1231,  1237,  1249,  1259,  1277,  1279,  1283,  1289,  1291,
   1297,  1301,  1303,  1307,  1319,  1321,  1327,  1361,  1367,  1373,
   1381,  1399,  1409,  1423,  1427,  1429,  1433,  1439,  1447,  1451,
   1453,  1459,  1471,  1481,  1483,  1487,  1489,  1493,  1499,  1511,
   1523,  1531,  1543,  1549,  1553,  1559,  1567,  1571,  1579,  1583,
   1597,  1601,  1607,  1609,  1613,  1619,  1621,  1627,  1637,  1657,
   1663,  1667,  1669,  1693,  1697,  1699,  1709,  1721,  1723,  1733,
   1741,  1747,  1753,  1759,  1777,  1783,  1787,  1789,  1801,  1811,
   1823,  1831,  1847,  1861,  1867,  1871,  1873,  1877,  1879,  1889,
   1901,  1907,  1913,  1931,  1933,  1949,  1951,  1973,  1979,  1987,
   1993,  1997,  1999,  2003,  2011,  2017,  2027,  2029,  2039,  2053,
   2063,  2069,  2081,  2083,  2087,  2089,  2099,  2111,  2113,  2129,
   2131,  2137,  2141,  2143,  2153,  2161,  2179,  2203,  2207,  2213,
   2221,  2237,  2239,  2243,  2251,  2267,  2269,  2273,  2281,  2287,
   2293,  2297,  2309,  2311,  2333,  2339,  2341,  2347,  2351,  2357,
   2371,  2377,  2381,  2383,  2389,  2393,  2399,  2411,  2417,  2423,
   2437,  2441,  2447,  2459,  2467,  2473,  2477,  2503,  2521,  2531,
   2539,  2543,  2549,  2551,  2557,  2579,  2591,  2593,  2609,  2617,
   2621,  2633,  2647,  2657,  2659,  2663,  2671,  2677,  2683,  2687,
   2689,  2693,  2699,  2707,  2711,  2713,  2719,  2729,  2731,  2741,
   2749,  2753,  2767,  2777,  2789,  2791,  2797,  2801,  2803,  2819,
   2833,  2837,  2843,  2851,  2857,  2861,  2879,  2887,  2897,  2903,
   2909,  2917,  2927,  2939,  2953,  2957,  2963,  2969,  2971,  2999,
   3001,  3011,  3019,  3023,  3037,  3041,  3049,  3061,  3067,  3079,
   3083,  3089,  3109,  3119,  3121,  3137,  3163,  3167,  3169,  3181,
   3187,  3191,  3203,  3209,  3217,  3221,  3229,  3251,  3253,  3257,
   3259,  3271,  3299,  3301,  3307,  3313,  3319,  3323,  3329,  3331,
   3343,  3347,  3359,  3361,  3371,  3373,  3389,  3391,  3407,  3413,
   3433,  3449,  3457,  3461,  3463,  3467,  3469,  3491,  3499,  3511,
   3517,  3527,  3529,  3533,  3539,  3541,  3547,  3557,  3559,  3571,
   3581,  3583,  3593,  3607,  3613,  3617,  3623,  3631,  3637,  3643,
   3659,  3671,  3673,  3677,  3691,  3697,  3701,  3709,  3719,  3727,
   3733,  3739,  3761,  3767,  3769,  3779,  3793,  3797,  3803,  3821,
   3823,  3833,  3847,  3851,  3853,  3863,  3877,  3881,  3889,  3907,
   3911,  3917,  3919,  3923,  3929,  3931,  3943,  3947,  3967,  3989,
   4001,  4003,  4007,  4013,  4019,  4021,  4027,  4049,  4051,  4057,
   4073,  4079,  4091,  4093,  4099,  4111,  4127,  4129,  4133,  4139,
   4153,  4157,  4159,  4177,  4201,  4211,  4217,  4219,  4229,  4231,
   4241,  4243,  4253,  4259,  4261,  4271,  4273,  4283,  4289,  4297,
   4327,  4337,  4339,  4349,  4357,  4363,  4373,  4391,  4397,  4409,
   4421,  4423,  4441,  4447,  4451,  4457,  4463,  4481,  4483,  4493,
   4507,  4513,  4517,  4519,  4523,  4547,  4549,  4561,  4567,  4583,
   4591,  4597,  4603,  4621,  4637,  4639,  4643,  4649,  4651,  4657,
   4663,  4673,  4679,  4691,  4703,  4721,  4723,  4729,  4733,  4751,
   4759,  4783,  4787,  4789,  4793,  4799,  4801,  4813,  4817,  4831,
   4861,  4871,  4877,  4889,  4903,  4909,  4919,  4931,  4933,  4937,
   4943,  4951,  4957,  4967,  4969,  4973,  4987,  4993,  4999,  5003,
   5009,  5011,  5021,  5023,  5039,  5051,  5059,  5077,  5081,  5087,
   5099,  5101,  5107,  5113,  5119,  5147,  5153,  5167,  5171,  5179,
   5189,  5197,  5209,  5227,  5231,  5233,  5237,  5261,  5273,  5279,
   5281,  5297,  5303,  5309,  5323,  5333,  5347,  5351,  5381,  5387,
   5393,  5399,  5407,  5413,  5417,  5419,  5431,  5437,  5441,  5443,
   5449,  5471,  5477,  5479,  5483,  5501,  5503,  5507,  5519,  5521,
   5527,  5531,  5557,  5563,  5569,  5573,  5581,  5591,  5623,  5639,
   5641,  5647,  5651,  5653,  5657,  5659,  5669,  5683,  5689,  5693,
   5701,  5711,  5717,  5737,  5741,  5743,  5749,  5779,  5783,  5791,
   5801,  5807,  5813,  5821,  5827,  5839,  5843,  5849,  5851,  5857,
   5861,  5867,  5869,  5879,  5881,  5897,  5903,  5923,  5927,  5939,
   5953,  5981,  5987,  6007,  6011,  6029,  6037,  6043,  6047,  6053,
   6067,  6073,  6079,  6089,  6091,  6101,  6113,  6121,  6131,  6133,
   6143,  6151,  6163,  6173,  6197,  6199,  6203,  6211,  6217,  6221,
   6229,  6247,  6257,  6263,  6269,  6271,  6277,  6287,  6299,  6301,
   6311,  6317,  6323,  6329,  6337,  6343,  6353,  6359,  6361,  6367,
