Pagina's

2018/04/16

primes less than 65536


//                    --μs--μs--μs--
//        n    pi(n)    p0  p1  p2  
//      255      54      1   1  1.05    p0 trial division
//      511      97      2   1  1.07    p1 Eratosthenes
//     1023     172      5   1  1.07    p2 500 lines
//     2047     309     12   2  1.07
//     4095     564     31   3  1.11 
//     8191    1028     80   6  1.17
//    16383    1900    191  12  1.27
//    32767    3512    456  26  1.50
//    65000    6493             2.02
//    65535    6542   1095  69  1.01

using System;  // i7-4790@3.6
class Program
{
    static void Main()
    {
        int i, j; ushort u; ushort[] p;
        var sw = new System.Diagnostics.Stopwatch();
        Console.WriteLine("  μs pi(n)    n");
        for (u = 0, i = 16; i >= 0; i--, u <<= 1, u |= 1)
        {
            p = primes0(u); sw.Restart();
            for (j = 1000; j > 0; j--) primes0(u);
            Console.Write("{0,4}", sw.ElapsedMilliseconds);
            Console.WriteLine(" {0,4} {1,5}", p.Length, u);
        }
        Console.WriteLine();
        for (u = 0, i = 16; i >= 0; i--, u <<= 1, u |= 1)
        {
            p = primes1(u); sw.Restart();
            for (j = 1000; j > 0; j--) primes1(u);
            Console.Write(sw.Elapsed);
            Console.WriteLine(" {0,4} {1,5}", p.Length, u);
        }
        Console.WriteLine();
        for (u = 0, i = 16; i >= 0; i--, u <<= 1, u |= 1)
        {
            p = primes2(u); sw.Restart();
            for (j = 1000000; j > 0; j--) primes2(u);
            Console.Write(sw.Elapsed);
            Console.WriteLine(" {0,4} {1,5}", p.Length, u);
        }
        Console.Read();
    }

    static ushort[] primes0(ushort u)  // trial division   
    {
        var p = new ushort[6542]; var w = new int[] { 4, 2, 4, 2, 4, 6, 2, 6 };
        int i = 2, j = 0, c = 0, m = u, r = 3, s = 25, d, n = 7;
        if (m > 1) p[c++] = 2; if (m > 2) p[c++] = 3; if (m > 4) p[c++] = 5;
        for (; n <= m; i = 2, n += w[j++])
        {
            if (j > 7) j = 0;
            if (s <= n) { r += 4; s = r * r; r -= 2; }
            while ((d = p[i]) <= r && n % d > 0) i++;
            if (d > r) p[c++] = (ushort)n;
        }
        Array.Resize(ref p, c); return p;
    }

    static ushort[] primes1(ushort u)  // Eratosthenes
    {
        if (u < 5) return u < 2 ? new ushort[0] : u < 3 ?
            new ushort[] { 2 } : new ushort[] { 2, 3 };
        int a = 1, b = 1, c = 1, m = u, d = m; d += d & 1; d >>= 1;
        var x = new int[c += d >> 5]; x[0] = 1 << 24;
        for (/* */; a < c; a += 7) x[a] = 0x08102040;
        for (a = 2; a < c; a += 7) x[a] = 0x40810204;
        for (a = 3; a < c; a += 7) x[a] = 0x04081020;
        for (a = 4; a < c; a += 7) x[a] = 0x20408102;
        for (a = 5; a < c; a += 7) x[a] = 0x02040810;
        for (a = 6; a < c; a += 7) x[a] = 0x10204081;
        for (a = 7; a < c; a += 7) x[a] = ~0x7EFDFBF7; a = 7;
        while ((c = (a += 0x5A28A6 >> (3 * (b++ & 7)) & 7) * a) <= m)
            if ((x[a >> 6] & 1 << (a >> 1)) == 0)
                for (c >>= 1; c < d; c += a) x[c >> 5] |= 1 << c;
        var p = new ushort[6542]; p[0] = 2; p[1] = 3; p[2] = 5;
        for (c = 3, a = 1; a + 30 <= m; )
        {
            if ((x[(a += 6) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 4) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 2) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 4) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 2) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 4) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 6) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
            if ((x[(a += 2) >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
        }
        for (b = 0; ; )
        {
            if ((a += 0x5A28A6 >> (3 * (b++ & 7)) & 7) > m) break;
            if ((x[a >> 6] & 1 << (a >> 1)) == 0) p[c++] = (ushort)a;
        }
        Array.Resize(ref p, c); return p;
    }

    static ushort[] primes2(ushort u)
    {
        ushort[] p = primes(); int i = Array.BinarySearch(p, u);
        Array.Resize(ref p, i < 0 ? ~i : i + 1); return p;
    }

