Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:35:43
Deutsch
29.Mar 2024

Polynom = x^2+35x-17

0. Sequence

1. Algorithm

2. Mathematical background

3. Correctness of the algorithm

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 17 = 17
f(1) = 19 = 19
f(2) = 57 = 3*19
f(3) = 97 = 97
f(4) = 139 = 139
f(5) = 183 = 3*61
f(6) = 229 = 229
f(7) = 277 = 277
f(8) = 327 = 3*109
f(9) = 379 = 379
f(10) = 433 = 433
f(11) = 489 = 3*163
f(12) = 547 = 547
f(13) = 607 = 607
f(14) = 669 = 3*223
f(15) = 733 = 733
f(16) = 799 = 17*47
f(17) = 867 = 3*17*17
f(18) = 937 = 937
f(19) = 1009 = 1009
f(20) = 1083 = 3*19*19
f(21) = 1159 = 19*61
f(22) = 1237 = 1237
f(23) = 1317 = 3*439
f(24) = 1399 = 1399
f(25) = 1483 = 1483
f(26) = 1569 = 3*523
f(27) = 1657 = 1657
f(28) = 1747 = 1747
f(29) = 1839 = 3*613
f(30) = 1933 = 1933
f(31) = 2029 = 2029
f(32) = 2127 = 3*709
f(33) = 2227 = 17*131
f(34) = 2329 = 17*137
f(35) = 2433 = 3*811
f(36) = 2539 = 2539
f(37) = 2647 = 2647
f(38) = 2757 = 3*919
f(39) = 2869 = 19*151
f(40) = 2983 = 19*157
f(41) = 3099 = 3*1033
f(42) = 3217 = 3217
f(43) = 3337 = 47*71
f(44) = 3459 = 3*1153
f(45) = 3583 = 3583
f(46) = 3709 = 3709
f(47) = 3837 = 3*1279
f(48) = 3967 = 3967
f(49) = 4099 = 4099
f(50) = 4233 = 3*17*83
f(51) = 4369 = 17*257
f(52) = 4507 = 4507
f(53) = 4647 = 3*1549
f(54) = 4789 = 4789
f(55) = 4933 = 4933
f(56) = 5079 = 3*1693
f(57) = 5227 = 5227
f(58) = 5377 = 19*283
f(59) = 5529 = 3*19*97
f(60) = 5683 = 5683
f(61) = 5839 = 5839
f(62) = 5997 = 3*1999
f(63) = 6157 = 47*131
f(64) = 6319 = 71*89
f(65) = 6483 = 3*2161
f(66) = 6649 = 61*109
f(67) = 6817 = 17*401
f(68) = 6987 = 3*17*137
f(69) = 7159 = 7159
f(70) = 7333 = 7333
f(71) = 7509 = 3*2503
f(72) = 7687 = 7687
f(73) = 7867 = 7867
f(74) = 8049 = 3*2683
f(75) = 8233 = 8233
f(76) = 8419 = 8419
f(77) = 8607 = 3*19*151
f(78) = 8797 = 19*463
f(79) = 8989 = 89*101
f(80) = 9183 = 3*3061
f(81) = 9379 = 83*113
f(82) = 9577 = 61*157
f(83) = 9777 = 3*3259
f(84) = 9979 = 17*587
f(85) = 10183 = 17*599
f(86) = 10389 = 3*3463
f(87) = 10597 = 10597
f(88) = 10807 = 101*107
f(89) = 11019 = 3*3673
f(90) = 11233 = 47*239
f(91) = 11449 = 107*107
f(92) = 11667 = 3*3889
f(93) = 11887 = 11887
f(94) = 12109 = 12109
f(95) = 12333 = 3*4111
f(96) = 12559 = 19*661
f(97) = 12787 = 19*673
f(98) = 13017 = 3*4339
f(99) = 13249 = 13249
f(100) = 13483 = 97*139

1. Algorithm

If you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.

