Inhaltsverzeichnis

Development of
Algorithmic Constructions

15:06:03
Deutsch
29.Mar 2024

Polynom = x^2+52x-79

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) = 79 = 79
f(1) = 13 = 13
f(2) = 29 = 29
f(3) = 43 = 43
f(4) = 145 = 5*29
f(5) = 103 = 103
f(6) = 269 = 269
f(7) = 167 = 167
f(8) = 401 = 401
f(9) = 235 = 5*47
f(10) = 541 = 541
f(11) = 307 = 307
f(12) = 689 = 13*53
f(13) = 383 = 383
f(14) = 845 = 5*13*13
f(15) = 463 = 463
f(16) = 1009 = 1009
f(17) = 547 = 547
f(18) = 1181 = 1181
f(19) = 635 = 5*127
f(20) = 1361 = 1361
f(21) = 727 = 727
f(22) = 1549 = 1549
f(23) = 823 = 823
f(24) = 1745 = 5*349
f(25) = 923 = 13*71
f(26) = 1949 = 1949
f(27) = 1027 = 13*79
f(28) = 2161 = 2161
f(29) = 1135 = 5*227
f(30) = 2381 = 2381
f(31) = 1247 = 29*43
f(32) = 2609 = 2609
f(33) = 1363 = 29*47
f(34) = 2845 = 5*569
f(35) = 1483 = 1483
f(36) = 3089 = 3089
f(37) = 1607 = 1607
f(38) = 3341 = 13*257
f(39) = 1735 = 5*347
f(40) = 3601 = 13*277
f(41) = 1867 = 1867
f(42) = 3869 = 53*73
f(43) = 2003 = 2003
f(44) = 4145 = 5*829
f(45) = 2143 = 2143
f(46) = 4429 = 43*103
f(47) = 2287 = 2287
f(48) = 4721 = 4721
f(49) = 2435 = 5*487
f(50) = 5021 = 5021
f(51) = 2587 = 13*199
f(52) = 5329 = 73*73
f(53) = 2743 = 13*211
f(54) = 5645 = 5*1129
f(55) = 2903 = 2903
f(56) = 5969 = 47*127
f(57) = 3067 = 3067
f(58) = 6301 = 6301
f(59) = 3235 = 5*647
f(60) = 6641 = 29*229
f(61) = 3407 = 3407
f(62) = 6989 = 29*241
f(63) = 3583 = 3583
f(64) = 7345 = 5*13*113
f(65) = 3763 = 53*71
f(66) = 7709 = 13*593
f(67) = 3947 = 3947
f(68) = 8081 = 8081
f(69) = 4135 = 5*827
f(70) = 8461 = 8461
f(71) = 4327 = 4327
f(72) = 8849 = 8849
f(73) = 4523 = 4523
f(74) = 9245 = 5*43*43
f(75) = 4723 = 4723
f(76) = 9649 = 9649
f(77) = 4927 = 13*379
f(78) = 10061 = 10061
f(79) = 5135 = 5*13*79
f(80) = 10481 = 47*223
f(81) = 5347 = 5347
f(82) = 10909 = 10909
f(83) = 5563 = 5563
f(84) = 11345 = 5*2269
f(85) = 5783 = 5783
f(86) = 11789 = 11789
f(87) = 6007 = 6007
f(88) = 12241 = 12241
f(89) = 6235 = 5*29*43
f(90) = 12701 = 13*977
f(91) = 6467 = 29*223
f(92) = 13169 = 13*1013
f(93) = 6703 = 6703
f(94) = 13645 = 5*2729
f(95) = 6943 = 53*131
f(96) = 14129 = 71*199
f(97) = 7187 = 7187
f(98) = 14621 = 14621
f(99) = 7435 = 5*1487
f(100) = 15121 = 15121

