Inhaltsverzeichnis

Development of
Algorithmic Constructions

00:43:14
Deutsch
29.Mar 2024

Polynom = x^2+55x-3

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) = 3 = 3
f(1) = 53 = 53
f(2) = 111 = 3*37
f(3) = 171 = 3*3*19
f(4) = 233 = 233
f(5) = 297 = 3*3*3*11
f(6) = 363 = 3*11*11
f(7) = 431 = 431
f(8) = 501 = 3*167
f(9) = 573 = 3*191
f(10) = 647 = 647
f(11) = 723 = 3*241
f(12) = 801 = 3*3*89
f(13) = 881 = 881
f(14) = 963 = 3*3*107
f(15) = 1047 = 3*349
f(16) = 1133 = 11*103
f(17) = 1221 = 3*11*37
f(18) = 1311 = 3*19*23
f(19) = 1403 = 23*61
f(20) = 1497 = 3*499
f(21) = 1593 = 3*3*3*59
f(22) = 1691 = 19*89
f(23) = 1791 = 3*3*199
f(24) = 1893 = 3*631
f(25) = 1997 = 1997
f(26) = 2103 = 3*701
f(27) = 2211 = 3*11*67
f(28) = 2321 = 11*211
f(29) = 2433 = 3*811
f(30) = 2547 = 3*3*283
f(31) = 2663 = 2663
f(32) = 2781 = 3*3*3*103
f(33) = 2901 = 3*967
f(34) = 3023 = 3023
f(35) = 3147 = 3*1049
f(36) = 3273 = 3*1091
f(37) = 3401 = 19*179
f(38) = 3531 = 3*11*107
f(39) = 3663 = 3*3*11*37
f(40) = 3797 = 3797
f(41) = 3933 = 3*3*19*23
f(42) = 4071 = 3*23*59
f(43) = 4211 = 4211
f(44) = 4353 = 3*1451
f(45) = 4497 = 3*1499
f(46) = 4643 = 4643
f(47) = 4791 = 3*1597
f(48) = 4941 = 3*3*3*3*61
f(49) = 5093 = 11*463
f(50) = 5247 = 3*3*11*53
f(51) = 5403 = 3*1801
f(52) = 5561 = 67*83
f(53) = 5721 = 3*1907
f(54) = 5883 = 3*37*53
f(55) = 6047 = 6047
f(56) = 6213 = 3*19*109
f(57) = 6381 = 3*3*709
f(58) = 6551 = 6551
f(59) = 6723 = 3*3*3*3*83
f(60) = 6897 = 3*11*11*19
f(61) = 7073 = 11*643
f(62) = 7251 = 3*2417
f(63) = 7431 = 3*2477
f(64) = 7613 = 23*331
f(65) = 7797 = 3*23*113
f(66) = 7983 = 3*3*887
f(67) = 8171 = 8171
f(68) = 8361 = 3*3*929
f(69) = 8553 = 3*2851
f(70) = 8747 = 8747
f(71) = 8943 = 3*11*271
f(72) = 9141 = 3*11*277
f(73) = 9341 = 9341
f(74) = 9543 = 3*3181
f(75) = 9747 = 3*3*3*19*19
f(76) = 9953 = 37*269
f(77) = 10161 = 3*3*1129
f(78) = 10371 = 3*3457
f(79) = 10583 = 19*557
f(80) = 10797 = 3*59*61
f(81) = 11013 = 3*3671
f(82) = 11231 = 11*1021
f(83) = 11451 = 3*11*347
f(84) = 11673 = 3*3*1297
f(85) = 11897 = 11897
f(86) = 12123 = 3*3*3*449
f(87) = 12351 = 3*23*179
f(88) = 12581 = 23*547
f(89) = 12813 = 3*4271
f(90) = 13047 = 3*4349
f(91) = 13283 = 37*359
f(92) = 13521 = 3*4507
f(93) = 13761 = 3*3*11*139
f(94) = 14003 = 11*19*67
f(95) = 14247 = 3*3*1583
f(96) = 14493 = 3*4831
f(97) = 14741 = 14741
f(98) = 14991 = 3*19*263
f(99) = 15243 = 3*5081
f(100) = 15497 = 15497