2. Mathematical background

Lemma: If p | f(x) then also p | f(x+p) and p | f(-x-b/a) a) p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0 mod p <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(x+p) b) if b = 0 mod a p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(-x-b/a) <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0 mod p <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(-x-b/a)

3. Correctness of the algorithm

The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1. a) First terms for the polynom f(x) = x^2+35x-17

f(0)=17
f(1)=19
f(2)=3
f(3)=97
f(4)=139
f(5)=61
f(6)=229
f(7)=277
f(8)=109
f(9)=379
f(10)=433
f(11)=163
f(12)=547
f(13)=607
f(14)=223
f(15)=733
f(16)=47
f(17)=1
f(18)=937
f(19)=1009
f(20)=1
f(21)=1
f(22)=1237
f(23)=439
f(24)=1399
f(25)=1483
f(26)=523
f(27)=1657
f(28)=1747
f(29)=613
f(30)=1933
f(31)=2029
f(32)=709
f(33)=131
f(34)=137
f(35)=811
f(36)=2539
f(37)=2647
f(38)=919
f(39)=151
f(40)=157
f(41)=1033
f(42)=3217
f(43)=71
f(44)=1153
f(45)=3583
f(46)=3709
f(47)=1279
f(48)=3967
f(49)=4099
f(50)=83
f(51)=257
f(52)=4507
f(53)=1549
f(54)=4789
f(55)=4933
f(56)=1693
f(57)=5227
f(58)=283
f(59)=1
f(60)=5683
f(61)=5839
f(62)=1999
f(63)=1
f(64)=89
f(65)=2161
f(66)=1
f(67)=401
f(68)=1
f(69)=7159
f(70)=7333
f(71)=2503
f(72)=7687
f(73)=7867
f(74)=2683
f(75)=8233
f(76)=8419
f(77)=1
f(78)=463
f(79)=101
f(80)=3061
f(81)=113
f(82)=1
f(83)=3259
f(84)=587
f(85)=599
f(86)=3463
f(87)=10597
f(88)=107
f(89)=3673
f(90)=239
f(91)=1
f(92)=3889
f(93)=11887
f(94)=12109
f(95)=4111
f(96)=661
f(97)=673
f(98)=4339
f(99)=13249

b) Substitution of the polynom
The polynom f(x)=x^2+35x-17 could be written as f(y)= y^2-323.25 with x=y-17.5

c) Backsubstitution Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x) f'(y)>(2y-1) with with y=x+17.5
f'(x)>2x+34