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+52x-79

f(0)=79
f(1)=13
f(2)=29
f(3)=43
f(4)=5
f(5)=103
f(6)=269
f(7)=167
f(8)=401
f(9)=47
f(10)=541
f(11)=307
f(12)=53
f(13)=383
f(14)=1
f(15)=463
f(16)=1009
f(17)=547
f(18)=1181
f(19)=127
f(20)=1361
f(21)=727
f(22)=1549
f(23)=823
f(24)=349
f(25)=71
f(26)=1949
f(27)=1
f(28)=2161
f(29)=227
f(30)=2381
f(31)=1
f(32)=2609
f(33)=1
f(34)=569
f(35)=1483
f(36)=3089
f(37)=1607
f(38)=257
f(39)=347
f(40)=277
f(41)=1867
f(42)=73
f(43)=2003
f(44)=829
f(45)=2143
f(46)=1
f(47)=2287
f(48)=4721
f(49)=487
f(50)=5021
f(51)=199
f(52)=1
f(53)=211
f(54)=1129
f(55)=2903
f(56)=1
f(57)=3067
f(58)=6301
f(59)=647
f(60)=229
f(61)=3407
f(62)=241
f(63)=3583
f(64)=113
f(65)=1
f(66)=593
f(67)=3947
f(68)=8081
f(69)=827
f(70)=8461
f(71)=4327
f(72)=8849
f(73)=4523
f(74)=1
f(75)=4723
f(76)=9649
f(77)=379
f(78)=10061
f(79)=1
f(80)=223
f(81)=5347
f(82)=10909
f(83)=5563
f(84)=2269
f(85)=5783
f(86)=11789
f(87)=6007
f(88)=12241
f(89)=1
f(90)=977
f(91)=1
f(92)=1013
f(93)=6703
f(94)=2729
f(95)=131
f(96)=1
f(97)=7187
f(98)=14621
f(99)=1487

b) Substitution of the polynom
The polynom f(x)=x^2+52x-79 could be written as f(y)= y^2-755 with x=y-26