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+55x-3

f(0)=3
f(1)=53
f(2)=37
f(3)=19
f(4)=233
f(5)=11
f(6)=1
f(7)=431
f(8)=167
f(9)=191
f(10)=647
f(11)=241
f(12)=89
f(13)=881
f(14)=107
f(15)=349
f(16)=103
f(17)=1
f(18)=23
f(19)=61
f(20)=499
f(21)=59
f(22)=1
f(23)=199
f(24)=631
f(25)=1997
f(26)=701
f(27)=67
f(28)=211
f(29)=811
f(30)=283
f(31)=2663
f(32)=1
f(33)=967
f(34)=3023
f(35)=1049
f(36)=1091
f(37)=179
f(38)=1
f(39)=1
f(40)=3797
f(41)=1
f(42)=1
f(43)=4211
f(44)=1451
f(45)=1499
f(46)=4643
f(47)=1597
f(48)=1
f(49)=463
f(50)=1
f(51)=1801
f(52)=83
f(53)=1907
f(54)=1
f(55)=6047
f(56)=109
f(57)=709
f(58)=6551
f(59)=1
f(60)=1
f(61)=643
f(62)=2417
f(63)=2477
f(64)=331
f(65)=113
f(66)=887
f(67)=8171
f(68)=929
f(69)=2851
f(70)=8747
f(71)=271
f(72)=277
f(73)=9341
f(74)=3181
f(75)=1
f(76)=269
f(77)=1129
f(78)=3457
f(79)=557
f(80)=1
f(81)=3671
f(82)=1021
f(83)=347
f(84)=1297
f(85)=11897
f(86)=449
f(87)=1
f(88)=547
f(89)=4271
f(90)=4349
f(91)=359
f(92)=4507
f(93)=139
f(94)=1
f(95)=1583
f(96)=4831
f(97)=14741
f(98)=263
f(99)=5081

b) Substitution of the polynom
The polynom f(x)=x^2+55x-3 could be written as f(y)= y^2-759.25 with x=y-27.5