4. Infinity of the sequence

The mathematical proof is analogue to the proof for the polynom f(x)=x^2+1

5. Sequence of the polynom with 1

17, 19, 3, 97, 139, 61, 229, 277, 109, 379, 433, 163, 547, 607, 223, 733, 47, 1, 937, 1009, 1, 1, 1237, 439, 1399, 1483, 523, 1657, 1747, 613, 1933, 2029, 709, 131, 137, 811, 2539, 2647, 919, 151, 157, 1033, 3217, 71, 1153, 3583, 3709, 1279, 3967, 4099, 83, 257, 4507, 1549, 4789, 4933, 1693, 5227, 283, 1, 5683, 5839, 1999, 1, 89, 2161, 1, 401, 1, 7159, 7333, 2503, 7687, 7867, 2683, 8233, 8419, 1, 463, 101, 3061, 113, 1, 3259, 587, 599, 3463, 10597, 107, 3673, 239, 1, 3889, 11887, 12109, 4111, 661, 673, 4339, 13249, 1, 269, 821, 14197, 4813, 14683, 14929, 5059, 15427, 15679, 1, 16189, 16447, 5569, 1, 907, 307, 1, 1061, 359, 18583, 18859, 6379, 19417, 19699, 6661, 20269, 337, 6949, 21139, 21433, 7243, 22027, 1, 397, 1, 1367, 167, 23857, 24169, 8161, 24799, 25117, 1, 25759, 26083, 8803, 26737, 27067, 9133, 27733, 28069, 557, 1, 1531, 9811, 1, 641, 10159, 30829, 31183, 10513, 191, 32257, 1, 32983, 33349, 11239, 383, 2027, 683, 1, 1873, 631, 1, 1, 12373, 37507, 37897, 12763, 1, 39079, 13159, 39877, 857, 1, 2417, 2441, 1, 419, 1, 757, 2293, 43987, 1, 1, 45259, 1, 431, 46549, 15661, 47419, 47857, 947, 1, 1, 233, 50077, 50527, 16993, 2707, 2731, 17449, 52807, 53269, 17911, 653, 54667, 18379, 55609, 3299, 1109, 57037, 509, 19333, 251, 541, 19819, 1, 3181, 1069, 1307, 61927, 20809, 62929, 1, 21313, 1, 3821, 1, 65983, 66499, 1, 67537, 68059, 22861, 69109, 839, 1231, 1, 71233, 1, 72307, 751, 1439, 4349, 1049, 281, 75577, 76129, 25561, 77239, 77797, 26119, 78919, 1303, 26683, 4243, 4273, 27253, 293, 4877, 1637, 84067, 84649, 28411, 85819, 1217, 617, 87589, 1, 1, 643, 89977, 1, 91183, 4831, 1621, 5471, 5507, 311, 94849, 95467, 32029, 997, 743, 32653, 317, 2111, 1, 100483, 101119, 1, 102397, 1, 1, 1, 929, 1, 106279, 1753, 35863, 108247, 108907, 36523, 110233, 110899, 37189, 112237, 112909, 37861, 114259, 6761, 2267, 6121, 1, 1, 1667, 119047, 1, 1451, 1361, 40609, 122527, 123229, 1, 1103, 769, 42019, 7457, 7499, 1, 941, 6823, 2287, 2789, 1481, 44179, 1877, 133999, 1, 135469, 136207, 1, 877, 138433, 2729, 8231, 140677, 47143, 142183, 142939, 2521, 7603, 145219, 48661, 146749, 147517, 49429, 149059, 1, 823, 3221, 8951, 2999, 153733, 809, 51769, 1123, 1039, 52561, 1, 1, 53359, 160879, 161683, 54163, 2677, 1, 54973, 9749, 1, 1187, 1543, 169009, 56611, 1087, 1, 1, 173149, 9157, 3067, 1, 176497, 59113, 178183, 179029, 3527, 10631, 1697, 60811, 183289, 1721, 1, 185869, 186733, 62533, 1, 189337, 1, 1, 1381, 64279, 1, 11447, 3833, 1499, 197257, 1, 199039, 199933, 66943, 3307, 202627, 67843, 2297, 205339, 68749, 10903, 1, 69661, 12347, 12401, 1, 1951, 1559, 71503, 2221, 3547, 1, 218233, 219169, 73369, 221047, 221989, 74311, 1709, 11833, 1, 13337, 1021, 76213, 1, 1, 1, 2801, 4967, 78139, 1, 3329, 79111, 1427, 239287, 1, 241249, 14249, 1, 12853, 1, 1, 247183, 248179, 83059, 2477, 251179, 84061, 5387, 254197, 1, 256219, 3623, 86083, 1, 1, 87103, 13807, 1, 88129, 265417, 266449, 1, 268519, 2473, 90199, 271639, 272683, 91243, 274777, 