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+26
f'(x)>2x+51

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

79, 13, 29, 43, 5, 103, 269, 167, 401, 47, 541, 307, 53, 383, 1, 463, 1009, 547, 1181, 127, 1361, 727, 1549, 823, 349, 71, 1949, 1, 2161, 227, 2381, 1, 2609, 1, 569, 1483, 3089, 1607, 257, 347, 277, 1867, 73, 2003, 829, 2143, 1, 2287, 4721, 487, 5021, 199, 1, 211, 1129, 2903, 1, 3067, 6301, 647, 229, 3407, 241, 3583, 113, 1, 593, 3947, 8081, 827, 8461, 4327, 8849, 4523, 1, 4723, 9649, 379, 10061, 1, 223, 5347, 10909, 5563, 2269, 5783, 11789, 6007, 12241, 1, 977, 1, 1013, 6703, 2729, 131, 1, 7187, 14621, 1487, 15121, 7687, 15629, 1, 3229, 631, 1, 8467, 1, 1747, 157, 9007, 18289, 9283, 3769, 1, 1493, 1, 1, 2027, 709, 10427, 21149, 10723, 4349, 151, 22349, 1, 22961, 179, 23581, 919, 563, 12263, 4969, 12583, 359, 12907, 26141, 2647, 26801, 13567, 2113, 13903, 433, 14243, 1, 503, 557, 1, 643, 15287, 197, 15643, 6329, 1231, 32369, 1259, 419, 3347, 787, 17107, 34589, 17483, 7069, 17863, 36109, 1, 2837, 3727, 2897, 1, 38449, 19423, 1, 461, 1381, 1, 1409, 4127, 41681, 1619, 42509, 1, 8669, 1, 44189, 22307, 617, 4547, 233, 23167, 46769, 23603, 733, 24043, 3733, 521, 677, 4987, 50321, 479, 51229, 601, 10429, 907, 53069, 1, 54001, 1, 54941, 1, 55889, 28183, 11369, 28663, 57809, 29147, 1367, 5927, 4597, 641, 4673, 271, 1, 31123, 1, 31627, 63761, 6427, 64781, 32647, 65809, 2551, 1, 2591, 2341, 1, 971, 6947, 70001, 35267, 71069, 35803, 1, 36343, 1, 36887, 5717, 7487, 1, 37987, 743, 38543, 293, 39103, 1, 39667, 79901, 619, 81041, 1, 82189, 1427, 1, 1447, 84509, 1, 1823, 8627, 86861, 599, 1, 1, 1373, 1, 1, 859, 1291, 9227, 317, 46747, 94109, 47363, 19069, 3691, 96589, 3739, 97841, 1, 877, 1061, 3461, 50503, 701, 1, 102929, 51787, 8017, 10487, 8117, 53087, 337, 1, 1, 54403, 109469, 1039, 1, 1, 883, 4339, 113489, 4391, 1, 1229, 1471, 1, 1, 11827, 353, 2063, 120349, 2087, 1873, 61223, 9473, 61927, 124561, 12527, 2377, 63347, 2963, 64063, 1, 64783, 1153, 5039, 2803, 1019, 133201, 1, 134669, 857, 373, 68443, 1, 1609, 139121, 1, 1, 70687, 1, 71443, 28729, 1, 811, 1, 146701, 14747, 2797, 74507, 3187, 5791, 30269, 5851, 152909, 76847, 1, 15527, 156061, 78427, 157649, 1, 31849, 1861, 12373, 1, 12497, 1, 2311, 2843, 165709, 1571, 33469, 1789, 1, 431, 170641, 1319, 4007, 6659, 173969, 87403, 35129, 1117, 1, 89087, 179021, 17987, 1423, 90787, 14033, 1, 2833, 92503, 185869, 1279, 6469, 1, 6529, 95107, 191089, 1811, 38569, 7451, 194609, 1, 4567, 19727, 2791, 99527, 2531, 1, 1, 101323, 203549, 1, 15797, 20627, 15937, 104047, 4447, 2441, 42169, 105883, 4013, 1, 214541, 1, 216401, 1, 1307, 8431, 44029, 110543, 222029, 111487, 223921, 1, 225821, 1597, 227729, 114343, 3533, 115303, 1, 499, 2267, 23447, 235441, 2749, 1, 119183, 47869, 120163, 1, 9319, 8389, 1879, 245261, 123127, 247249, 124123, 1, 1, 5843, 126127, 1, 1, 1, 128147, 19793, 1, 51869, 130183, 261389, 1, 2011, 1, 3637, 1877, 267569, 10331, 1, 1, 271729, 4703, 273821, 27487, 275921, 1753, 278029, 2969, 1303, 140603, 21713, 141667, 1, 28547, 