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+27.5
f'(x)>2x+54

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

3, 53, 37, 19, 233, 11, 1, 431, 167, 191, 647, 241, 89, 881, 107, 349, 103, 1, 23, 61, 499, 59, 1, 199, 631, 1997, 701, 67, 211, 811, 283, 2663, 1, 967, 3023, 1049, 1091, 179, 1, 1, 3797, 1, 1, 4211, 1451, 1499, 4643, 1597, 1, 463, 1, 1801, 83, 1907, 1, 6047, 109, 709, 6551, 1, 1, 643, 2417, 2477, 331, 113, 887, 8171, 929, 2851, 8747, 271, 277, 9341, 3181, 1, 269, 1129, 3457, 557, 1, 3671, 1021, 347, 1297, 11897, 449, 1, 547, 4271, 4349, 359, 4507, 139, 1, 1583, 4831, 14741, 263, 5081, 15497, 1, 593, 307, 1, 509, 151, 1, 5867, 293, 1, 1, 18701, 1, 6421, 1777, 601, 353, 20411, 1, 2333, 1, 2399, 7297, 22193, 7499, 691, 1, 1, 1, 173, 2707, 433, 1087, 367, 1, 1367, 797, 1, 457, 337, 1, 1, 9437, 9551, 1, 9781, 3299, 2731, 1, 1, 1637, 10487, 10607, 32183, 571, 1, 1447, 3739, 1031, 1, 1, 11717, 35531, 11971, 1, 36683, 1373, 12487, 37853, 1, 1171, 39041, 13147, 1, 1, 4517, 13687, 1, 607, 613, 1, 1307, 1613, 43973, 4933, 223, 1223, 15227, 809, 523, 15661, 479, 229, 1789, 16249, 49193, 16547, 1, 50543, 1, 5717, 1, 1, 1601, 239, 17921, 18077, 2879, 1, 1, 56123, 1, 1, 5233, 1759, 1, 59021, 1, 1, 60497, 251, 1, 61991, 563, 1, 1, 1123, 1, 65033, 7283, 1, 1, 22367, 22541, 68147, 2081, 1, 677, 1, 23599, 71333, 23957, 24137, 72953, 24499, 1, 6781, 1, 1097, 4013, 25601, 1, 1163, 26161, 8783, 79613, 1, 2447, 7393, 27299, 743, 1567, 1, 3119, 4463, 1, 28657, 86561, 1, 2659, 1, 1289, 9949, 827, 3361, 823, 91961, 1, 1, 8527, 2861, 10559, 95651, 1, 32299, 97523, 32717, 1733, 1117, 33349, 1, 1, 11329, 1487, 1, 34631, 34847, 2843, 35281, 11833, 1291, 1, 1, 109133, 36599, 36821, 5849, 1, 12497, 113147, 12647, 38167, 1, 3511, 733, 1987, 1709, 1, 2251, 1, 40231, 6389, 40697, 1, 1, 2179, 13879, 125621, 4679, 42349, 127763, 379, 43067, 751, 1, 1327, 2239, 14759, 1, 5839, 1, 1, 136511, 45751, 1, 12613, 1409, 46747, 1, 1277, 47501, 143261, 787, 16087, 145547, 5419, 4457, 13441, 1, 2621, 1, 1, 1, 1, 17033, 51361, 154871, 1, 1, 157247, 1, 1, 1, 17827, 911, 162053, 54287, 1, 1, 5009, 18457, 166931, 1, 1, 2777, 2467, 1, 171881, 57571, 1753, 1, 19469, 58687, 176903, 1, 1009, 179441, 3163, 6709, 181997, 1847, 1, 184571, 1, 62099, 187163, 1, 1, 1, 1, 63841, 17491, 5857, 1061, 195047, 1, 1151, 197711, 22067, 66499, 1, 1, 1, 1, 1283, 7589, 1, 1, 1033, 1, 69821, 3049, 9187, 1, 2153, 1, 7963, 1, 947, 3821, 72911, 2053, 487, 1, 1, 1, 74797, 225341, 75431, 1, 228203, 76387, 1, 3449, 1, 1, 1, 78317, 4139, 236897, 2143, 26539, 239831, 991, 80599, 2269, 1, 7417, 245753, 1, 27527, 1, 27749, 1, 4127, 1, 1, 1, 1, 9511, 1, 28759, 2341, 260873, 87299, 87641, 263951, 88327, 2687, 2207, 9929, 1, 1, 90401, 90749, 4079, 4813, 1, 276443, 2803, 1, 12157, 93557, 93911, 282797, 94621, 1, 286001, 1, 1, 26293, 