275827, 1, 16349, 1777, 1, 5981, 14851, 4969, 2657, 2053, 95479, 2687, 288583, 96553, 290737, 291817, 1097, 293983, 1, 5807, 297247, 298339, 1, 1, 301627, 1, 15991, 304933, 102013, 307147, 1, 103123, 4373, 1, 104239, 18461, 1, 105361, 317209, 318337, 1283, 1, 321733, 1, 17053, 1, 1783, 327433, 328579, 1, 330877, 1, 1, 1, 2969, 1, 1, 1319, 113383, 1787, 1433, 114553, 5653, 18211, 6091, 348367, 1193, 116911, 351919, 20771, 6947, 4283, 7589, 119293, 2621, 360277, 1, 3739, 363889, 1, 366307, 367519, 6469, 19471, 1553, 1229, 21977, 1297, 125353, 377287, 4253, 126583, 380983, 1, 127819, 384697, 385939, 129061, 2383, 8291, 1, 20641, 20707, 1, 23291, 4463, 1, 6553, 2927, 134089, 403537, 1277, 1, 407359, 408637, 1, 1453, 1, 1289, 8831, 1, 1, 418933, 1, 1979, 422827, 424129, 141811, 426739, 428047, 1, 1601, 431983, 1489, 1, 2887, 145753, 25799, 1, 7741, 23293, 443899, 148411, 446569, 447907, 149749, 9587, 451933, 1, 454627, 455977, 152443, 458683, 460039, 1, 1, 464119, 155161, 24571, 1, 1, 470959, 472333, 1, 2039, 476467, 3389, 4241, 7879, 160669, 483397, 28517, 9533, 3229, 1, 1, 1, 1, 8677, 5573, 3797, 166273, 500233, 2153, 2749, 2203, 5009, 169111, 29927, 30011, 170539, 3691, 1, 3659, 5813, 7307, 9127, 27457, 523129, 174859, 526027, 1, 1, 530389, 531847, 10457, 1, 536233, 179233, 539167, 4127, 1, 4987, 11597, 182179, 28843, 28921, 183661, 552469, 2207, 185149, 556939, 1, 10979, 1, 5261, 188143, 2963, 567439, 3109, 5881, 571969, 191161, 1, 1597, 10141, 2309, 7001, 194203, 34361, 1, 195733, 588733, 590269, 197269, 1, 594889, 198811, 2113, 1, 1, 5333, 1, 10627, 31963, 608857, 11969, 35999, 613549, 1, 13121, 3793, 1, 5701, 622987, 208189, 8819, 627733, 2357, 630907, 632497, 2179, 33457, 1973, 12527, 640477, 642079, 214561, 6389, 646897, 1913, 2347, 651733, 2447, 654967, 656587, 1, 1, 661459, 2663, 1, 2063, 11719, 1987, 671257, 224299, 1, 4049, 225943, 2843, 681127, 227593, 684433, 686089, 229249, 1, 11329, 1, 40847, 696067, 12241, 1, 1, 1, 2741, 706117, 3323, 4519, 4363, 237619, 8609, 6571, 1, 3011, 1, 14177, 724729, 7489, 242713, 1, 1, 244423, 6869, 736699, 5237, 3319, 741859, 3491, 1, 747037, 249589, 1, 44249, 251323, 755707, 1, 253063, 760933, 1, 13411, 766177, 12589, 256561, 771439, 16451, 258319, 776719, 8747, 15299, 1, 783787, 1, 787333, 789109, 2333, 792667, 794449, 1, 1, 799807, 267199, 803389, 1, 268993, 808777, 47681, 1, 814183, 815989, 2699, 819607, 17477, 2003, 825049, 826867, 276229, 43711, 1, 2753, 835987, 837817, 279883, 49499, 1, 281719, 846997, 848839, 1, 3823, 3019, 1, 18257, 859933, 3461, 2777, 45553, 15217, 869233, 12269, 17117, 51461, 2819, 4801, 1, 3853, 1, 3803, 1, 1, 1, 2251, 298513, 897433, 2837, 1, 47533, 1, 17783, 1, 910747, 2791, 914569, 3407, 306133, 3581, 922237, 2879, 6133, 2423, 2897, 1, 4889, 2381, 2903, 2909, 313849, 10601, 1, 6719, 1, 951277, 317743, 955183, 957139, 319699, 11579, 963019, 321661, 13619, 968917, 19037, 1, 51307, 17137, 5861, 10111, 327583, 984733, 986719, 2371, 3947, 992689, 2531, 16339, 6361, 1, 1002679, 1, 19739, 14207, 10007, 1, 53407, 1016749, 339589, 7451, 12323, 1, 1026859, 1, 343639, 1032949,