    static ushort[] primes()
    {
        #region primes < 2^16
        return new ushort[] {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,
25031,25033,25037,25057,25073,25087,25097,25111,25117,25121,25127,25147,25153,
25163,25169,25171,25183,25189,25219,25229,25237,25243,25247,25253,25261,25301,
25303,25307,25309,25321,25339,25343,25349,25357,25367,25373,25391,25409,25411,
25423,25439,25447,25453,25457,25463,25469,25471,25523,25537,25541,25561,25577,
25579,25583,25589,25601,25603,25609,25621,25633,25639,25643,25657,25667,25673,
25679,25693,25703,25717,25733,25741,25747,25759,25763,25771,25793,25799,25801,
25819,25841,25847,25849,25867,25873,25889,25903,25913,25919,25931,25933,25939,
25943,25951,25969,25981,25997,25999,26003,26017,26021,26029,26041,26053,26083,
26099,26107,26111,26113,26119,26141,26153,26161,26171,26177,26183,26189,26203,
26209,26227,26237,26249,26251,26261,26263,26267,26293,26297,26309,26317,26321,
26339,26347,26357,26371,26387,26393,26399,26407,26417,26423,26431,26437,26449,
26459,26479,26489,26497,26501,26513,26539,26557,26561,26573,26591,26597,26627,
26633,26641,26647,26669,26681,26683,26687,26693,26699,26701,26711,26713,26717,
26723,26729,26731,26737,26759,26777,26783,26801,26813,26821,26833,26839,26849,
26861,26863,26879,26881,26891,26893,26903,26921,26927,26947,26951,26953,26959,
26981,26987,26993,27011,27017,27031,27043,27059,27061,27067,27073,27077,27091,
27103,27107,27109,27127,27143,27179,27191,27197,27211,27239,27241,27253,27259,
27271,27277,27281,27283,27299,27329,27337,27361,27367,27397,27407,27409,27427,
27431,27437,27449,27457,27479,27481,27487,27509,27527,27529,27539,27541,27551,
27581,27583,27611,27617,27631,27647,27653,27673,27689,27691,27697,27701,27733,
27737,27739,27743,27749,27751,27763,27767,27773,27779,27791,27793,27799,27803,
27809,27817,27823,27827,27847,27851,27883,27893,27901,27917,27919,27941,27943,
27947,27953,27961,27967,27983,27997,28001,28019,28027,28031,28051,28057,28069,
28081,28087,28097,28099,28109,28111,28123,28151,28163,28181,28183,28201,28211,
28219,28229,28277,28279,28283,28289,28297,28307,28309,28319,28349,28351,28387,
28393,28403,28409,28411,28429,28433,28439,28447,28463,28477,28493,28499,28513,
28517,28537,28541,28547,28549,28559,28571,28573,28579,28591,28597,28603,28607,
28619,28621,28627,28631,28643,28649,28657,28661,28663,28669,28687,28697,28703,
28711,28723,28729,28751,28753,28759,28771,28789,28793,28807,28813,28817,28837,
28843,28859,28867,28871,28879,28901,28909,28921,28927,28933,28949,28961,28979,
29009,29017,29021,29023,29027,29033,29059,29063,29077,29101,29123,29129,29131,
29137,29147,29153,29167,29173,29179,29191,29201,29207,29209,29221,29231,29243,
29251,29269,29287,29297,29303,29311,29327,29333,29339,29347,29363,29383,29387,
29389,29399,29401,29411,29423,29429,29437,29443,29453,29473,29483,29501,29527,
29531,29537,29567,29569,29573,29581,29587,29599,29611,29629,29633,29641,29663,
29669,29671,29683,29717,29723,29741,29753,29759,29761,29789,29803,29819,29833,
29837,29851,29863,29867,29873,29879,29881,29917,29921,29927,29947,29959,29983,
29989,30011,30013,30029,30047,30059,30071,30089,30091,30097,30103,30109,30113,
30119,30133,30137,30139,30161,30169,30181,30187,30197,30203,30211,30223,30241,
30253,30259,30269,30271,30293,30307,30313,30319,30323,30341,30347,30367,30389,
30391,30403,30427,30431,30449,30467,30469,30491,30493,30497,30509,30517,30529,
30539,30553,30557,30559,30577,30593,30631,30637,30643,30649,30661,30671,30677,
30689,30697,30703,30707,30713,30727,30757,30763,30773,30781,30803,30809,30817,
30829,30839,30841,30851,30853,30859,30869,30871,30881,30893,30911,30931,30937,
30941,30949,30971,30977,30983,31013,31019,31033,31039,31051,31063,31069,31079,
31081,31091,31121,31123,31139,31147,31151,31153,31159,31177,31181,31183,31189,
31193,31219,31223,31231,31237,31247,31249,31253,31259,31267,31271,31277,31307,
31319,31321,31327,31333,31337,31357,31379,31387,31391,31393,31397,31469,31477,
31481,31489,31511,31513,31517,31531,31541,31543,31547,31567,31573,31583,31601,
31607,31627,31643,31649,31657,31663,31667,31687,31699,31721,31723,31727,31729,
31741,31751,31769,31771,31793,31799,31817,31847,31849,31859,31873,31883,31891,
31907,31957,31963,31973,31981,31991,32003,32009,32027,32029,32051,32057,32059,
32063,32069,32077,32083,32089,32099,32117,32119,32141,32143,32159,32173,32183,
32189,32191,32203,32213,32233,32237,32251,32257,32261,32297,32299,32303,32309,
32321,32323,32327,32341,32353,32359,32363,32369,32371,32377,32381,32401,32411,
32413,32423,32429,32441,32443,32467,32479,32491,32497,32503,32507,32531,32533,
32537,32561,32563,32569,32573,32579,32587,32603,32609,32611,32621,32633,32647,
32653,32687,32693,32707,32713,32717,32719,32749,32771,32779,32783,32789,32797,
32801,32803,32831,32833,32839,32843,32869,32887,32909,32911,32917,32933,32939,
32941,32957,32969,32971,32983,32987,32993,32999,33013,33023,33029,33037,33049,
33053,33071,33073,33083,33091,33107,33113,33119,33149,33151,33161,33179,33181,
33191,33199,33203,33211,33223,33247,33287,33289,33301,33311,33317,33329,33331,
33343,33347,33349,33353,33359,33377,33391,33403,33409,33413,33427,33457,33461,
33469,33479,33487,33493,33503,33521,33529,33533,33547,33563,33569,33577,33581,
33587,33589,33599,33601,33613,33617,33619,33623,33629,33637,33641,33647,33679,
33703,33713,33721,33739,33749,33751,33757,33767,33769,33773,33791,33797,33809,
33811,33827,33829,33851,33857,33863,33871,33889,33893,33911,33923,33931,33937,
33941,33961,33967,33997,34019,34031,34033,34039,34057,34061,34123,34127,34129,
34141,34147,34157,34159,34171,34183,34211,34213,34217,34231,34253,34259,34261,
34267,34273,34283,34297,34301,34303,34313,34319,34327,34337,34351,34361,34367,
34369,34381,34403,34421,34429,34439,34457,34469,34471,34483,34487,34499,34501,
34511,34513,34519,34537,34543,34549,34583,34589,34591,34603,34607,34613,34631,
34649,34651,34667,34673,34679,34687,34693,34703,34721,34729,34739,34747,34757,
34759,34763,34781,34807,34819,34841,34843,34847,34849,34871,34877,34883,34897,
34913,34919,34939,34949,34961,34963,34981,35023,35027,35051,35053,35059,35069,
35081,35083,35089,35099,35107,35111,35117,35129,35141,35149,35153,35159,35171,
35201,35221,35227,35251,35257,35267,35279,35281,35291,35311,35317,35323,35327,
35339,35353,35363,35381,35393,35401,35407,35419,35423,35437,35447,35449,35461,
35491,35507,35509,35521,35527,35531,35533,35537,35543,35569,35573,35591,35593,
35597,35603,35617,35671,35677,35729,35731,35747,35753,35759,35771,35797,35801,
35803,35809,35831,35837,35839,35851,35863,35869,35879,35897,35899,35911,35923,
35933,35951,35963,35969,35977,35983,35993,35999,36007,36011,36013,36017,36037,
36061,36067,36073,36083,36097,36107,36109,36131,36137,36151,36161,36187,36191,
36209,36217,36229,36241,36251,36263,36269,36277,36293,36299,36307,36313,36319,
36341,36343,36353,36373,36383,36389,36433,36451,36457,36467,36469,36473,36479,
36493,36497,36523,36527,36529,36541,36551,36559,36563,36571,36583,36587,36599,
36607,36629,36637,36643,36653,36671,36677,36683,36691,36697,36709,36713,36721,
36739,36749,36761,36767,36779,36781,36787,36791,36793,36809,36821,36833,36847,
36857,36871,36877,36887,36899,36901,36913,36919,36923,36929,36931,36943,36947,
36973,36979,36997,37003,37013,37019,37021,37039,37049,37057,37061,37087,37097,
37117,37123,37139,37159,37171,37181,37189,37199,37201,37217,37223,37243,37253,
37273,37277,37307,37309,37313,37321,37337,37339,37357,37361,37363,37369,37379,
37397,37409,37423,37441,37447,37463,37483,37489,37493,37501,37507,37511,37517,
37529,37537,37547,37549,37561,37567,37571,37573,37579,37589,37591,37607,37619,
37633,37643,37649,37657,37663,37691,37693,37699,37717,37747,37781,37783,37799,
37811,37813,37831,37847,37853,37861,37871,37879,37889,37897,37907,37951,37957,