1, 97127, 4957, 97849, 1723, 2617, 10993, 1, 8081, 1493, 9127, 27481, 4397, 1471, 305603, 1, 102607, 308933, 103349, 1, 312281, 9497, 1, 1, 577, 105967, 319031, 106721, 107099, 322433, 107857, 1, 2693, 1, 1229, 1, 4789, 1, 1, 111301, 1, 336221, 1973, 10259, 1, 1, 1, 2273, 114799, 12799, 346751, 1, 116371, 1, 10651, 10687, 5801, 6229, 1721, 15541, 1, 1, 2017, 120749, 121151, 33151, 11087, 40787, 6949, 41057, 1, 371951, 2347, 124799, 375623, 125617, 1, 34483, 42283, 127261, 1, 5569, 1, 386747, 1, 1, 390491, 1319, 11909, 3617, 131837, 132257, 398033, 1, 44507, 21149, 44789, 3643, 36877, 1, 136067, 1511, 1, 5087, 17971, 2003, 138637, 417203, 1, 12721, 38281, 140797, 1, 424997, 1, 142537, 1, 2351, 1, 11699, 1, 1, 4241, 1, 1, 8317, 147377, 6427, 1, 148711, 16573, 40801, 4547, 1, 12239, 151397, 151847, 1, 1483, 1, 4229, 17123, 14051, 1, 1453, 1, 7691, 156841, 52433, 2243, 2293, 6899, 25127, 1, 14551, 481571, 1, 17939, 8233, 1, 162847, 489941, 163781, 164249, 1, 15017, 55219, 1, 1, 167071, 502631, 1, 168491, 22039, 1, 1, 46471, 56957, 1, 8737, 172307, 172787, 519803, 1, 1, 7823, 5309, 15971, 2767, 176651, 9323, 2287, 1, 1609, 28277, 739, 7829, 2141, 1, 1, 546101, 1, 61007, 1879, 1, 1, 2789, 3041, 1, 2677, 2791, 1, 6337, 3307, 1, 1, 757, 3229, 573047, 1, 1, 577601, 1, 193549, 582173, 1889, 195077, 11071, 10321, 65537, 1, 1, 10429, 3947, 199181, 199697, 600641, 200731, 1, 9923, 67429, 1, 2411, 8863, 204371, 1, 205417, 3613, 619397, 1, 207517, 1, 1, 19009, 1, 210157, 70229, 633653, 70583, 212281, 3823, 1, 213881, 58477, 19541, 1, 1, 1951, 3559, 652913, 11483, 218717, 5821, 219799, 1, 1, 1, 3313, 667547, 223061, 4219, 1, 6073, 75083, 12781, 1, 20627, 8221, 1, 1, 687311, 1, 25579, 692297, 1, 231877, 63391, 1, 233549, 702323, 3847, 1, 2099, 26261, 1, 1, 238037, 1, 3433, 239731, 1, 1, 3499, 1741, 727703, 243137, 243707, 732833, 22259, 1, 737981, 82189, 247141, 39113, 248291, 248867, 748331, 13159, 1, 1, 2543, 1, 758753, 4783, 11047, 1, 255247, 1, 769247, 85667, 23417, 1, 13619, 1, 1, 260527, 1, 785123, 87433, 1, 790451, 24007, 24061, 795797, 265861, 88819, 1, 1, 1, 806543, 1, 1, 1, 24659, 90617, 43019, 91019, 1, 3593, 274871, 3319, 3079, 1, 2801, 75793, 1, 279127, 10111, 7577, 1, 1933, 12269, 1, 850247, 1, 1, 1, 1, 286499, 861353, 1, 1, 866933, 2609, 4919, 7211, 26497, 292091, 878147, 15439, 1, 883781, 98407, 1, 38671, 1, 27067, 81373, 298999, 99877, 3769, 1, 301531, 906497, 302801, 1, 8369, 27701, 1, 2557, 1, 307267, 1, 1, 1, 929501, 13499, 1, 85027, 9467, 8461, 9137, 314351, 1, 946943, 316297, 105649, 50147, 3929, 1, 87151, 1, 320861, 18199, 322171, 107609, 3581, 1, 324799, 42451, 1, 1, 1, 1, 36529, 988271, 5791, 2927, 994247, 332081, 1, 90931, 1, 1, 1006253, 1, 336757, 1, 1, 1, 5689, 340117, 1, 4049, 1, 18043, 3517, 344171, 5147, 1, 346207, 38543, 1042703, 1, 31721, 1, 1, 350981,