286541, 143807, 288689, 144883, 58169, 145963, 2593, 147047, 1289, 1, 2887, 1, 4219, 150323, 2081, 151423, 1, 1201, 1451, 30727, 1, 154747, 23893, 155863, 4813, 1, 1, 1, 1171, 31847, 319601, 160367, 6073, 161503, 1, 12511, 1657, 1, 328721, 32987, 7043, 1, 1297, 1, 67129, 5807, 337969, 169567, 26177, 34147, 26357, 1669, 1, 2371, 1, 174263, 349709, 175447, 352081, 35327, 354461, 13679, 6733, 1, 71849, 1, 361649, 1, 2411, 36527, 4639, 1, 12721, 185063, 1, 186283, 28753, 187507, 376241, 37747, 8807, 189967, 381169, 2693, 1, 3631, 386129, 1, 388621, 2999, 1723, 196187, 393629, 1, 79229, 4621, 398669, 199967, 401201, 1, 2389, 6983, 31253, 7027, 81769, 205063, 411409, 206347, 413981, 1, 8863, 208927, 1, 1, 84349, 1, 5813, 212827, 1, 1, 429581, 215447, 5471, 1, 86969, 218083, 1, 219407, 33857, 1, 15269, 222067, 15361, 223403, 1907, 224743, 450829, 226087, 1, 3499, 1, 17599, 458929, 230143, 1, 231503, 1993, 691, 467101, 1, 469841, 235607, 36353, 236983, 1, 238363, 2671, 5101, 480881, 1663, 1, 8363, 486449, 1, 97849, 1, 11443, 18979, 9337, 49627, 1, 1, 500509, 250963, 100669, 252383, 2399, 719, 39157, 51047, 1, 1, 514769, 1, 103529, 1, 520529, 260987, 18049, 1, 18149, 1, 529229, 20411, 1499, 1, 1, 268267, 538001, 739, 2383, 3433, 543889, 272683, 1, 274163, 42293, 1, 1, 1, 555761, 278627, 1601, 751, 112349, 281623, 5483, 1, 567761, 1, 570781, 5399, 573809, 1, 2683, 4073, 1, 290707, 1, 1, 45077, 293767, 1, 1, 118429, 296843, 595229, 1, 1, 1, 4591, 301487, 604529, 23311, 2293, 23431, 21061, 306167, 21169, 61547, 2767, 6581, 14423, 310883, 124669, 1, 48193, 3049, 48437, 63127, 4831, 317227, 8713, 318823, 127849, 4513, 3847, 7489, 645661, 1, 4133, 1, 8933, 326863, 2473, 1, 658589, 11383, 5857, 66347, 8419, 333367, 51413, 1, 10333, 336643, 674929, 338287, 678221, 67987, 681521, 341587, 684829, 343243, 1, 1, 9739, 1, 14783, 1, 1741, 349907, 701489, 1, 4861, 353263, 24421, 4493, 1, 71327, 1, 358327, 718349, 360023, 144349, 361723, 725149, 1, 728561, 1, 731981, 28219, 15647, 28351, 147769, 1, 742289, 7019, 745741, 74747, 2557, 375467, 4507, 13007, 11633, 1, 1, 1913, 17747, 76487, 3181, 384187, 770129, 385943, 1, 1, 777169, 29959, 2543, 1, 14797, 393007, 1, 9181, 158269, 396563, 2713, 398347, 798481, 1, 1, 1, 2137, 1, 5581, 1, 812849, 5737, 1, 1, 5431, 3137, 823709, 31751, 165469, 31891, 830989, 416407, 834641, 1, 15817, 9769, 4231, 2357, 169129, 2699, 1, 2017, 65617, 85487, 856721, 1, 18307, 14867, 172829, 433003, 20183, 434867, 6863, 6719, 875341, 33739, 879089, 8311, 1, 442363, 886609, 6257, 12197, 89227, 894161, 448027, 69073, 1931, 13873, 451823, 19267, 1, 12457, 91127, 31489, 457547, 1, 459463, 4283, 35491, 2917, 1, 3613, 93047, 4733, 1, 1, 1, 188029, 3709, 944029, 1, 1, 1, 1, 476887, 955729, 478843, 191929, 480803, 1, 482767, 967501, 3343, 971441, 1, 975389, 37591, 195869, 490663, 18553, 492647, 8737, 98927, 991261, 1, 4463, 1, 15373, 6337, 1, 11689, 5113, 100927, 1011281, 506647, 1015309, 2281, 203869, 510683, 1023389, 39439, 1, 7919, 35569, 7079, 24083, 518803, 1, 1, 1, 522887, 1047821, 104987,