37963,37967,37987,37991,37993,37997,38011,38039,38047,38053,38069,38083,38113,
38119,38149,38153,38167,38177,38183,38189,38197,38201,38219,38231,38237,38239,
38261,38273,38281,38287,38299,38303,38317,38321,38327,38329,38333,38351,38371,
38377,38393,38431,38447,38449,38453,38459,38461,38501,38543,38557,38561,38567,
38569,38593,38603,38609,38611,38629,38639,38651,38653,38669,38671,38677,38693,
38699,38707,38711,38713,38723,38729,38737,38747,38749,38767,38783,38791,38803,
38821,38833,38839,38851,38861,38867,38873,38891,38903,38917,38921,38923,38933,
38953,38959,38971,38977,38993,39019,39023,39041,39043,39047,39079,39089,39097,
39103,39107,39113,39119,39133,39139,39157,39161,39163,39181,39191,39199,39209,
39217,39227,39229,39233,39239,39241,39251,39293,39301,39313,39317,39323,39341,
39343,39359,39367,39371,39373,39383,39397,39409,39419,39439,39443,39451,39461,
39499,39503,39509,39511,39521,39541,39551,39563,39569,39581,39607,39619,39623,
39631,39659,39667,39671,39679,39703,39709,39719,39727,39733,39749,39761,39769,
39779,39791,39799,39821,39827,39829,39839,39841,39847,39857,39863,39869,39877,
39883,39887,39901,39929,39937,39953,39971,39979,39983,39989,40009,40013,40031,
40037,40039,40063,40087,40093,40099,40111,40123,40127,40129,40151,40153,40163,
40169,40177,40189,40193,40213,40231,40237,40241,40253,40277,40283,40289,40343,
40351,40357,40361,40387,40423,40427,40429,40433,40459,40471,40483,40487,40493,
40499,40507,40519,40529,40531,40543,40559,40577,40583,40591,40597,40609,40627,
40637,40639,40693,40697,40699,40709,40739,40751,40759,40763,40771,40787,40801,
40813,40819,40823,40829,40841,40847,40849,40853,40867,40879,40883,40897,40903,
40927,40933,40939,40949,40961,40973,40993,41011,41017,41023,41039,41047,41051,
41057,41077,41081,41113,41117,41131,41141,41143,41149,41161,41177,41179,41183,
41189,41201,41203,41213,41221,41227,41231,41233,41243,41257,41263,41269,41281,
41299,41333,41341,41351,41357,41381,41387,41389,41399,41411,41413,41443,41453,
41467,41479,41491,41507,41513,41519,41521,41539,41543,41549,41579,41593,41597,
41603,41609,41611,41617,41621,41627,41641,41647,41651,41659,41669,41681,41687,
41719,41729,41737,41759,41761,41771,41777,41801,41809,41813,41843,41849,41851,
41863,41879,41887,41893,41897,41903,41911,41927,41941,41947,41953,41957,41959,
41969,41981,41983,41999,42013,42017,42019,42023,42043,42061,42071,42073,42083,
42089,42101,42131,42139,42157,42169,42179,42181,42187,42193,42197,42209,42221,
42223,42227,42239,42257,42281,42283,42293,42299,42307,42323,42331,42337,42349,
42359,42373,42379,42391,42397,42403,42407,42409,42433,42437,42443,42451,42457,
42461,42463,42467,42473,42487,42491,42499,42509,42533,42557,42569,42571,42577,
42589,42611,42641,42643,42649,42667,42677,42683,42689,42697,42701,42703,42709,
42719,42727,42737,42743,42751,42767,42773,42787,42793,42797,42821,42829,42839,
42841,42853,42859,42863,42899,42901,42923,42929,42937,42943,42953,42961,42967,
42979,42989,43003,43013,43019,43037,43049,43051,43063,43067,43093,43103,43117,
43133,43151,43159,43177,43189,43201,43207,43223,43237,43261,43271,43283,43291,
43313,43319,43321,43331,43391,43397,43399,43403,43411,43427,43441,43451,43457,
43481,43487,43499,43517,43541,43543,43573,43577,43579,43591,43597,43607,43609,
43613,43627,43633,43649,43651,43661,43669,43691,43711,43717,43721,43753,43759,
43777,43781,43783,43787,43789,43793,43801,43853,43867,43889,43891,43913,43933,
43943,43951,43961,43963,43969,43973,43987,43991,43997,44017,44021,44027,44029,
44041,44053,44059,44071,44087,44089,44101,44111,44119,44123,44129,44131,44159,
44171,44179,44189,44201,44203,44207,44221,44249,44257,44263,44267,44269,44273,
44279,44281,44293,44351,44357,44371,44381,44383,44389,44417,44449,44453,44483,
44491,44497,44501,44507,44519,44531,44533,44537,44543,44549,44563,44579,44587,
44617,44621,44623,44633,44641,44647,44651,44657,44683,44687,44699,44701,44711,
44729,44741,44753,44771,44773,44777,44789,44797,44809,44819,44839,44843,44851,
44867,44879,44887,44893,44909,44917,44927,44939,44953,44959,44963,44971,44983,
44987,45007,45013,45053,45061,45077,45083,45119,45121,45127,45131,45137,45139,
45161,45179,45181,45191,45197,45233,45247,45259,45263,45281,45289,45293,45307,
45317,45319,45329,45337,45341,45343,45361,45377,45389,45403,45413,45427,45433,
45439,45481,45491,45497,45503,45523,45533,45541,45553,45557,45569,45587,45589,
45599,45613,45631,45641,45659,45667,45673,45677,45691,45697,45707,45737,45751,
45757,45763,45767,45779,45817,45821,45823,45827,45833,45841,45853,45863,45869,
45887,45893,45943,45949,45953,45959,45971,45979,45989,46021,46027,46049,46051,
46061,46073,46091,46093,46099,46103,46133,46141,46147,46153,46171,46181,46183,
46187,46199,46219,46229,46237,46261,46271,46273,46279,46301,46307,46309,46327,
46337,46349,46351,46381,46399,46411,46439,46441,46447,46451,46457,46471,46477,
46489,46499,46507,46511,46523,46549,46559,46567,46573,46589,46591,46601,46619,
46633,46639,46643,46649,46663,46679,46681,46687,46691,46703,46723,46727,46747,
46751,46757,46769,46771,46807,46811,46817,46819,46829,46831,46853,46861,46867,
46877,46889,46901,46919,46933,46957,46993,46997,47017,47041,47051,47057,47059,
47087,47093,47111,47119,47123,47129,47137,47143,47147,47149,47161,47189,47207,
47221,47237,47251,47269,47279,47287,47293,47297,47303,47309,47317,47339,47351,
47353,47363,47381,47387,47389,47407,47417,47419,47431,47441,47459,47491,47497,
47501,47507,47513,47521,47527,47533,47543,47563,47569,47581,47591,47599,47609,
47623,47629,47639,47653,47657,47659,47681,47699,47701,47711,47713,47717,47737,
47741,47743,47777,47779,47791,47797,47807,47809,47819,47837,47843,47857,47869,
47881,47903,47911,47917,47933,47939,47947,47951,47963,47969,47977,47981,48017,
48023,48029,48049,48073,48079,48091,48109,48119,48121,48131,48157,48163,48179,
48187,48193,48197,48221,48239,48247,48259,48271,48281,48299,48311,48313,48337,
48341,48353,48371,48383,48397,48407,48409,48413,48437,48449,48463,48473,48479,
48481,48487,48491,48497,48523,48527,48533,48539,48541,48563,48571,48589,48593,
48611,48619,48623,48647,48649,48661,48673,48677,48679,48731,48733,48751,48757,
48761,48767,48779,48781,48787,48799,48809,48817,48821,48823,48847,48857,48859,
48869,48871,48883,48889,48907,48947,48953,48973,48989,48991,49003,49009,49019,
49031,49033,49037,49043,49057,49069,49081,49103,49109,49117,49121,49123,49139,
49157,49169,49171,49177,49193,49199,49201,49207,49211,49223,49253,49261,49277,
49279,49297,49307,49331,49333,49339,49363,49367,49369,49391,49393,49409,49411,
49417,49429,49433,49451,49459,49463,49477,49481,49499,49523,49529,49531,49537,
49547,49549,49559,49597,49603,49613,49627,49633,49639,49663,49667,49669,49681,
49697,49711,49727,49739,49741,49747,49757,49783,49787,49789,49801,49807,49811,
49823,49831,49843,49853,49871,49877,49891,49919,49921,49927,49937,49939,49943,
49957,49991,49993,49999,50021,50023,50033,50047,50051,50053,50069,50077,50087,
50093,50101,50111,50119,50123,50129,50131,50147,50153,50159,50177,50207,50221,
50227,50231,50261,50263,50273,50287,50291,50311,50321,50329,50333,50341,50359,
50363,50377,50383,50387,50411,50417,50423,50441,50459,50461,50497,50503,50513,
50527,50539,50543,50549,50551,50581,50587,50591,50593,50599,50627,50647,50651,
50671,50683,50707,50723,50741,50753,50767,50773,50777,50789,50821,50833,50839,
50849,50857,50867,50873,50891,50893,50909,50923,50929,50951,50957,50969,50971,
50989,50993,51001,51031,51043,51047,51059,51061,51071,51109,51131,51133,51137,
51151,51157,51169,51193,51197,51199,51203,51217,51229,51239,51241,51257,51263,
51283,51287,51307,51329,51341,51343,51347,51349,51361,51383,51407,51413,51419,
51421,51427,51431,51437,51439,51449,51461,51473,51479,51481,51487,51503,51511,
51517,51521,51539,51551,51563,51577,51581,51593,51599,51607,51613,51631,51637,