6. Sequence of the polynom (only primes)

3, 53, 37, 19, 233, 11, 431, 167, 191, 647, 241, 89, 881, 107, 349, 103, 23, 61, 499, 59, 199, 631, 1997, 701, 67, 211, 811, 283, 2663, 967, 3023, 1049, 1091, 179, 3797, 4211, 1451, 1499, 4643, 1597, 463, 1801, 83, 1907, 6047, 109, 709, 6551, 643, 2417, 2477, 331, 113, 887, 8171, 929, 2851, 8747, 271, 277, 9341, 3181, 269, 1129, 3457, 557, 3671, 1021, 347, 1297, 11897, 449, 547, 4271, 4349, 359, 4507, 139, 1583, 4831, 14741, 263, 5081, 15497, 593, 307, 509, 151, 5867, 293, 18701, 6421, 1777, 601, 353, 20411, 2333, 2399, 7297, 22193, 7499, 691, 173, 2707, 433, 1087, 367, 1367, 797, 457, 337, 9437, 9551, 9781, 3299, 2731, 1637, 10487, 10607, 32183, 571, 1447, 3739, 1031, 11717, 35531, 11971, 36683, 1373, 12487, 37853, 1171, 39041, 13147, 4517, 13687, 607, 613, 1307, 1613, 43973, 4933, 223, 1223, 15227, 809, 523, 15661, 479, 229, 1789, 16249, 49193, 16547, 50543, 5717, 1601, 239, 17921, 18077, 2879, 56123, 5233, 1759, 59021, 60497, 251, 61991, 563, 1123, 65033, 7283, 22367, 22541, 68147, 2081, 677, 23599, 71333, 23957, 24137, 72953, 24499, 6781, 1097, 4013, 25601, 1163, 26161, 8783, 79613, 2447, 7393, 27299, 743, 1567, 3119, 4463, 28657, 86561, 2659, 1289, 9949, 827, 3361, 823, 91961, 8527, 2861, 10559, 95651, 32299, 97523, 32717, 1733, 1117, 33349, 11329, 1487, 34631, 34847, 2843, 35281, 11833, 1291, 109133, 36599, 36821, 5849, 12497, 113147, 12647, 38167, 3511, 733, 1987, 1709, 2251, 40231, 6389, 40697, 2179, 13879, 125621, 4679, 42349, 127763, 379, 43067, 751, 1327, 2239, 14759, 5839, 136511, 45751, 12613, 1409, 46747, 1277, 47501, 143261, 787, 16087, 145547, 5419, 4457, 13441, 2621, 17033, 51361, 154871, 157247, 17827, 911, 162053, 54287, 5009, 18457, 166931, 2777, 2467, 171881, 57571, 1753, 19469, 58687, 176903, 1009, 179441, 3163, 6709, 181997, 1847, 184571, 62099, 187163, 63841, 17491, 5857, 1061, 195047, 1151, 197711, 22067, 66499, 1283, 7589, 1033, 69821, 3049, 9187, 2153, 7963, 947, 3821, 72911, 2053, 487, 74797, 225341, 75431, 228203, 76387, 3449, 78317, 4139, 236897, 2143, 26539, 239831, 991, 80599, 2269, 7417, 245753, 27527, 27749, 4127, 9511, 28759, 2341, 260873, 87299, 87641, 263951, 88327, 2687, 2207, 9929, 90401, 90749, 4079, 4813, 276443, 2803, 12157, 93557, 93911, 282797, 94621, 286001, 26293, 97127, 4957, 97849, 1723, 2617, 10993, 8081, 1493, 9127, 27481, 4397, 1471, 305603, 102607, 308933, 103349, 