6. Sequence of the polynom (only primes)

79, 13, 29, 43, 5, 103, 269, 167, 401, 47, 541, 307, 53, 383, 463, 1009, 547, 1181, 127, 1361, 727, 1549, 823, 349, 71, 1949, 2161, 227, 2381, 2609, 569, 1483, 3089, 1607, 257, 347, 277, 1867, 73, 2003, 829, 2143, 2287, 4721, 487, 5021, 199, 211, 1129, 2903, 3067, 6301, 647, 229, 3407, 241, 3583, 113, 593, 3947, 8081, 827, 8461, 4327, 8849, 4523, 4723, 9649, 379, 10061, 223, 5347, 10909, 5563, 2269, 5783, 11789, 6007, 12241, 977, 1013, 6703, 2729, 131, 7187, 14621, 1487, 15121, 7687, 15629, 3229, 631, 8467, 1747, 157, 9007, 18289, 9283, 3769, 1493, 2027, 709, 10427, 21149, 10723, 4349, 151, 22349, 22961, 179, 23581, 919, 563, 12263, 4969, 12583, 359, 12907, 26141, 2647, 26801, 13567, 2113, 13903, 433, 14243, 503, 557, 643, 15287, 197, 15643, 6329, 1231, 32369, 1259, 419, 3347, 787, 17107, 34589, 17483, 7069, 17863, 36109, 2837, 3727, 2897, 38449, 19423, 461, 1381, 1409, 4127, 41681, 1619, 42509, 8669, 44189, 22307, 617, 4547, 233, 23167, 46769, 23603, 733, 24043, 3733, 521, 677, 4987, 50321, 479, 51229, 601, 10429, 907, 53069, 54001, 54941, 55889, 28183, 11369, 28663, 57809, 29147, 1367, 5927, 4597, 641, 4673, 271, 31123, 31627, 63761, 6427, 64781, 32647, 65809, 2551, 2591, 2341, 971, 6947, 70001, 35267, 71069, 35803, 36343, 36887, 5717, 7487, 37987, 743, 38543, 293, 39103, 39667, 79901, 619, 81041, 82189, 1427, 1447, 84509, 1823, 8627, 86861, 599, 1373, 859, 1291, 9227, 317, 46747, 94109, 47363, 19069, 3691, 96589, 3739, 97841, 877, 1061, 3461, 50503, 701, 102929, 51787, 8017, 10487, 8117, 53087, 337, 54403, 109469, 1039, 883, 4339, 113489, 4391, 1229, 1471, 11827, 353, 2063, 120349, 2087, 1873, 61223, 9473, 61927, 124561, 12527, 2377, 63347, 2963, 64063, 64783, 1153, 5039, 2803, 1019, 133201, 134669, 857, 373, 68443, 1609, 139121, 70687, 71443, 28729, 811, 146701, 14747, 2797, 74507, 3187, 5791, 30269, 5851, 152909, 76847, 15527, 156061, 78427, 157649, 31849, 1861, 12373, 12497, 2311, 2843, 165709, 1571, 33469, 1789, 431, 170641, 1319, 4007, 6659, 173969, 87403, 35129, 1117, 89087, 179021, 17987, 1423, 90787, 14033, 2833, 92503, 185869, 1279, 6469, 6529, 95107, 191089, 1811, 38569, 7451, 194609, 4567, 19727, 2791, 99527, 2531, 101323, 203549, 15797, 20627, 15937, 104047, 4447, 2441, 42169, 105883, 4013, 214541, 216401, 1307, 8431, 44029, 110543, 222029, 111487, 223921, 225821, 1597, 227729, 114343, 3533, 115303, 499, 2267, 23447, 235441, 2749, 119183, 47869, 120163, 9319, 8389, 1879, 245261, 123127, 247249, 124123, 5843, 126127, 128147, 19793, 51869, 130183, 261389, 2011, 3637, 1877, 267569, 10331, 271729, 4703, 273821, 27487, 275921, 1753, 278029, 2969, 1303, 140603, 21713, 141667, 28547, 286541, 143807, 288689, 144883, 58169, 145963, 2593, 147047, 1289, 2887, 4219, 150323, 