51647,51659,51673,51679,51683,51691,51713,51719,51721,51749,51767,51769,51787,
51797,51803,51817,51827,51829,51839,51853,51859,51869,51871,51893,51899,51907,
51913,51929,51941,51949,51971,51973,51977,51991,52009,52021,52027,52051,52057,
52067,52069,52081,52103,52121,52127,52147,52153,52163,52177,52181,52183,52189,
52201,52223,52237,52249,52253,52259,52267,52289,52291,52301,52313,52321,52361,
52363,52369,52379,52387,52391,52433,52453,52457,52489,52501,52511,52517,52529,
52541,52543,52553,52561,52567,52571,52579,52583,52609,52627,52631,52639,52667,
52673,52691,52697,52709,52711,52721,52727,52733,52747,52757,52769,52783,52807,
52813,52817,52837,52859,52861,52879,52883,52889,52901,52903,52919,52937,52951,
52957,52963,52967,52973,52981,52999,53003,53017,53047,53051,53069,53077,53087,
53089,53093,53101,53113,53117,53129,53147,53149,53161,53171,53173,53189,53197,
53201,53231,53233,53239,53267,53269,53279,53281,53299,53309,53323,53327,53353,
53359,53377,53381,53401,53407,53411,53419,53437,53441,53453,53479,53503,53507,
53527,53549,53551,53569,53591,53593,53597,53609,53611,53617,53623,53629,53633,
53639,53653,53657,53681,53693,53699,53717,53719,53731,53759,53773,53777,53783,
53791,53813,53819,53831,53849,53857,53861,53881,53887,53891,53897,53899,53917,
53923,53927,53939,53951,53959,53987,53993,54001,54011,54013,54037,54049,54059,
54083,54091,54101,54121,54133,54139,54151,54163,54167,54181,54193,54217,54251,
54269,54277,54287,54293,54311,54319,54323,54331,54347,54361,54367,54371,54377,
54401,54403,54409,54413,54419,54421,54437,54443,54449,54469,54493,54497,54499,
54503,54517,54521,54539,54541,54547,54559,54563,54577,54581,54583,54601,54617,
54623,54629,54631,54647,54667,54673,54679,54709,54713,54721,54727,54751,54767,
54773,54779,54787,54799,54829,54833,54851,54869,54877,54881,54907,54917,54919,
54941,54949,54959,54973,54979,54983,55001,55009,55021,55049,55051,55057,55061,
55073,55079,55103,55109,55117,55127,55147,55163,55171,55201,55207,55213,55217,
55219,55229,55243,55249,55259,55291,55313,55331,55333,55337,55339,55343,55351,
55373,55381,55399,55411,55439,55441,55457,55469,55487,55501,55511,55529,55541,
55547,55579,55589,55603,55609,55619,55621,55631,55633,55639,55661,55663,55667,
55673,55681,55691,55697,55711,55717,55721,55733,55763,55787,55793,55799,55807,
55813,55817,55819,55823,55829,55837,55843,55849,55871,55889,55897,55901,55903,
55921,55927,55931,55933,55949,55967,55987,55997,56003,56009,56039,56041,56053,
56081,56087,56093,56099,56101,56113,56123,56131,56149,56167,56171,56179,56197,
56207,56209,56237,56239,56249,56263,56267,56269,56299,56311,56333,56359,56369,
56377,56383,56393,56401,56417,56431,56437,56443,56453,56467,56473,56477,56479,
56489,56501,56503,56509,56519,56527,56531,56533,56543,56569,56591,56597,56599,
56611,56629,56633,56659,56663,56671,56681,56687,56701,56711,56713,56731,56737,
56747,56767,56773,56779,56783,56807,56809,56813,56821,56827,56843,56857,56873,
56891,56893,56897,56909,56911,56921,56923,56929,56941,56951,56957,56963,56983,
56989,56993,56999,57037,57041,57047,57059,57073,57077,57089,57097,57107,57119,
57131,57139,57143,57149,57163,57173,57179,57191,57193,57203,57221,57223,57241,
57251,57259,57269,57271,57283,57287,57301,57329,57331,57347,57349,57367,57373,
57383,57389,57397,57413,57427,57457,57467,57487,57493,57503,57527,57529,57557,
57559,57571,57587,57593,57601,57637,57641,57649,57653,57667,57679,57689,57697,
57709,57713,57719,57727,57731,57737,57751,57773,57781,57787,57791,57793,57803,
57809,57829,57839,57847,57853,57859,57881,57899,57901,57917,57923,57943,57947,
57973,57977,57991,58013,58027,58031,58043,58049,58057,58061,58067,58073,58099,
58109,58111,58129,58147,58151,58153,58169,58171,58189,58193,58199,58207,58211,
58217,58229,58231,58237,58243,58271,58309,58313,58321,58337,58363,58367,58369,
58379,58391,58393,58403,58411,58417,58427,58439,58441,58451,58453,58477,58481,
58511,58537,58543,58549,58567,58573,58579,58601,58603,58613,58631,58657,58661,
58679,58687,58693,58699,58711,58727,58733,58741,58757,58763,58771,58787,58789,
58831,58889,58897,58901,58907,58909,58913,58921,58937,58943,58963,58967,58979,
58991,58997,59009,59011,59021,59023,59029,59051,59053,59063,59069,59077,59083,
59093,59107,59113,59119,59123,59141,59149,59159,59167,59183,59197,59207,59209,
59219,59221,59233,59239,59243,59263,59273,59281,59333,59341,59351,59357,59359,
59369,59377,59387,59393,59399,59407,59417,59419,59441,59443,59447,59453,59467,
59471,59473,59497,59509,59513,59539,59557,59561,59567,59581,59611,59617,59621,
59627,59629,59651,59659,59663,59669,59671,59693,59699,59707,59723,59729,59743,
59747,59753,59771,59779,59791,59797,59809,59833,59863,59879,59887,59921,59929,
59951,59957,59971,59981,59999,60013,60017,60029,60037,60041,60077,60083,60089,
60091,60101,60103,60107,60127,60133,60139,60149,60161,60167,60169,60209,60217,
60223,60251,60257,60259,60271,60289,60293,60317,60331,60337,60343,60353,60373,
60383,60397,60413,60427,60443,60449,60457,60493,60497,60509,60521,60527,60539,
60589,60601,60607,60611,60617,60623,60631,60637,60647,60649,60659,60661,60679,
60689,60703,60719,60727,60733,60737,60757,60761,60763,60773,60779,60793,60811,
60821,60859,60869,60887,60889,60899,60901,60913,60917,60919,60923,60937,60943,
60953,60961,61001,61007,61027,61031,61043,61051,61057,61091,61099,61121,61129,
61141,61151,61153,61169,61211,61223,61231,61253,61261,61283,61291,61297,61331,
61333,61339,61343,61357,61363,61379,61381,61403,61409,61417,61441,61463,61469,
61471,61483,61487,61493,61507,61511,61519,61543,61547,61553,61559,61561,61583,
61603,61609,61613,61627,61631,61637,61643,61651,61657,61667,61673,61681,61687,
61703,61717,61723,61729,61751,61757,61781,61813,61819,61837,61843,61861,61871,
61879,61909,61927,61933,61949,61961,61967,61979,61981,61987,61991,62003,62011,
62017,62039,62047,62053,62057,62071,62081,62099,62119,62129,62131,62137,62141,
62143,62171,62189,62191,62201,62207,62213,62219,62233,62273,62297,62299,62303,
62311,62323,62327,62347,62351,62383,62401,62417,62423,62459,62467,62473,62477,
62483,62497,62501,62507,62533,62539,62549,62563,62581,62591,62597,62603,62617,
62627,62633,62639,62653,62659,62683,62687,62701,62723,62731,62743,62753,62761,
62773,62791,62801,62819,62827,62851,62861,62869,62873,62897,62903,62921,62927,
62929,62939,62969,62971,62981,62983,62987,62989,63029,63031,63059,63067,63073,
63079,63097,63103,63113,63127,63131,63149,63179,63197,63199,63211,63241,63247,
63277,63281,63299,63311,63313,63317,63331,63337,63347,63353,63361,63367,63377,
63389,63391,63397,63409,63419,63421,63439,63443,63463,63467,63473,63487,63493,
63499,63521,63527,63533,63541,63559,63577,63587,63589,63599,63601,63607,63611,
63617,63629,63647,63649,63659,63667,63671,63689,63691,63697,63703,63709,63719,
63727,63737,63743,63761,63773,63781,63793,63799,63803,63809,63823,63839,63841,
63853,63857,63863,63901,63907,63913,63929,63949,63977,63997,64007,64013,64019,
64033,64037,64063,64067,64081,64091,64109,64123,64151,64153,64157,64171,64187,
64189,64217,64223,64231,64237,64271,64279,64283,64301,64303,64319,64327,64333,
64373,64381,64399,64403,64433,64439,64451,64453,64483,64489,64499,64513,64553,
64567,64577,64579,64591,64601,64609,64613,64621,64627,64633,64661,64663,64667,
64679,64693,64709,64717,64747,64763,64781,64783,64793,64811,64817,64849,64853,
64871,64877,64879,64891,64901,64919,64921,64927,64937,64951,64969,64997,65003,
65011,65027,65029,65033,65053,65063,65071,65089,65099,65101,65111,65119,65123,
65129,65141,65147,65167,65171,65173,65179,65183,65203,65213,65239,65257,65267,
65269,65287,65293,65309,65323,65327,65353,65357,65371,65381,65393,65407,65413,
65419,65423,65437,65447,65449,65479,65497,65519,65521};
        #endregion primes < 2^16
    }
}