312281, 9497, 577, 105967, 319031, 106721, 107099, 322433, 107857, 2693, 1229, 4789, 111301, 336221, 1973, 10259, 2273, 114799, 12799, 346751, 116371, 10651, 10687, 5801, 6229, 1721, 15541, 2017, 120749, 121151, 33151, 11087, 40787, 6949, 41057, 371951, 2347, 124799, 375623, 125617, 34483, 42283, 127261, 5569, 386747, 390491, 1319, 11909, 3617, 131837, 132257, 398033, 44507, 21149, 44789, 3643, 36877, 136067, 1511, 5087, 17971, 2003, 138637, 417203, 12721, 38281, 140797, 424997, 142537, 2351, 11699, 4241, 8317, 147377, 6427, 148711, 16573, 40801, 4547, 12239, 151397, 151847, 1483, 4229, 17123, 14051, 1453, 7691, 156841, 52433, 2243, 2293, 6899, 25127, 14551, 481571, 17939, 8233, 162847, 489941, 163781, 164249, 15017, 55219, 167071, 502631, 168491, 22039, 46471, 56957, 8737, 172307, 172787, 519803, 7823, 5309, 15971, 2767, 176651, 9323, 2287, 1609, 28277, 739, 7829, 2141, 546101, 61007, 1879, 2789, 3041, 2677, 2791, 6337, 3307, 757, 3229, 573047, 577601, 193549, 582173, 1889, 195077, 11071, 10321, 65537, 10429, 3947, 199181, 199697, 600641, 200731, 9923, 67429, 2411, 8863, 204371, 205417, 3613, 619397, 207517, 19009, 210157, 70229, 633653, 70583, 212281, 3823, 213881, 58477, 19541, 1951, 3559, 652913, 11483, 218717, 5821, 219799, 3313, 667547, 223061, 4219, 6073, 75083, 12781, 20627, 8221, 687311, 25579, 692297, 231877, 63391, 233549, 702323, 3847, 2099, 26261, 238037, 3433, 239731, 3499, 1741, 727703, 243137, 243707, 732833, 22259, 737981, 82189, 247141, 39113, 248291, 248867, 748331, 13159, 2543, 758753, 4783, 11047, 255247, 769247, 85667, 23417, 13619, 260527, 785123, 87433, 790451, 24007, 24061, 795797, 265861, 88819, 806543, 24659, 90617, 43019, 91019, 3593, 274871, 3319, 3079, 2801, 75793, 279127, 10111, 7577, 1933, 12269, 850247, 286499, 861353, 866933, 2609, 4919, 7211, 26497, 292091, 878147, 15439, 883781, 98407, 38671, 27067, 81373, 298999, 99877, 3769, 301531, 906497, 302801, 8369, 27701, 2557, 307267, 929501, 13499, 85027, 9467, 8461, 9137, 314351, 946943, 316297, 105649, 50147, 3929, 87151, 320861, 18199, 322171, 107609, 3581, 324799, 42451, 36529, 988271, 5791, 2927, 994247, 332081, 90931, 1006253, 336757, 5689, 340117, 4049, 18043, 3517, 344171, 5147, 346207, 38543, 1042703, 31721, 350981,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+55x-3 and
the reducible primes which appear as divisor for the first time
p | x^2+55x-3 and p < x^2+55x-3