2081, 151423, 1201, 1451, 30727, 154747, 23893, 155863, 4813, 1171, 31847, 319601, 160367, 6073, 161503, 12511, 1657, 328721, 32987, 7043, 1297, 67129, 5807, 337969, 169567, 26177, 34147, 26357, 1669, 2371, 174263, 349709, 175447, 352081, 35327, 354461, 13679, 6733, 71849, 361649, 2411, 36527, 4639, 12721, 185063, 186283, 28753, 187507, 376241, 37747, 8807, 189967, 381169, 2693, 3631, 386129, 388621, 2999, 1723, 196187, 393629, 79229, 4621, 398669, 199967, 401201, 2389, 6983, 31253, 7027, 81769, 205063, 411409, 206347, 413981, 8863, 208927, 84349, 5813, 212827, 429581, 215447, 5471, 86969, 218083, 219407, 33857, 15269, 222067, 15361, 223403, 1907, 224743, 450829, 226087, 3499, 17599, 458929, 230143, 231503, 1993, 691, 467101, 469841, 235607, 36353, 236983, 238363, 2671, 5101, 480881, 1663, 8363, 486449, 97849, 11443, 18979, 9337, 49627, 500509, 250963, 100669, 252383, 2399, 719, 39157, 51047, 514769, 103529, 520529, 260987, 18049, 18149, 529229, 20411, 1499, 268267, 538001, 739, 2383, 3433, 543889, 272683, 274163, 42293, 555761, 278627, 1601, 751, 112349, 281623, 5483, 567761, 570781, 5399, 573809, 2683, 4073, 290707, 45077, 293767, 118429, 296843, 595229, 4591, 301487, 604529, 23311, 2293, 23431, 21061, 306167, 21169, 61547, 2767, 6581, 14423, 310883, 124669, 48193, 3049, 48437, 63127, 4831, 317227, 8713, 318823, 127849, 4513, 3847, 7489, 645661, 4133, 8933, 326863, 2473, 658589, 11383, 5857, 66347, 8419, 333367, 51413, 10333, 336643, 674929, 338287, 678221, 67987, 681521, 341587, 684829, 343243, 9739, 14783, 1741, 349907, 701489, 4861, 353263, 24421, 4493, 71327, 358327, 718349, 360023, 144349, 361723, 725149, 728561, 731981, 28219, 15647, 28351, 147769, 742289, 7019, 745741, 74747, 2557, 375467, 4507, 13007, 11633, 1913, 17747, 76487, 3181, 384187, 770129, 385943, 777169, 29959, 2543, 14797, 393007, 9181, 158269, 396563, 2713, 398347, 798481, 2137, 5581, 812849, 5737, 5431, 3137, 823709, 31751, 165469, 31891, 830989, 416407, 834641, 15817, 9769, 4231, 2357, 169129, 2699, 2017, 65617, 85487, 856721, 18307, 14867, 172829, 433003, 20183, 434867, 6863, 6719, 875341, 33739, 879089, 8311, 442363, 886609, 6257, 12197, 89227, 894161, 448027, 69073, 1931, 13873, 451823, 19267, 12457, 91127, 31489, 457547, 459463, 4283, 35491, 2917, 3613, 93047, 4733, 188029, 3709, 944029, 476887, 955729, 478843, 191929, 480803, 482767, 967501, 3343, 971441, 975389, 37591, 195869, 490663, 18553, 492647, 8737, 98927, 991261, 4463, 15373, 6337, 11689, 5113, 100927, 1011281, 506647, 1015309, 2281, 203869, 510683, 1023389, 39439, 7919, 35569, 7079, 24083, 518803, 522887, 1047821, 104987,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+52x-79 and
the reducible primes which appear as divisor for the first time
p | x^2+52x-79 and p < x^2+52x-79