6. Sequence of the polynom (only primes)

17, 19, 3, 97, 139, 61, 229, 277, 109, 379, 433, 163, 547, 607, 223, 733, 47, 937, 1009, 1237, 439, 1399, 1483, 523, 1657, 1747, 613, 1933, 2029, 709, 131, 137, 811, 2539, 2647, 919, 151, 157, 1033, 3217, 71, 1153, 3583, 3709, 1279, 3967, 4099, 83, 257, 4507, 1549, 4789, 4933, 1693, 5227, 283, 5683, 5839, 1999, 89, 2161, 401, 7159, 7333, 2503, 7687, 7867, 2683, 8233, 8419, 463, 101, 3061, 113, 3259, 587, 599, 3463, 10597, 107, 3673, 239, 3889, 11887, 12109, 4111, 661, 673, 4339, 13249, 269, 821, 14197, 4813, 14683, 14929, 5059, 15427, 15679, 16189, 16447, 5569, 907, 307, 1061, 359, 18583, 18859, 6379, 19417, 19699, 6661, 20269, 337, 6949, 21139, 21433, 7243, 22027, 397, 1367, 167, 23857, 24169, 8161, 24799, 25117, 25759, 26083, 8803, 26737, 27067, 9133, 27733, 28069, 557, 1531, 9811, 641, 10159, 30829, 31183, 10513, 191, 32257, 32983, 33349, 11239, 383, 2027, 683, 1873, 631, 12373, 37507, 37897, 12763, 39079, 13159, 39877, 857, 2417, 2441, 419, 757, 2293, 43987, 45259, 431, 46549, 15661, 47419, 47857, 947, 233, 50077, 50527, 16993, 2707, 2731, 17449, 52807, 53269, 17911, 653, 54667, 18379, 55609, 3299, 1109, 57037, 509, 19333, 251, 541, 19819, 3181, 1069, 1307, 61927, 20809, 62929, 21313, 3821, 65983, 66499, 67537, 68059, 22861, 69109, 839, 1231, 71233, 72307, 751, 1439, 4349, 1049, 281, 75577, 76129, 25561, 77239, 77797, 26119, 78919, 1303, 26683, 4243, 4273, 27253, 293, 4877, 1637, 84067, 84649, 28411, 85819, 1217, 617, 87589, 643, 89977, 91183, 4831, 1621, 5471, 5507, 311, 94849, 95467, 32029, 997, 743, 32653, 317, 2111, 100483, 101119, 102397, 929, 106279, 1753, 35863, 108247, 108907, 36523, 110233, 110899, 37189, 112237, 112909, 37861, 114259, 6761, 2267, 6121, 1667, 119047, 1451, 1361, 40609, 122527, 123229, 1103, 769, 42019, 7457, 7499, 941, 6823, 2287, 2789, 1481, 44179, 1877, 133999, 135469, 136207, 877, 138433, 2729, 8231, 140677, 47143, 142183, 142939, 2521, 7603, 145219, 48661, 146749, 147517, 49429, 149059, 823, 3221, 8951, 2999, 153733, 809, 51769, 1123, 1039, 52561, 53359, 160879, 161683, 54163, 2677, 54973, 9749, 1187, 1543, 169009, 56611, 1087, 173149, 9157, 3067, 176497, 59113, 178183, 179029, 3527, 10631, 1697, 60811, 183289, 1721, 185869, 186733, 62533, 189337, 1381, 64279, 11447, 3833, 1499, 197257, 199039, 199933, 66943, 3307, 202627, 67843, 2297, 205339, 68749, 10903, 69661, 12347, 12401, 1951, 1559, 71503, 2221, 3547, 218233, 219169, 73369, 221047, 221989, 74311, 1709, 11833, 13337, 1021, 76213, 2801, 4967, 78139, 3329, 79111, 1427, 239287, 241249, 14249, 12853, 247183, 248179, 83059, 2477, 251179, 84061, 5387, 254197, 256219, 3623, 86083, 87103, 13807, 88129, 265417, 266449, 268519, 2473, 90199, 271639, 272683, 91243, 274777, 275827, 16349, 1777, 5981, 14851, 4969, 2657, 2053, 95479, 2687, 288583, 96553, 290737, 291817, 1097, 