   6373,  6379,  6389,  6397,  6421,  6427,  6449,  6451,  6469,  6473,
   6481,  6491,  6521,  6529,  6547,  6551,  6553,  6563,  6569,  6571,
   6577,  6581,  6599,  6607,  6619,  6637,  6653,  6659,  6661,  6673,
   6679,  6689,  6691,  6701,  6703,  6709,  6719,  6733,  6737,  6761,
   6763,  6779,  6781,  6791,  6793,  6803,  6823,  6827,  6829,  6833,
   6841,  6857,  6863,  6869,  6871,  6883,  6899,  6907,  6911,  6917,
   6947,  6949,  6959,  6961,  6967,  6971,  6977,  6983,  6991,  6997,
   7001,  7013,  7019,  7027,  7039,  7043,  7057,  7069,  7079,  7103,
   7109,  7121,  7127,  7129,  7151,  7159,  7177,  7187,  7193,  7207,
   7211,  7213,  7219,  7229,  7237,  7243,  7247,  7253,  7283,  7297,
   7307,  7309,  7321,  7331,  7333,  7349,  7351,  7369,  7393,  7411,
   7417,  7433,  7451,  7457,  7459,  7477,  7481,  7487,  7489,  7499,
   7507,  7517,  7523,  7529,  7537,  7541,  7547,  7549,  7559,  7561,
   7573,  7577,  7583,  7589,  7591,  7603,  7607,  7621,  7639,  7643,
   7649,  7669,  7673,  7681,  7687,  7691,  7699,  7703,  7717,  7723,
   7727,  7741,  7753,  7757,  7759,  7789,  7793,  7817,  7823,  7829,
   7841,  7853,  7867,  7873,  7877,  7879,  7883,  7901,  7907,  7919,
   7927,  7933,  7937,  7949,  7951,  7963,  7993,  8009,  8011,  8017,
   8039,  8053,  8059,  8069,  8081,  8087,  8089,  8093,  8101,  8111,
   8117,  8123,  8147,  8161,  8167,  8171,  8179,  8191,  8209,  8219,
   8221,  8231,  8233,  8237,  8243,  8263,  8269,  8273,  8287,  8291,
   8293,  8297,  8311,  8317,  8329,  8353,  8363,  8369,  8377,  8387,
   8389,  8419,  8423,  8429,  8431,  8443,  8447,  8461,  8467,  8501,
   8513,  8521,  8527,  8537,  8539,  8543,  8563,  8573,  8581,  8597,
   8599,  8609,  8623,  8627,  8629,  8641,  8647,  8663,  8669,  8677,
   8681,  8689,  8693,  8699,  8707,  8713,  8719,  8731,  8737,  8741,
   8747,  8753,  8761,  8779,  8783,  8803,  8807,  8819,  8821,  8831,
   8837,  8839,  8849,  8861,  8863,  8867,  8887,  8893,  8923,  8929,
   8933,  8941,  8951,  8963,  8969,  8971,  8999,  9001,  9007,  9011,
   9013,  9029,  9041,  9043,  9049,  9059,  9067,  9091,  9103,  9109,
   9127,  9133,  9137,  9151,  9157,  9161,  9173,  9181,  9187,  9199,
   9203,  9209,  9221,  9227,  9239,  9241,  9257,  9277,  9281,  9283,
   9293,  9311,  9319,  9323,  9337,  9341,  9343,  9349,  9371,  9377,
   9391,  9397,  9403,  9413,  9419,  9421,  9431,  9433,  9437,  9439,
   9461,  9463,  9467,  9473,  9479,  9491,  9497,  9511,  9521,  9533,
   9539,  9547,  9551,  9587,  9601,  9613,  9619,  9623,  9629,  9631,
   9643,  9649,  9661,  9677,  9679,  9689,  9697,  9719,  9721,  9733,
   9739,  9743,  9749,  9767,  9769,  9781,  9787,  9791,  9803,  9811,
   9817,  9829,  9833,  9839,  9851,  9857,  9859,  9871,  9883,  9887,
   9901,  9907,  9923,  9929,  9931,  9941,  9949,  9967,  9973, 10007,
  10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099,
  10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177,
  10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271,
  10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343,
  10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459,
  10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567,
  10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657,
  10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739,
  10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859,
  10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949,
  10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059,
  11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149,
  11159, 11161, 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251,
  11257, 11261, 11273, 11279, 11287, 11299, 11311, 11317, 11321, 11329,
  11351, 11353, 11369, 11383, 11393, 11399, 11411, 11423, 11437, 11443,
  11447, 11467, 11471, 11483, 11489, 11491, 11497, 11503, 11519, 11527,
  11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657,
  11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777,
  11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833,
  11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933,
  11939, 11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011,
  12037, 12041, 12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109,
  12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197, 12203, 12211,
  12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289,
  12301, 12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401,