2017/09/25

Direct Search Factorization


// MathWorld: DSF is the simplest (and most simple-minded) prime factorization algorithm.
// It consists of searching for factors of a number by systemetically performing trial divisions,
// usually using a sequence of increasing numbers. Multiples of small primes are commonly excluded
// to reduce the number of trial divisors, but just including them is sometimes faster than the time
// required to exclude them. DSF is very inefficient, and can be used only with fairly small numbers.

using System;                             // i7-4790@3.6GHz
using System.Collections.Generic;
class Program
{
    static void Main()
    {
        findPF(~0uL);                     // warm up
        time(4294967291);                 // largest prime < 2^32          50 µs
        time(4294967291 * 4294967291uL);  //                       6.5 s  
        time(3183958073 * 5793651691uL);  // 2 primes              4.8 s
        time(~0uL - 58);                  // largest prime < 2^64  6.5 s
        time(~0uL);                       // (2^64)-1                     100 µs 
        Console.Read();
    }

    static void time(ulong u)
    {
        Console.Write(" n " + u + "\n" + " t ");
        var sw = System.Diagnostics.Stopwatch.StartNew();
        List<ulong> pf = findPF(u);
        Console.Write(sw.Elapsed + "\n" + " f ");
        foreach (var v in pf) Console.Write(v + " ");
        Console.Write("\n" + "\n");
    }

    static List<ulong> findPF(ulong n)
    {
        var pf = new List<ulong>();
        if (n < 4) { if (n > 0) pf.Add(n); return pf; }
        while (n % 2 == 0) { n /= 2; pf.Add(2); }
        while (n % 3 == 0) { n /= 3; pf.Add(3); }
        while (n % 5 == 0) { n /= 5; pf.Add(5); }
        if (n == 1) return pf;
        ulong d = 1; uint b = 0, rt = sqrt(n);
        if (n <= ~0u) { findPF((uint)n, rt, 1, 0, pf); return pf; }
        for (; ; b = 0x5A28A6)
        {
            while (b > 0)
            {
                d += b & 7; b >>= 3; if (d > rt) { pf.Add(n); return pf; }
                if (n % d == 0)
                {
                    n /= d; pf.Add(d); while (n % d == 0) { n /= d; pf.Add(d); }
                    if (n == 1) return pf;
                    rt = sqrt(n);
                    if (n <= ~0u) { findPF((uint)n, rt, (uint)d, b, pf); return pf; }
                }
            }
            while (d + 30 <= rt && n % (d + 06) > 0 && n % (d + 10) > 0
                                && n % (d + 12) > 0 && n % (d + 16) > 0
                                && n % (d + 18) > 0 && n % (d + 22) > 0
                                && n % (d + 28) > 0 && n % (d + 30) > 0) d += 30;
        }
    }

    static void findPF(uint n, uint rt, uint d, uint b, List<ulong> pf)
    {
        for (; ; b = 0x5A28A6)
        {
            while (b > 0)
            {
                d += b & 7; b >>= 3; if (d > rt) { pf.Add(n); return; }
                if (n % d == 0)
                {
                    n /= d; pf.Add(d); while (n % d == 0) { n /= d; pf.Add(d); }
                    if (n == 1) return;
                    rt = (uint)Math.Sqrt(n);
                }
            }
            while (d + 30 <= rt && n % (d + 06) > 0 && n % (d + 10) > 0
                                && n % (d + 12) > 0 && n % (d + 16) > 0
                                && n % (d + 18) > 0 && n % (d + 22) > 0
                                && n % (d + 28) > 0 && n % (d + 30) > 0) d += 30;
        }
    }

    static uint sqrt(ulong n0)
    {
        if (n0 < 1uL << 52) return (uint)Math.Sqrt(n0);
        ulong n1 = n0 - 1 >> 24; int s = 25;
        if (n1 > 255) { n1 >>= 4; s = 29; }
        if (n1 > 15) { n1 >>= 2; s |= 2; }
        if (n1 > 3) s++;
        ulong r0 = 1uL << s, r1 = r0 + (n0 >> s) >> 1;
        while (r1 < r0) { r0 = r1; r1 = r0 + n0 / r0 >> 1; }
        return (uint)r0;
    }
}