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)
11011651.1000000.6000001.1000000.0000000.0000000.000000
21008528570.8500000.2800000.8500007.7272734.66666711.400000
31.0007741716030.7740000.1710000.7740009.1058836.10714310.578947
410.0007.4811.2646.2170.7481000.1264000.7481009.6653757.39181310.310116
5100.00073.6089.79663.8120.7360800.0979600.7360809.8393267.75000010.264114
61.000.000728.38380.391647.9920.7283830.0803910.7283839.8954328.20651210.154704
710.000.0007.232.188678.8976.553.2910.7232190.0678900.7232199.9291008.44493810.113228
8100.000.00071.922.5655.889.56266.033.0030.7192260.0588960.7192269.9447878.67519210.076312
91.000.000.000716.126.58851.993.516664.133.0720.7161270.0519940.7161279.9569118.82807810.057592
1010.000.000.0007.137.049.065465.276.9696.671.772.0960.7137050.0465280.7137059.9661848.94875010.045836


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)
123211.5000001.0000000.5000000.0000000.0000000.000000
245321.2500000.7500000.5000001.6666671.5000002.000000
389541.1250000.6250000.5000001.8000001.6666672.000000
41616791.0000000.4375000.5625001.7777781.4000002.250000
5323014160.9375000.4375000.5000001.8750002.0000001.777778
6645618380.8750000.2812500.5937501.8666671.2857142.375000
712810533720.8203120.2578120.5625001.8750001.8333331.894737
8256209561530.8164060.2187500.5976561.9904761.6969702.125000
95124051013040.7910160.1972660.5937501.9377991.8035711.986928
101.0247921736190.7734380.1689450.6044921.9555561.7128712.036184
112.0481.5653221.2430.7641600.1572270.6069341.9760101.8612722.008078
124.0963.1085842.5240.7587890.1425780.6162111.9859421.8136652.030571
138.1926.1481.0675.0810.7504880.1302490.6202391.9781211.8270552.013074
1416.38412.2081.96610.2420.7451170.1199950.6251221.9856861.8425492.015745
1532.76824.2453.61720.6280.7398990.1103820.6295171.9859931.8397762.014060
1665.53648.3346.70541.6290.7375180.1023100.6352081.9935661.8537462.018082
17131.07296.37212.55483.8180.7352600.0957790.6394811.9938761.8723342.013452
18262.144192.04523.576168.4690.7325940.0899350.6426581.9927471.8779672.009938
19524.288382.82644.394338.4320.7301830.0846750.6455081.9934181.8830172.008868
201.048.576763.67183.992679.6790.7282930.0801010.6481921.9948251.8919672.008318
212.097.1521.523.941158.9151.365.0260.7266720.0757770.6508951.9955461.8920252.008339
224.194.3043.040.897302.7592.738.1380.7250060.0721830.6528231.9954171.9051632.005924
238.388.6086.069.845576.3955.493.4500.7235820.0687120.6548701.9960711.9038082.006272
2416.777.21612.117.0631.101.36611.015.6970.7222330.0656470.6565871.9962721.9107832.005242
2533.554.43224.191.9882.110.20422.081.7840.7209770.0628890.6580881.9965221.9159882.004574
2667.108.86448.307.0254.044.49744.262.5280.7198310.0602680.6595631.9968191.9166382.004482
27134.217.72896.473.2967.774.43888.698.8580.7187820.0579240.6608581.9970861.9222262.003927
28268.435.456192.681.40714.960.648177.720.7590.7177940.0557330.6620611.9972511.9243382.003642
29536.870.912384.869.91328.830.301356.039.6120.7168760.0537010.6631751.9974421.9270762.003366
301.073.741.824768.845.07655.628.131713.216.9450.7160430.0518080.6642351.9976751.9295022.003196
312.147.483.6481.536.023.777107.462.6381.428.561.1390.7152670.0500410.6652261.9978331.9318042.002983
324.294.967.2963.068.897.697207.854.6002.861.043.0970.7145330.0483950.6661391.9979491.9342032.002745
338.589.934.5926.131.904.011402.459.0925.729.444.9190.7138480.0468520.6669951.9980801.9362532.002572
3417.179.869.18412.252.799.793780.094.47611.472.705.3170.7132070.0454070.6677991.9982051.9383202.002411