293983, 5807, 297247, 298339, 301627, 15991, 304933, 102013, 307147, 103123, 4373, 104239, 18461, 105361, 317209, 318337, 1283, 321733, 17053, 1783, 327433, 328579, 330877, 2969, 1319, 113383, 1787, 1433, 114553, 5653, 18211, 6091, 348367, 1193, 116911, 351919, 20771, 6947, 4283, 7589, 119293, 2621, 360277, 3739, 363889, 366307, 367519, 6469, 19471, 1553, 1229, 21977, 1297, 125353, 377287, 4253, 126583, 380983, 127819, 384697, 385939, 129061, 2383, 8291, 20641, 20707, 23291, 4463, 6553, 2927, 134089, 403537, 1277, 407359, 408637, 1453, 1289, 8831, 418933, 1979, 422827, 424129, 141811, 426739, 428047, 1601, 431983, 1489, 2887, 145753, 25799, 7741, 23293, 443899, 148411, 446569, 447907, 149749, 9587, 451933, 454627, 455977, 152443, 458683, 460039, 464119, 155161, 24571, 470959, 472333, 2039, 476467, 3389, 4241, 7879, 160669, 483397, 28517, 9533, 3229, 8677, 5573, 3797, 166273, 500233, 2153, 2749, 2203, 5009, 169111, 29927, 30011, 170539, 3691, 3659, 5813, 7307, 9127, 27457, 523129, 174859, 526027, 530389, 531847, 10457, 536233, 179233, 539167, 4127, 4987, 11597, 182179, 28843, 28921, 183661, 552469, 2207, 185149, 556939, 10979, 5261, 188143, 2963, 567439, 3109, 5881, 571969, 191161, 1597, 10141, 2309, 7001, 194203, 34361, 195733, 588733, 590269, 197269, 594889, 198811, 2113, 5333, 10627, 31963, 608857, 11969, 35999, 613549, 13121, 3793, 5701, 622987, 208189, 8819, 627733, 2357, 630907, 632497, 2179, 33457, 1973, 12527, 640477, 642079, 214561, 6389, 646897, 1913, 2347, 651733, 2447, 654967, 656587, 661459, 2663, 2063, 11719, 1987, 671257, 224299, 4049, 225943, 2843, 681127, 227593, 684433, 686089, 229249, 11329, 40847, 696067, 12241, 2741, 706117, 3323, 4519, 4363, 237619, 8609, 6571, 3011, 14177, 724729, 7489, 242713, 244423, 6869, 736699, 5237, 3319, 741859, 3491, 747037, 249589, 44249, 251323, 755707, 253063, 760933, 13411, 766177, 12589, 256561, 771439, 16451, 258319, 776719, 8747, 15299, 783787, 787333, 789109, 2333, 792667, 794449, 799807, 267199, 803389, 268993, 808777, 47681, 814183, 815989, 2699, 819607, 17477, 2003, 825049, 826867, 276229, 43711, 2753, 835987, 837817, 279883, 49499, 281719, 846997, 848839, 3823, 3019, 18257, 859933, 3461, 2777, 45553, 15217, 869233, 12269, 17117, 51461, 2819, 4801, 3853, 3803, 2251, 298513, 897433, 2837, 47533, 17783, 910747, 2791, 914569, 3407, 306133, 3581, 922237, 2879, 6133, 2423, 2897, 4889, 2381, 2903, 2909, 313849, 10601, 6719, 951277, 317743, 955183, 957139, 319699, 11579, 963019, 321661, 13619, 968917, 19037, 51307, 17137, 5861, 10111, 327583, 984733, 986719, 2371, 3947, 992689, 2531, 16339, 6361, 1002679, 19739, 14207, 10007, 53407, 1016749, 339589, 7451, 12323, 1026859, 343639, 1032949,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+35x-17 and
the reducible primes which appear as divisor for the first time
p | x^2+35x-17 and p < x^2+35x-17