///////////////////////////////////////////////////////////////////////////////////////////////////
//                                                                                               //
//                                  Eight threads, four cores.                                   //          
//                                                                                               //
///////////////////////////////////////////////////////////////////////////////////////////////////

using System;
using System.Collections.Generic;
using System.Threading.Tasks;
class Program
{
    static void Main()
    {
        findPF(~0uL);                     // warm up
        time(4294967291);                 // largest prime < 2^32          90 µs
        time(4294967291 * 4294967291uL);  //                              110 µs 
        time(3183958073 * 5793651691uL);  // 2 primes              0.9 s
        time(~0uL - 58);                  // largest prime < 2^64  1.1 s
        time(~0uL);                       // (2^64)-1                      80 µs 
        Console.Read();
    }

    static void time(ulong u)
    {
        Console.Write(" n " + u + "\n" + " t ");
        var sw = System.Diagnostics.Stopwatch.StartNew();
        List<ulong> pf = findPF(u);
        Console.Write(sw.Elapsed + "\n" + " f ");
        foreach (var v in pf) Console.Write(v + " ");
        Console.Write("\n" + "\n");
    }

    static List<ulong> findPF(ulong n)
    {
        var pf = new List<ulong>();
        if (n < 4) { if (n > 0) pf.Add(n); return pf; }
        while (n % 2 == 0) { n /= 2; pf.Add(2); }
        while (n % 3 == 0) { n /= 3; pf.Add(3); }
        while (n % 5 == 0) { n /= 5; pf.Add(5); }
        if (n > 1) { recFindF(n, pf); pf.Sort(); } return pf;
    }

    static void recFindF(ulong n, List<ulong> pf)
    {
        ulong f = findF(n); if (f == n) pf.Add(n);
        else { recFindF(f, pf); recFindF(n / f, pf); }
    }

    static ulong findF(ulong n)
    {
        uint r = sqrt(n); if (n % r == 0) return r;
        var locker = new object();
        Parallel.For(0, 8, k =>
        {
            ulong d0 = k == 0 ? 07u : k == 1 ? 11u : k == 2 ? 13u : k == 3 ? 17u :
                       k == 4 ? 19u : k == 5 ? 23u : k == 6 ? 29u : 31;
            ulong d1 = d0 + 30, d2 = d0 + 60, d3 = d0 + 90;
            while (d0 <= r)
            {
                if (n % d0 == 0 || n % d1 == 0 || n % d2 == 0 || n % d3 == 0)
                    lock (locker)
                    {
                        if (r == 0) return; r = 0;
                        n = n % d0 == 0 ? d0 : n % d1 == 0 ? d1 : n % d2 == 0 ? d2 : d3;
                        return;
                    }
                d0 += 120; d1 += 120; d2 += 120; d3 += 120;
            }
        }); return n;
    }

    static uint sqrt(ulong n0)
    {
        if (n0 < 1uL << 52) return (uint)Math.Sqrt(n0);
        ulong n1 = n0 - 1 >> 24; int s = 25;
        if (n1 > 255) { n1 >>= 4; s = 29; }
        if (n1 > 15) { n1 >>= 2; s |= 2; }
        if (n1 > 3) s++;
        ulong r0 = 1uL << s, r1 = r0 + (n0 >> s) >> 1;
        while (r1 < r0) { r0 = r1; r1 = r0 + n0 / r0 >> 1; }
        return (uint)r0;
    }
}


///////////////////////////////////////////////////////////////////////////////////////////////////
//                                                                                               //
//                                               ?                                               //
//                                                                                               //
///////////////////////////////////////////////////////////////////////////////////////////////////

using System;
using System.Collections.Generic;
using System.Threading.Tasks;
class Program
{
    static void Main()
    {
        sqrt(1uL << 52);
        ulong[] n = new ulong[] { ~0u - 4, 0, ~0uL - 172, ~0uL - 58, ~0uL };
        n[1] = n[0] * n[0];
        findPF(n[0]); time(n[0]);  // largest prime < 2^32          50 µs
        findPF(n[1]); time(n[1]);  //                               80 µs 
        findPF(n[2]); time(n[2]);  // 2 primes              0.78 s
        findPF(n[3]); time(n[3]);  // largest prime < 2^64  1.05 s
        findPF(n[4]); time(n[4]);  // (2^64)-1                      70 µs 
        Console.Read();
    }

    static void time(ulong u)
    {
        Console.Write(" n " + u + "\n" + " t ");
        var sw = System.Diagnostics.Stopwatch.StartNew();
        List<ulong> pf = findPF(u);
        Console.Write(sw.Elapsed + "\n" + " f ");
        foreach (var v in pf) Console.Write(v + " ");
        Console.Write("\n" + "\n");
    }

    static List<ulong> findPF(ulong n)
    {
        var pf = new List<ulong>();
        if (n < 4) { if (n > 0) pf.Add(n); return pf; }
        while (n % 2 == 0) { n /= 2; pf.Add(2); }
        while (n % 3 == 0) { n /= 3; pf.Add(3); }
        while (n % 5 == 0) { n /= 5; pf.Add(5); }
        if (n > 1)
            if (n <= ~0u) findPF((uint)n, pf);
            else { recFindF(n, pf, 1); pf.Sort(); }
        return pf;
    }

    static void findPF(uint n, List<ulong> pf)
    {
        for (uint d = 1, r = (uint)Math.Sqrt(n), b = 0; ; b = 0x5A28A6)
        {
            while (b > 0)
            {
                d += b & 7; b >>= 3; if (d > r) { pf.Add(n); return; }
                if (n % d == 0)
                {
                    n /= d; pf.Add(d); while (n % d == 0) { n /= d; pf.Add(d); }
                    if (n == 1) return;
                    r = (uint)Math.Sqrt(n);
                }
            }
            while (d + 30 <= r && n % (d + 06) > 0 && n % (d + 10) > 0
                               && n % (d + 12) > 0 && n % (d + 16) > 0
                               && n % (d + 18) > 0 && n % (d + 22) > 0
                               && n % (d + 28) > 0 && n % (d + 30) > 0) d += 30;
        }
    }

    static void recFindF(ulong n, List<ulong> pf, int i)
    {
        ulong f = findF(n);
        if (f == n) { while (i > 0) { pf.Add(n); i--; } }
        else if (n / f == f) { recFindF(f, pf, i << 1); }
        else { recFindF(f, pf, i); recFindF(n / f, pf, i); }
    }

    static ulong findF(ulong n0)
    {
        if (n0 <= ~0u) return findF((uint)n0);
        uint r = sqrt(n0); if (n0 % r == 0) return r;
        var locker = new object();
        Parallel.For(0, 8, k =>
        {
            ulong d0 = k == 0 ? 07u : k == 1 ? 11u : k == 2 ? 13u : k == 3 ? 17u :
                       k == 4 ? 19u : k == 5 ? 23u : k == 6 ? 29u : 31,
                  d1 = d0 + 30, d2 = d0 + 60, d3 = d0 + 90, n1 = n0;
            for (; d0 <= r; d0 += 120, d1 += 120, d2 += 120, d3 += 120)
                if (n1 % d0 == 0 || n1 % d1 == 0 || n1 % d2 == 0 || n1 % d3 == 0)
                    lock (locker)
                    {
                        if (r == 0) return; r = 0;
                        n0 = n0 % d0 == 0 ? d0 : n0 % d1 == 0 ? d1 : n0 % d2 == 0 ? d2 : d3;
                        return;
                    }
        }); return n0;
    }

    static uint findF(uint n0)
    {
        uint r = (uint)Math.Sqrt(n0); if (n0 % r == 0) return r;
        var locker = new object();
        Parallel.For(0, 8, k =>
        {
            uint d0 = k == 0 ? 07u : k == 1 ? 11u : k == 2 ? 13u : k == 3 ? 17u :
                      k == 4 ? 19u : k == 5 ? 23u : k == 6 ? 29u : 31,
                 d1 = d0 + 30, d2 = d0 + 60, d3 = d0 + 90, n1 = n0;
            for (; d0 <= r; d0 += 120, d1 += 120, d2 += 120, d3 += 120)
                if (n1 % d0 == 0 || n1 % d1 == 0 || n1 % d2 == 0 || n1 % d3 == 0)
                    lock (locker)
                    {
                        if (r == 0) return; r = 0;
                        n0 = n0 % d0 == 0 ? d0 : n0 % d1 == 0 ? d1 : n0 % d2 == 0 ? d2 : d3;
                        return;
                    }
        }); return n0;
    }

    static uint sqrt(ulong n0)
    {
        if (n0 < 1uL << 52) return (uint)Math.Sqrt(n0);
        uint n1 = (uint)(n0 >> 48); int s = 25;
        if (n1 > 255u) { n1 >>= 8; s = 29; }
        if (n1 > 15u) { n1 >>= 4; s |= 2; }
        if (n1 > 3u) s++;
        ulong r0 = 1uL << s, r1 = r0 + (n0 >> s) >> 1;
        while (r1 < r0) { r0 = r1; r1 = r0 + n0 / r0 >> 1; }
        return (uint)r0;
    }
}