  12409, 12413, 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487,
  12491, 12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553,
  12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641,
  12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739,
  12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829,
  12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923,
  12941, 12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007,
  13009, 13033, 13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109,
  13121, 13127, 13147, 13151, 13159, 13163, 13171, 13177, 13183, 13187,
  13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309,
  13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411,
  13417, 13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499,
  13513, 13523, 13537, 13553, 13567, 13577, 13591, 13597, 13613, 13619,
  13627, 13633, 13649, 13669, 13679, 13681, 13687, 13691, 13693, 13697,
  13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763, 13781,
  13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879,
  13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967,
  13997, 13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081,
  14083, 14087, 14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197,
  14207, 14221, 14243, 14249, 14251, 14281, 14293, 14303, 14321, 14323,
  14327, 14341, 14347, 14369, 14387, 14389, 14401, 14407, 14411, 14419,
  14423, 14431, 14437, 14447, 14449, 14461, 14479, 14489, 14503, 14519,
  14533, 14537, 14543, 14549, 14551, 14557, 14561, 14563, 14591, 14593,
  14621, 14627, 14629, 14633, 14639, 14653, 14657, 14669, 14683, 14699,
  14713, 14717, 14723, 14731, 14737, 14741, 14747, 14753, 14759, 14767,
  14771, 14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843, 14851,
  14867, 14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947,
  14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073,
  15077, 15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149,
  15161, 15173, 15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259,
  15263, 15269, 15271, 15277, 15287, 15289, 15299, 15307, 15313, 15319,
  15329, 15331, 15349, 15359, 15361, 15373, 15377, 15383, 15391, 15401,
  15413, 15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497,
  15511, 15527, 15541, 15551, 15559, 15569, 15581, 15583, 15601, 15607,
  15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679,
  15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773,
  15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881,
  15887, 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971,
  15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069,
  16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183,
  16187, 16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267,
  16273, 16301, 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381,
  16411, 16417, 16421, 16427, 16433, 16447, 16451, 16453, 16477, 16481,
  16487, 16493, 16519, 16529, 16547, 16553, 16561, 16567, 16573, 16603,
  16607, 16619, 16631, 16633, 16649, 16651, 16657, 16661, 16673, 16691,
  16693, 16699, 16703, 16729, 16741, 16747, 16759, 16763, 16787, 16811,
  16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903,
  16921, 16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987, 16993,
  17011, 17021, 17027, 17029, 17033, 17041, 17047, 17053, 17077, 17093,
  17099, 17107, 17117, 17123, 17137, 17159, 17167, 17183, 17189, 17191,
  17203, 17207, 17209, 17231, 17239, 17257, 17291, 17293, 17299, 17317,
  17321, 17327, 17333, 17341, 17351, 17359, 17377, 17383, 17387, 17389,
  17393, 17401, 17417, 17419, 17431, 17443, 17449, 17467, 17471, 17477,
  17483, 17489, 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573,
  17579, 17581, 17597, 17599, 17609, 17623, 17627, 17657, 17659, 17669,
  17681, 17683, 17707, 17713, 17729, 17737, 17747, 17749, 17761, 17783,
  17789, 17791, 17807, 17827, 17837, 17839, 17851, 17863, 17881, 17891,
  17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957, 17959, 17971,
  17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049, 18059,
  18061, 18077, 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143,
  18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233,
  18251, 18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313,
  18329, 18341, 18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427,
  18433, 18439, 18443, 18451, 18457, 18461, 18481, 18493, 18503, 18517,
  18521, 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637,