To avoid confusion with the number of primes:
I did not count the primes <= A
but I counted the primes appending the x and therefore the x <= A

ABCDEFGHIJK
exponent =log10 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
11011921.1000000.9000001.1000000.0000000.0000000.000000
21008444400.8400000.4400000.8400007.6363644.88888920.000000
31.0007992985010.7990000.2980000.7990009.5119056.77272712.525000
410.0007.6452.1395.5060.7645000.2139000.7645009.5682117.17785210.990020
5100.00074.94516.66258.2830.7494500.1666200.7494509.8031407.78962110.585361
61.000.000739.715136.171603.5440.7397150.1361710.7397159.8701058.17254810.355404
710.000.0007.329.8061.151.0656.178.7410.7329810.1151070.7329819.9089598.45308510.237432
8100.000.00072.781.7419.977.07362.804.6680.7278170.0997710.7278179.9295608.66768810.164639
91.000.000.000723.783.56688.058.518635.725.0480.7237840.0880590.7237849.9445758.82608810.122258
1010.000.000.0007.205.909.547788.073.2226.417.836.3250.7205910.0788070.7205919.9558908.94942610.095303


ABCDEFGHIJK
exponent =log2 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
123301.5000001.5000000.0000000.0000000.0000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
389721.1250000.8750000.2500001.8000001.400000inf
416171251.0625000.7500000.3125001.8888891.7142862.500000
532302190.9375000.6562500.2812501.7647061.7500001.800000
6645734230.8906250.5312500.3593751.9000001.6190482.555556
712810956530.8515620.4375000.4140621.9122811.6470592.304348
82562081001080.8125000.3906250.4218751.9082571.7857142.037736
95124111732380.8027340.3378910.4648441.9759621.7300002.203704
101.0248203075130.8007810.2998050.5009771.9951341.7745672.155462
112.0481.6025421.0600.7822270.2646480.5175781.9536591.7654722.066277
124.0963.1839742.2090.7771000.2377930.5393071.9868911.7970482.083962
138.1926.2971.7754.5220.7686770.2166750.5520021.9783221.8223822.047080
1416.38412.4643.2999.1650.7607420.2013550.5593871.9793551.8585922.026758
1532.76824.7886.11518.6730.7564700.1866150.5698551.9887681.8535922.037425
1665.53649.30111.35037.9510.7522740.1731870.5790861.9889061.8560922.032400
17131.07298.03321.32476.7090.7479320.1626890.5852431.9884591.8787672.021264
18262.144195.31940.032155.2870.7450830.1527100.5923731.9923801.8773212.024365
19524.288388.91575.294313.6210.7417960.1436120.5981851.9911791.8808452.019622
201.048.576775.470142.326633.1440.7395460.1357330.6038131.9939321.8902702.018819
212.097.1521.546.194269.6661.276.5280.7372830.1285870.6086961.9938801.8947062.016173
224.194.3043.084.042512.4942.571.5480.7352930.1221880.6131051.9946021.9004772.014486
238.388.6086.152.263977.5285.174.7350.7334070.1165300.6168761.9948701.9073942.012304
2416.777.21612.276.0331.865.57710.410.4560.7317090.1111970.6205121.9953691.9084642.011785
2533.554.43224.498.0533.572.35720.925.6960.7300990.1064650.6236341.9956001.9148812.010065
2667.108.86448.895.5206.853.93442.041.5860.7286000.1021320.6264681.9958941.9186032.009089
27134.217.72897.606.33113.169.46684.436.8650.7272240.0981200.6291041.9962221.9214462.008413
28268.435.456194.873.45725.338.957169.534.5000.7259600.0943950.6315651.9965251.9240692.007826
29536.870.912389.111.52948.825.532340.285.9970.7247770.0909450.6338321.9967401.9268962.007179
301.073.741.824777.040.49994.209.185682.831.3140.7236750.0877390.6359361.9969611.9295072.006639
312.147.483.6481.551.872.227182.004.5091.369.867.7180.7226470.0847520.6378941.9971571.9319192.006158
324.294.967.2963.099.605.866352.033.9302.747.571.9360.7216830.0819640.6397191.9973331.9342042.005721
338.589.934.5926.191.465.407681.675.8015.509.789.6060.7207810.0793580.6414241.9975011.9363922.005330
3417.179.869.18412.368.365.3081.321.320.68711.047.044.6210.7199340.0769110.6430231.9976481.9383422.004985