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
122110011
243211011
385232021
4167433031
53214686071
664187118091
7128331320150171
8256561937270281
95121013566480521
101.02417362111870851
112.04832211620615201691
124.09658419838627303101
138.1921.06734971851605501
1416.3841.9666521.31494301.0221
1532.7683.6171.2112.4061.78701.8291
1665.5366.7052.2384.4673.33303.3711
17131.07212.5544.1918.3636.24506.3081
18262.14423.5767.79715.77911.827011.7481
19524.28844.39414.67129.72322.239022.1541
201.048.57683.99227.83656.15641.986042.0051
212.097.152158.91552.804106.11179.248079.6661
224.194.304302.759100.699202.060151.0540151.7041
238.388.608576.395191.623384.772287.8740288.5201
2416.777.2161.101.366366.891734.475550.5780550.7871
2533.554.4322.110.204703.1811.407.0231.054.30801.055.8951
2667.108.8644.044.4971.347.4862.697.0112.021.71402.022.7821
27134.217.7287.774.4382.592.4605.181.9783.886.75603.887.6811
28268.435.45614.960.6484.987.1459.973.5037.479.52407.481.1231
29536.870.91228.830.3019.610.05919.220.24214.414.935014.415.3651
301.073.741.82455.628.13118.541.86037.086.27127.810.573027.817.5571
312.147.483.648107.462.63835.822.33771.640.30153.731.237053.731.4001
324.294.967.296207.854.60069.285.798138.568.802103.936.6230103.917.9761
338.589.934.592402.459.092134.161.631268.297.461201.234.2120201.224.8791
3417.179.869.184780.094.476260.029.550520.064.926390.035.2810390.059.1941


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
121100010
242200110
384310112
4169540225
532161060439
664382414410618
71287244288241228
8256153846920512557
95123041681364210352107
101.024619355264104194109212
112.0481.243695548211397241394
124.0962.5241.3601.164480800480764
138.1925.0812.7372.3449951.5589821.546
1416.38410.2425.4124.8302.0193.0992.0493.075
1532.76820.62810.9039.7254.1516.1464.1956.136
1665.53641.62921.92219.7078.57312.2088.69612.152
17131.07283.81844.01739.80117.58924.37917.65024.200
18262.144168.46988.20780.26236.01948.32235.82048.308
19524.288338.432176.458161.97472.90096.38573.04196.106
201.048.576679.679353.697325.982148.051192.043147.900191.685
212.097.1521.365.026708.859656.167299.626382.530300.359382.511
224.194.3042.738.1381.420.3741.317.764605.804762.603606.158763.573
238.388.6085.493.4502.845.0002.648.4501.224.0591.523.1891.224.1531.522.049
2416.777.21611.015.6975.695.5595.320.1382.470.4783.038.6762.468.8273.037.716
2533.554.43222.081.78411.397.33210.684.4524.977.4456.062.5594.976.0376.065.743
2667.108.86444.262.52822.814.05321.448.47510.023.92012.105.00110.025.37812.108.229
27134.217.72888.698.85845.663.19243.035.66620.177.25924.170.00520.174.44724.177.147
28268.435.456177.720.75991.392.84686.327.91340.588.57948.264.28240.591.21948.276.679
29536.870.912356.039.612182.904.904173.134.70881.606.23196.398.38481.624.02396.410.974
301.073.741.824713.216.945366.003.665347.213.280164.044.209192.565.109164.042.851192.564.776
312.147.483.6481.428.561.139732.431.857696.129.282329.615.481384.658.632329.637.065384.649.961
324.294.967.2962.861.043.0971.465.632.7191.395.410.378662.076.121768.440.828662.096.765768.429.383
338.589.934.5925.729.444.9192.932.697.8902.796.747.0291.329.467.3371.535.276.0741.329.487.9281.535.213.580
3417.179.869.18411.472.705.3175.868.051.1765.604.654.1412.668.877.5923.067.505.6822.668.889.7093.067.432.334


8. Check for existing Integer Sequences by OEIS

Found in Database : 79, 13, 29, 43, 5, 103, 269, 167, 401, 47, 541, 307, 53, 383, 1, 463, 1009, 547, 1181, 127,
Found in Database : 79, 13, 29, 43, 5, 103, 269, 167, 401, 47, 541, 307, 53, 383, 463, 1009, 547, 1181, 127, 1361, 727, 1549, 823, 349, 71, 1949, 2161, 227, 2381, 2609, 569, 1483, 3089, 1607, 257, 347,
Found in Database : 5, 13, 29, 43, 47, 53, 71, 73, 79, 103, 113, 127, 131,