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)
11010551.0000000.5000000.5000000.0000000.0000000.000000
21007920590.7900000.2000000.5900007.9000004.00000011.800000
31.0006791255540.6790000.1250000.5540008.5949366.2500009.389831
410.0006.8898656.0240.6889000.0865000.60240010.1458026.92000010.873646
5100.00069.1816.71262.4690.6918100.0671200.62469010.0422417.75953810.370020
61.000.000692.70054.627638.0730.6927000.0546270.63807310.0128658.13870710.214234
710.000.0006.926.457463.0926.463.3650.6926460.0463090.6463369.9992168.47734610.129507
8100.000.00069.253.4964.013.63365.239.8630.6925350.0401360.6523999.9984028.66703110.093792
91.000.000.000692.496.58935.413.634657.082.9550.6924970.0354140.6570839.9994468.82333610.071801
1010.000.000.0006.924.919.599316.909.9086.608.009.6910.6924920.0316910.6608019.9999338.94881210.056583


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
388441.0000000.5000000.5000001.6000001.3333332.000000
416166101.0000000.3750000.6250002.0000001.5000002.500000
532288200.8750000.2500000.6250001.7500001.3333332.000000
6644914350.7656250.2187500.5468751.7500001.7500001.750000
71289623730.7500000.1796880.5703121.9591841.6428572.085714
8256180401400.7031250.1562500.5468751.8750001.7391301.917808
9512350732770.6835940.1425780.5410161.9444441.8250001.978571
101.0246961295670.6796880.1259770.5537111.9885711.7671232.046932
112.0481.4012261.1750.6840820.1103520.5737302.0129311.7519382.072310
124.0962.8343852.4490.6918950.0939940.5979002.0228411.7035402.084255
138.1925.6587264.9320.6906740.0886230.6020511.9964711.8857142.013883
1416.38411.3321.30610.0260.6916500.0797120.6119382.0028281.7988982.032847
1532.76822.6402.43820.2020.6909180.0744020.6165161.9978821.8667692.014961
1665.53645.3244.58040.7440.6915890.0698850.6217042.0019431.8785892.016830
17131.07290.7858.56882.2170.6926350.0653690.6272662.0030231.8707422.017892
18262.144181.59516.119165.4760.6927300.0614890.6312412.0002751.8813022.012674
19524.288363.20830.169333.0390.6927640.0575430.6352212.0000991.8716422.012612
201.048.576726.33857.108669.2300.6926900.0544620.6382271.9997851.8929362.009464
212.097.1521.452.934108.2681.344.6660.6928130.0516260.6411872.0003551.8958462.009273
224.194.3042.905.244206.0712.699.1730.6926640.0491310.6435331.9995701.9033422.007318
238.388.6085.810.144393.0955.417.0490.6926230.0468610.6457631.9998821.9075712.006929
2416.777.21611.619.401750.65910.868.7420.6925700.0447430.6478281.9998471.9096122.006395
2533.554.43223.237.9421.437.12321.800.8190.6925450.0428300.6497151.9999261.9144822.005827
2667.108.86446.475.5972.756.68743.718.9100.6925400.0410780.6514631.9999881.9181982.005379
27134.217.72892.947.7845.297.84587.649.9390.6925150.0394720.6530431.9999271.9218162.004852
28268.435.456185.898.15310.190.757175.707.3960.6925250.0379640.6545612.0000281.9235662.004649
29536.870.912371.782.03419.634.891352.147.1430.6924980.0365730.6559251.9999231.9267352.004168
301.073.741.824743.561.97337.887.495705.674.4780.6924960.0352850.6572111.9999941.9296002.003919
312.147.483.6481.487.120.52273.193.2601.413.927.2620.6924940.0340830.6584111.9999951.9318582.003654
324.294.967.2962.974.243.894141.561.1892.832.682.7050.6924950.0329600.6595352.0000021.9340742.003415
338.589.934.5925.948.456.890274.119.3225.674.337.5680.6924920.0319120.6605801.9999901.9364022.003167
3417.179.869.18411.896.932.058531.313.84011.365.618.2180.6924930.0309270.6615662.0000031.9382582.002986
3534.359.738.36823.793.956.0761.030.820.07722.763.135.9990.6924950.0300010.6624942.0000081.9401342.002806
3668.719.476.73647.588.148.9542.001.690.89845.586.458.0560.6924990.0291280.6633702.0000101.9418432.002644


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
122010110
243021110
384031111
4166052112
5328072123
664140132336
7128230225666
8256400391010119
95127307218201817
101.024129012830333333
112.048226022460565555
124.0963850383103989292
138.1927260724191182178175
1416.3841.30601.304333332313328
1532.7682.43802.436603611610614
1665.5364.58004.5781.1451.1531.1181.164
17131.0728.56808.5662.1462.1172.1292.176
18262.14416.119016.1174.0294.0064.0424.042
19524.28830.169030.1677.6217.5377.4267.585
201.048.57657.108057.10614.34314.34214.18914.234
212.097.152108.2680108.26627.16627.05426.96727.081
224.194.304206.0710206.06951.62051.72251.26551.464
238.388.608393.0950393.09398.28698.52998.16798.113
2416.777.216750.6590750.657187.723187.959187.881187.096
2533.554.4321.437.12301.437.121359.308359.774359.321358.720
2667.108.8642.756.68702.756.685688.760689.823689.111688.993
27134.217.7285.297.84505.297.8431.324.5851.325.8981.324.3651.322.997
28268.435.45610.190.757010.190.7552.547.8872.549.2052.547.7122.545.953
29536.870.91219.634.891019.634.8894.907.3004.911.0294.908.9894.907.573
301.073.741.82437.887.495037.887.4939.471.8759.473.3859.471.5829.470.653
312.147.483.64873.193.260073.193.25818.297.77218.296.34618.300.51718.298.625
324.294.967.296141.561.1890141.561.18735.383.76235.394.71035.393.12135.389.596
338.589.934.592274.119.3220274.119.32068.527.48468.533.87268.526.00768.531.959
3417.179.869.184531.313.8400531.313.838132.815.943132.822.783132.833.070132.842.044
3534.359.738.3681.030.820.07701.030.820.075257.694.483257.699.441257.694.451257.731.702
3668.719.476.7362.001.690.89802.001.690.896500.399.160500.414.412500.425.965500.451.361