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p=f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
123112100
245313200
387513220
416121014431
532211917662
6643432181196
71285654112171512
825610098123292721
9512173171141474540
101.024307305175808072
112.0485425401131136147128
124.0969749721235255248236
138.1921.7751.7731442445455433
1416.3843.2993.2971835805850809
1532.7686.1156.11311.5411.5001.5351.539
1665.53611.35011.34812.8102.8592.8192.862
17131.07221.32421.32215.3025.3305.3045.388
18262.14440.03240.03019.99710.0099.93710.089
19524.28875.29475.292118.78618.74218.83918.927
201.048.576142.326142.324135.60235.63635.62535.463
212.097.152269.666269.664167.31367.21867.76467.371
224.194.304512.494512.4921128.096127.815128.418128.165
238.388.608977.528977.5261243.946244.500244.660244.422
2416.777.2161.865.5771.865.5751465.936466.714466.632466.295
2533.554.4323.572.3573.572.3551892.843893.182892.572893.760
2667.108.8646.853.9346.853.93211.713.9141.713.9611.712.1661.713.893
27134.217.72813.169.46613.169.46413.293.1133.292.6363.291.6623.292.055
28268.435.45625.338.95725.338.95516.333.6476.334.9896.336.1756.334.146
29536.870.91248.825.53248.825.530112.204.66912.208.17512.206.55212.206.136
301.073.741.82494.209.18594.209.183123.552.50323.550.62123.552.96723.553.094
312.147.483.648182.004.509182.004.507145.502.06445.496.92345.506.45345.499.069
324.294.967.296352.033.930352.033.928188.010.70888.007.77288.007.04788.008.403
338.589.934.592681.675.801681.675.7991170.420.613170.419.177170.413.318170.422.693
3417.179.869.1841.321.320.6871.321.320.6851330.330.103330.321.767330.330.144330.338.673


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
240000000
382200020
4165410122
5329810243
664231944577
712853411211131514
8256108753323283225
95122381548456586460
101.024513301212125134137117
112.0481.060616444259268269264
124.0962.2091.234975532543566568
138.1924.5222.5052.0171.0921.1111.1371.182
1416.3849.1655.0614.1042.2592.2862.2862.334
1532.76818.67310.2268.4474.6354.6734.6454.720
1665.53637.95120.60917.3429.4689.5179.5049.462
17131.07276.70941.39035.31919.20419.27319.16819.064
18262.144155.28783.45771.83038.88438.80938.81638.778
19524.288313.621167.591146.03078.39478.42178.43478.372
201.048.576633.144336.959296.185158.523158.090158.247158.284
212.097.1521.276.528677.710598.818319.421319.114319.120318.873
224.194.3042.571.5481.360.7981.210.750642.630642.523643.043643.352
238.388.6085.174.7352.730.4552.444.2801.293.5041.293.2481.294.1311.293.852
2416.777.21610.410.4565.479.8464.930.6102.601.0182.603.3442.602.6372.603.457
2533.554.43220.925.69610.991.3849.934.3125.230.5615.230.0765.232.2085.232.851
2667.108.86442.041.58622.036.63320.004.95310.509.61210.507.02910.511.78210.513.163
27134.217.72884.436.86544.174.90240.261.96321.108.07321.106.12021.110.41421.112.258
28268.435.456169.534.50088.531.37781.003.12342.382.77042.379.17542.390.30942.382.246
29536.870.912340.285.997177.419.233162.866.76485.070.15285.069.66585.078.36485.067.816
301.073.741.824682.831.314355.477.680327.353.634170.721.128170.705.833170.711.300170.693.053
312.147.483.6481.369.867.718712.144.393657.723.325342.478.947342.470.567342.470.461342.447.743
324.294.967.2962.747.571.9361.426.489.3031.321.082.633686.892.624686.907.885686.899.078686.872.349
338.589.934.5925.509.789.6062.857.057.6692.652.731.9371.377.404.5041.377.473.5991.377.469.6481.377.441.855
3417.179.869.18411.047.044.6215.721.734.9505.325.309.6712.761.737.2442.761.752.8602.761.756.9842.761.797.533


8. Check for existing Integer Sequences by OEIS

Found in Database : 17, 19, 3, 97, 139, 61, 229, 277, 109, 379, 433, 163, 547, 607, 223, 733, 47, 1, 937, 1009,
Found in Database : 17, 19, 3, 97, 139, 61, 229, 277, 109, 379, 433, 163, 547, 607, 223, 733, 47, 937, 1009, 1237, 439, 1399, 1483, 523, 1657, 1747, 613, 1933, 2029, 709, 131, 137, 811, 2539, 2647, 919, 151,
Found in Database : 3, 17, 19, 47, 61, 71, 83, 89, 97, 101, 107, 109, 113, 131, 137, 139,