2016/08/14

Odd prime sieve


/* 105,000,000 Odd primes in 1 second on an i7-4790. It's almost a sieve of Eratosthenes.
Composites are marked by 24 threads running in cache friendly ranges, after they're counted,
a prime array of exactly the right size is filled with ... primes. Bit twiddling hacks used:
"count bits set, in parallel" and "count trailing zero bits with a de Bruijn sequence".
...Code colorized with: C# Syntax Highlighter...                  125,000,000 ? scroll down
 
            Old times      New times    (in seconds).  
      n      x      p       x      p    p(rimes) <= n < 2^32
    1e9   0.50   0.66    0.31   0.42    x: odd composites, marked in 
    2e9   1.02   1.31    0.63   0.84       essentially a bit array.
    4e9   2.13   2.65    1.29   1.70    best of 5
    ~0u   2.28   2.87    1.39   1.82    best of 5
*/
using System;
using System.Threading.Tasks;
using sw = System.Diagnostics.Stopwatch;
class oddPrimes
{
    static sw sw = new sw();
    static void Main()
    {
        uint m = (uint)1e9;
        sw.Start(); uint[] p = getPrimes(m);
        Console.Write(p.Length + " odd primes <= " + m);
        Console.Read();
    }

    static uint[] getPrimes(uint m) { return countPrimes(buildX(m)); }

    static uint[] buildX(uint m)  // odd composites
    {
        uint i, j, t, r; uint[] x, r0, r1;
        t = (uint)Environment.ProcessorCount * 3; r = 32768 * 6;
        m -= m / ~0u; m += m & 1; m /= 2;
        x = new uint[(m >> 5) + 1];
        x[m >> 5] = ~0u << (int)m;
        if (m > 32) x[0] = 0x9b4b3491;
        for (i = j = 0; i < m; i += j += 4) x[i >> 5] |= 1u << (int)i;
        r0 = new uint[t + 1]; r1 = new uint[t + 1];
        for (i = 0; i <= t; i++) { r0[i] = i * r; r1[i] = (i + 1) * r; } r *= t;
        while (r0[0] < m)
        {
            Parallel.For(0, t, k =>
            {
                uint b, c, d, e, f, g, m0, m1;
                if ((m0 = r0[k]) >= m) return;
                if ((m1 = r1[k]) > m) m1 = m;
                for (d = 3, b = e = 4; ; b += e += 4, d += 2)
                {
                    if ((c = b + d) >= m1) break;
                    if ((x[d >> 6] & (1 << (int)(d >> 1))) == 0)
                    {
                        if (c < m0) if ((c += (m0 - c) / d * d) < m0) c += d;
                        for (g = d << 1, f = c + d; f < m1; c += g, f += g)
                        { x[c >> 5] |= 1u << (int)c; x[f >> 5] |= 1u << (int)f; }
                        for (; c < m1; c += g) x[c >> 5] |= 1u << (int)c;
                    }
                }
            });
            for (j = 0; j <= t; j++) { r0[j] += r; r1[j] += r; }
        }
        time("x");
        return x;
    }

    static uint[] countPrimes(uint[] x)
    {
        int t = Environment.ProcessorCount; uint[] c = new uint[t];
        byte[] b = new byte[x.Length]; uint n;
        Parallel.For(0, t, k =>
        {
            uint xi, ck = 0;
            for (int i = x.Length - 1 - k; i >= 0; i -= t)
            {
                xi = x[i]; xi -= (xi >> 1) & 0x55555555;
                xi = (xi & 0x33333333) + (xi >> 2 & 0x33333333);
                ck += b[i] = (byte)(32 - ((xi + (xi >> 4) & 0xf0f0f0f) * 0x1010101 >> 24));
            }
            c[k] = ck;
        });
        for (n = c[0], t--; t > 0; t--) n += c[t];
        time("c");
        return buildPrimes(b, x, n);
    }

    static uint[] buildPrimes(byte[] b, uint[] x, uint n)
    {
        int j = b.Length - 1; uint[] c = new uint[8]; uint[] p = new uint[n];
        sbyte[] bruijn = {01,02,29,03,30,15,25,04,31,23,21,16,26,18,05,09,
                          32,28,14,24,22,20,17,08,27,13,19,07,12,06,11,10};
        for (uint s = 0, k = 1; k < 8 && k <= j; k++) c[k] += s += b[k - 1];
        Parallel.For(0, 8, k =>
        {
            int s, i = k; uint ck = c[k], ci, u, v; long xi;
            if (k > 0 && ck == 0) return;
            for (u = (uint)(k << 6) - 1; ; u += 512)
            {
                v = u; ci = ck; xi = ~x[i];
                while (xi > 0)
                {
                    s = bruijn[((uint)((xi & -xi) * 0x077cb531)) >> 27];
                    p[ci++] = v += (uint)s << 1;
                    xi >>= s;
                }
                i += 8; if (i > j) break;
                ck += (uint)(b[i - 1] + b[i - 2] + b[i - 3] + b[i - 4] +
                             b[i - 5] + b[i - 6] + b[i - 7] + b[i - 8]);
            }
        });
        time("p");
        return p;
    }

    static void time(string s) { Console.WriteLine(sw.ElapsedMilliseconds + " ms " + s); }
}
/* Nothing new under the sun, pre-sieving 3, 5 & 7.
First I did it the old way (scroll down), but a hard
coded table works faster. Actually the table can be
made "on the fly", with a few more small primes?

            Old times      New times (in seconds).
      n      x      p       x      p
    1e9   0.31   0.42    0.24   0.34
    2e9   0.63   0.84    0.48   0.68
    4e9   1.29   1.70    0.99   1.50 (average of 10 runs)
    ~0u   1.39   1.82    1.07   1.60 (average of 10000 ;)
*/
using System;
using System.Threading.Tasks;
using sw = System.Diagnostics.Stopwatch;
class oddPrimes
{
    static sw sw = new sw();
    static void Main()
    {
        uint m = (uint)1e9;
        sw.Start(); uint[] p = getPrimes(m);
        Console.Write(p.Length + " odd primes <= " + m);
        Console.Read();
    }

    static uint[] getPrimes(uint m) { return countPrimes(buildX(m)); }

    static uint[] buildX(uint m)
    {
        uint i, j, t, r; uint[] x, r0, r1;
        t = (uint)Environment.ProcessorCount * 3; r = 32768 * 6;
        m -= m / ~0u; m += m & 1; m >>= 1;
        x = new uint[(m >> 5) + 1]; x[0] = 0x9b4b3491;
        preMark((int)m >> 5, x);
        for (i = j = 0; i < m; i += j += 4) x[i >> 5] |= 1u << (int)i;
        r0 = new uint[t]; r1 = new uint[t];
        for (i = 0; i < t; i++) { r0[i] = i * r; r1[i] = (i + 1) * r; } r *= t;
        while (r0[0] < m)
        {
            Parallel.For(0, t, k =>
            {
                uint b, c, d, e, f, g, m0, m1;
                if ((m0 = r0[k]) >= m) return;
                if ((m1 = r1[k]) > m) m1 = m;
                for (d = 11, b = 60, e = 20; ; b += e += 4, d += 2)
                {
                    if ((c = b + d) >= m1) break;
                    if ((x[d >> 6] & (1 << (int)(d >> 1))) == 0)
                    {
                        if (c < m0) if ((c += (m0 - c) / d * d) < m0) c += d;
                        for (g = d << 1, f = c + d; f < m1; c += g, f += g)
                        { x[c >> 5] |= 1u << (int)c; x[f >> 5] |= 1u << (int)f; }
                        for (; c < m1; c += g) x[c >> 5] |= 1u << (int)c;
                    }
                }
            });
            for (j = 0; j < t; j++) { r0[j] += r; r1[j] += r; }
        }
        x[m >> 5] |= ~0u << (int)m;
        time("x");
        return x;
    }