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
384220211
41610552323
532201372945
66435201551677
712873373616241716
8256140677329413634
951227713813959716978
101.024567290277130147145145
112.0481.175602573282305299289
124.0962.4491.2601.189604627617601
138.1924.9322.5062.4261.2151.2491.2311.237
1416.38410.0265.0664.9602.5022.5562.4842.484
1532.76820.20210.2209.9824.9955.1335.0475.027
1665.53640.74420.68920.05510.17110.26310.11110.199
17131.07282.21741.65240.56520.48920.60720.55120.570
18262.144165.47683.91681.56041.08941.43741.49341.457
19524.288333.039168.834164.20582.92783.36883.27183.473
201.048.576669.230339.039330.191166.925167.497167.407167.401
212.097.1521.344.666681.324663.342335.480336.358336.573336.255
224.194.3042.699.1731.365.8791.333.294674.476675.217674.960674.520
238.388.6085.417.0492.739.3672.677.6821.353.3791.354.7961.354.9941.353.880
2416.777.21610.868.7425.495.7515.372.9912.715.2142.717.8372.718.6012.717.090
2533.554.43221.800.81911.013.35310.787.4665.450.3965.449.8275.449.3965.451.200
2667.108.86443.718.91022.075.34621.643.56410.932.77810.928.65110.929.02510.928.456
27134.217.72887.649.93944.239.05643.410.88321.911.63821.917.73921.912.77221.907.790
28268.435.456175.707.39688.640.07687.067.32043.929.59443.928.04943.929.68043.920.073
29536.870.912352.147.143177.594.585174.552.55888.039.22388.040.28488.034.42488.033.212
301.073.741.824705.674.478355.762.044349.912.434176.427.492176.407.883176.411.494176.427.609
312.147.483.6481.413.927.262712.631.038701.296.224353.502.163353.459.934353.468.718353.496.447
324.294.967.2962.832.682.7051.427.285.6511.405.397.054708.169.903708.171.136708.155.924708.185.742
338.589.934.5925.674.337.5682.858.324.4112.816.013.1571.418.584.8891.418.546.9621.418.567.8261.418.637.891
3417.179.869.18411.365.618.2185.723.782.6535.641.835.5652.841.391.9252.841.414.8222.841.397.7732.841.413.698
3534.359.738.36822.763.135.99911.460.855.19011.302.280.8095.690.791.9025.690.808.5625.690.709.9955.690.825.540
3668.719.476.73645.586.458.05622.946.999.95022.639.458.10611.396.652.53411.396.664.57311.396.553.64111.396.587.308


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 53, 37, 19, 233, 11, 1, 431, 167, 191, 647, 241, 89, 881, 107, 349, 103, 1, 23, 61,
Found in Database : 3, 53, 37, 19, 233, 11, 431, 167, 191, 647, 241, 89, 881, 107, 349, 103, 23, 61, 499, 59, 199, 631, 1997, 701, 67, 211, 811, 283, 2663, 967, 3023, 1049, 1091, 179,
Found in Database : 3, 11, 19, 23, 37, 53, 59, 61, 67, 83, 89, 103, 107, 109, 113, 139,