  18661, 18671, 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749,
  18757, 18773, 18787, 18793, 18797, 18803, 18839, 18859, 18869, 18899,
  18911, 18913, 18917, 18919, 18947, 18959, 18973, 18979, 19001, 19009,
  19013, 19031, 19037, 19051, 19069, 19073, 19079, 19081, 19087, 19121,
  19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211, 19213, 19219,
  19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319,
  19333, 19373, 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423,
  19427, 19429, 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477,
  19483, 19489, 19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571,
  19577, 19583, 19597, 19603, 19609, 19661, 19681, 19687, 19697, 19699,
  19709, 19717, 19727, 19739, 19751, 19753, 19759, 19763, 19777, 19793,
  19801, 19813, 19819, 19841, 19843, 19853, 19861, 19867, 19889, 19891,
  19913, 19919, 19927, 19937, 19949, 19961, 19963, 19973, 19979, 19991,
  19993, 19997, 20011, 20021, 20023, 20029, 20047, 20051, 20063, 20071,
  20089, 20101, 20107, 20113, 20117, 20123, 20129, 20143, 20147, 20149,
  20161, 20173, 20177, 20183, 20201, 20219, 20231, 20233, 20249, 20261,
  20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357,
  20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443,
  20477, 20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551,
  20563, 20593, 20599, 20611, 20627, 20639, 20641, 20663, 20681, 20693,
  20707, 20717, 20719, 20731, 20743, 20747, 20749, 20753, 20759, 20771,
  20773, 20789, 20807, 20809, 20849, 20857, 20873, 20879, 20887, 20897,
  20899, 20903, 20921, 20929, 20939, 20947, 20959, 20963, 20981, 20983,
  21001, 21011, 21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067,
  21089, 21101, 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169,
  21179, 21187, 21191, 21193, 21211, 21221, 21227, 21247, 21269, 21277,
  21283, 21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379, 21383,
  21391, 21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, 21491,
  21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, 21559, 21563,
  21569, 21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647,
  21649, 21661, 21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751,
  21757, 21767, 21773, 21787, 21799, 21803, 21817, 21821, 21839, 21841,
  21851, 21859, 21863, 21871, 21881, 21893, 21911, 21929, 21937, 21943,
  21961, 21977, 21991, 21997, 22003, 22013, 22027, 22031, 22037, 22039,
  22051, 22063, 22067, 22073, 22079, 22091, 22093, 22109, 22111, 22123,
  22129, 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229,
  22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307,
  22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433, 22441,
  22447, 22453, 22469, 22481, 22483, 22501, 22511, 22531, 22541, 22543,
  22549, 22567, 22571, 22573, 22613, 22619, 22621, 22637, 22639, 22643,
  22651, 22669, 22679, 22691, 22697, 22699, 22709, 22717, 22721, 22727,
  22739, 22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817,
  22853, 22859, 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943,
  22961, 22963, 22973, 22993, 23003, 23011, 23017, 23021, 23027, 23029,
  23039, 23041, 23053, 23057, 23059, 23063, 23071, 23081, 23087, 23099,
  23117, 23131, 23143, 23159, 23167, 23173, 23189, 23197, 23201, 23203,
  23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297, 23311, 23321,
  23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447,
  23459, 23473, 23497, 23509, 23531, 23537, 23539, 23549, 23557, 23561,
  23563, 23567, 23581, 23593, 23599, 23603, 23609, 23623, 23627, 23629,
  23633, 23663, 23669, 23671, 23677, 23687, 23689, 23719, 23741, 23743,
  23747, 23753, 23761, 23767, 23773, 23789, 23801, 23813, 23819, 23827,
  23831, 23833, 23857, 23869, 23873, 23879, 23887, 23893, 23899, 23909,
  23911, 23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007,
  24019, 24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091,
  24097, 24103, 24107, 24109, 24113, 24121, 24133, 24137, 24151, 24169,
  24179, 24181, 24197, 24203, 24223, 24229, 24239, 24247, 24251, 24281,
  24317, 24329, 24337, 24359, 24371, 24373, 24379, 24391, 24407, 24413,
  24419, 24421, 24439, 24443, 24469, 24473, 24481, 24499, 24509, 24517,
  24527, 24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631, 24659,
  24671, 24677, 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767,
  24781, 24793, 24799, 24809, 24821, 24841, 24847, 24851, 24859, 24877,
  24889, 24907, 24917, 24919, 24923, 24943, 24953, 24967, 24971, 24977,
  24979, 24989, 25013
]