    static void preMark(int m, uint[] x)
    {
        uint[] a =  // 3*5*7 uints = 420 bytes
        {
            0x6e92ed65,0x59a5b34d,0x96693cf2,0x25dacb36,0x4b669add,0xd279e4b3,0xb5966d2c,  // No
            0xcd35ba4b,0xf3c96696,0x2cda59a4,0x6b74976b,0x92cd2d9a,0xb4b349e7,0xe92ed659,  // co
            0x9a5b34d6,0x6693cf25,0x5dacb369,0xb669add2,0x279e4b34,0x5966d2cd,0xd35ba4bb,  // de
            0x3c96696c,0xcda59a4f,0xb74976b2,0x2cd2d9a6,0x4b349e79,0x92ed659b,0xa5b34d6e,  // an
            0x693cf259,0xdacb3696,0x669add25,0x79e4b34b,0x966d2cd2,0x35ba4bb5,0xc96696cd,  // aly
            0xda59a4f3,0x74976b2c,0xcd2d9a6b,0xb349e792,0x2ed659b4,0x5b34d6e9,0x93cf259a,  // sis
            0xacb36966,0x69add25d,0x9e4b34b6,0x66d2cd27,0x5ba4bb59,0x96696cd3,0xa59a4f3c,  // iss
            0x4976b2cd,0xd2d9a6b7,0x349e792c,0xed659b4b,0xb34d6e92,0x3cf259a5,0xcb369669,  // ue
            0x9add25da,0xe4b34b66,0x6d2cd279,0xba4bb596,0x6696cd35,0x59a4f3c9,0x976b2cda,  // s
            0x2d9a6b74,0x49e792cd,0xd659b4b3,0x34d6e92e,0xcf259a5b,0xb3696693,0xadd25dac,  // we
            0x4b34b669,0xd2cd279e,0xa4bb5966,0x696cd35b,0x9a4f3c96,0x76b2cda5,0xd9a6b749,  // re
            0x9e792cd2,0x659b4b34,0x4d6e92ed,0xf259a5b3,0x3696693c,0xdd25dacb,0xb34b669a,  // de
            0x2cd279e4,0x4bb5966d,0x96cd35ba,0xa4f3c966,0x6b2cda59,0x9a6b7497,0xe792cd2d,  // tec
            0x59b4b349,0xd6e92ed6,0x259a5b34,0x696693cf,0xd25dacb3,0x34b669ad,0xcd279e4b,  // te
            0xbb5966d2,0x6cd35ba4,0x4f3c9669,0xb2cda59a,0xa6b74976,0x792cd2d9,0x9b4b349e,  // d.
        };
        Parallel.For(0, 8, k =>
        {
            for (int i, j, m0 = 1 + k * 105, m1 = m0 + 104; m0 <= m; m0 += 840, m1 += 840)
            { if (m1 > m) m1 = m; j = 0; i = m0; while (i <= m1) x[i++] = a[j++]; }
        });
    }

    static uint[] countPrimes(uint[] x)
    {
        uint[] c = new uint[8]; byte[] b = new byte[x.Length];
        Parallel.For(0, 8, k =>
        {
            uint xi, ck = 0;
            for (int i = x.Length - 1 - k; i >= 0; i -= 8)
            {
                xi = x[i]; xi -= xi >> 1 & 0x55555555;
                xi = (xi & 0x33333333) + (xi >> 2 & 0x33333333);
                ck += b[i] = (byte)(32 - ((xi + (xi >> 4) & 0xf0f0f0f) * 0x1010101 >> 24));
            }
            c[k] = ck;
        });
        time("c");
        return buildPrimes(b, x, c[0] + c[1] + c[2] + c[3] + c[4] + c[5] + c[6] + c[7]);
    }

    static uint[] buildPrimes(byte[] b, uint[] x, uint n)
    {
        int j = b.Length - 1; byte[] c = new byte[8]; uint[] p = new uint[n];
        sbyte[] bruijn = {01,02,29,03,30,15,25,04,31,23,21,16,26,18,05,09,
                          32,28,14,24,22,20,17,08,27,13,19,07,12,06,11,10};
        for (byte s = 0, k = 1; k < 8 && k <= j; k++) c[k] += s += b[k - 1];
        Parallel.For(0, 8, k =>
        {
            int s, i = k; uint ck = c[k], ci, u, v; long xi;
            if (k > 0 && ck == 0) return;
            for (u = (uint)(k << 6) - 1; ; u += 512)
            {
                v = u; ci = ck; xi = ~x[i];
                while (xi > 0)
                {
                    s = bruijn[((uint)((xi & -xi) * 0x077cb531)) >> 27];
                    p[ci++] = v += (uint)s << 1;
                    xi >>= s;
                }
                i += 8; if (i > j) break;
                ck += (uint)(b[i - 1] + b[i - 2] + b[i - 3] + b[i - 4] +
                             b[i - 5] + b[i - 6] + b[i - 7] + b[i - 8]);
            }
        });
        time("p");
        return p;
    }

    static void time(string s) { Console.WriteLine(sw.ElapsedMilliseconds + " ms " + s); }
}

/*
    static void preMark(int m, uint[] x)  // the old way
    {
        Parallel.For(0, 3, k =>
        {
            if (k == 0) for (int i = 1; i <= m; i += 3) x[i] = 0x24924924;
            if (k == 1) for (int i = 2; i <= m; i += 3) x[i] = 0x49249249;
            if (k == 2) for (int i = 3; i <= m; i += 3) x[i] = 0x92492492;
        });
        Parallel.For(0, 5, k =>
        {
            if (k == 0) for (int i = 1; i <= m; i += 5) x[i] |= 0x42108421;
            if (k == 1) for (int i = 2; i <= m; i += 5) x[i] |= 0x10842108;
            if (k == 2) for (int i = 3; i <= m; i += 5) x[i] |= 0x84210842;
            if (k == 3) for (int i = 4; i <= m; i += 5) x[i] |= 0x21084210;
            if (k == 4) for (int i = 5; i <= m; i += 5) x[i] |= 0x08421084;
        });                                                                
        Parallel.For(0, 7, k =>                                            
        {                                                                  
            if (k == 0) for (int i = 1; i <= m; i += 7) x[i] |= 0x08102040;
            if (k == 1) for (int i = 2; i <= m; i += 7) x[i] |= 0x40810204;
            if (k == 2) for (int i = 3; i <= m; i += 7) x[i] |= 0x04081020;
            if (k == 3) for (int i = 4; i <= m; i += 7) x[i] |= 0x20408102;
            if (k == 4) for (int i = 5; i <= m; i += 7) x[i] |= 0x02040810;
            if (k == 5) for (int i = 6; i <= m; i += 7) x[i] |= 0x10204081;
            if (k == 6) for (int i = 7; i <= m; i += 7) x[i] |= 0x81020408;
        });
    }
*/