这是我的递归例程，该例程查找积小于给定数字的不同素数的所有组合。这段代码是pythonic的-如果您需要帮助将其转换为另一种语言，请告诉我。我前段时间针对另一个问题对此进行了编程，因此您不需要为每个组合返回四个项中的两个-您只需要为素数和标记mu乘积即可为该组合加还是减

def prime_combinations(n):
    """Generate combinations of distinct primes where their product is
    less than n. Each yielded item is a 4-tuple containing:
    - the combination in increasing order,
    - k where the last and largest prime in the combination is the k'th
      prime (where 2 is the 1st prime),
    - the product of the primes in the combination,
    - the Moebius mu function of the product (i.e. 1 if the combination
      has an even number of primes, -1 if an odd number of primes).
    Items are yielded in lexicographical order. The first item yielded
    is the empty combination ((), 0, 1, 1).
    """

    def primecombos(prefix, ndx, prod, mu):
        yield prefix, ndx, prod, mu
        while True:
            newprime = primes_2_25000[ndx]
            newprod = prod * newprime
            if newprod >= n:
                return
            ndx += 1
            yield from primecombos(prefix + (newprime,), ndx, newprod, -mu)

    if 1 < n <= primes_2_25000[-1] and n == int(n):
        yield from primecombos((), 0, 1, 1)

鉴于这些事情，解决您的问题的代码很短：

def relatively_prime_quadruplets(n):
    result = 0
    for _, _, prime_prod, mu in prime_combinations(n // 4 + 1):
        cnt = n // prime_prod
        result += mu * cnt * (cnt - 1) * (cnt - 2) * (cnt - 3) // 24
    return result

将所有这些放到一个模块中，您就可以找到解决方案。通过与蛮力解决方案进行比较，我测试了这段代码中从n到4的所有值100。我在100停了下来，因为强行程序变慢了（4.07秒）。我的代码速度更快：100的运行时间为23.5微秒。对于最大值100000，它以23.1毫秒为单位运行。这似乎是线性执行的复杂性，这正是我所期望的。

以下是您需要的测试代码：

import functools
import math 
import itertools

def gcd_iter(iterable):
    """Return the greatest common divisor of the numbers in an
    iterable."""
    return functools.reduce(math.gcd, iterable, 0)

def brute(n):
    return sum(1 for c in itertools.combinations(range(1, n+1), 4) if gcd_iter(c) == 1)

if __name__ == "__main__":
    for n in range(4, 101):
        print(n, relatively_prime_quadruplets(n) == brute(n))

查找GCD等于1的四联体数

2 个答案: