Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:49:42
Deutsch
28.Mar 2024

Polynom = x^2-16x+191

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) = 191 = 191
f(1) = 11 = 11
f(2) = 163 = 163
f(3) = 19 = 19
f(4) = 143 = 11*13
f(5) = 17 = 17
f(6) = 131 = 131
f(7) = 1 = 1
f(8) = 127 = 127
f(9) = 1 = 1
f(10) = 131 = 131
f(11) = 17 = 17
f(12) = 143 = 11*13
f(13) = 19 = 19
f(14) = 163 = 163
f(15) = 11 = 11
f(16) = 191 = 191
f(17) = 13 = 13
f(18) = 227 = 227
f(19) = 31 = 31
f(20) = 271 = 271
f(21) = 37 = 37
f(22) = 323 = 17*19
f(23) = 11 = 11
f(24) = 383 = 383
f(25) = 13 = 13
f(26) = 451 = 11*41
f(27) = 61 = 61
f(28) = 527 = 17*31
f(29) = 71 = 71
f(30) = 611 = 13*47
f(31) = 41 = 41
f(32) = 703 = 19*37
f(33) = 47 = 47
f(34) = 803 = 11*73
f(35) = 107 = 107
f(36) = 911 = 911
f(37) = 121 = 11*11
f(38) = 1027 = 13*79
f(39) = 17 = 17
f(40) = 1151 = 1151
f(41) = 19 = 19
f(42) = 1283 = 1283
f(43) = 169 = 13*13
f(44) = 1423 = 1423
f(45) = 187 = 11*17
f(46) = 1571 = 1571
f(47) = 103 = 103
f(48) = 1727 = 11*157
f(49) = 113 = 113
f(50) = 1891 = 31*61
f(51) = 247 = 13*19
f(52) = 2063 = 2063
f(53) = 269 = 269
f(54) = 2243 = 2243
f(55) = 73 = 73
f(56) = 2431 = 11*13*17
f(57) = 79 = 79
f(58) = 2627 = 37*71
f(59) = 341 = 11*31
f(60) = 2831 = 19*149
f(61) = 367 = 367
f(62) = 3043 = 17*179
f(63) = 197 = 197
f(64) = 3263 = 13*251
f(65) = 211 = 211
f(66) = 3491 = 3491
f(67) = 451 = 11*41
f(68) = 3727 = 3727
f(69) = 481 = 13*37
f(70) = 3971 = 11*19*19
f(71) = 1 = 1
f(72) = 4223 = 41*103
f(73) = 17 = 17
f(74) = 4483 = 4483
f(75) = 577 = 577
f(76) = 4751 = 4751
f(77) = 611 = 13*47
f(78) = 5027 = 11*457
f(79) = 323 = 17*19
f(80) = 5311 = 47*113
f(81) = 341 = 11*31
f(82) = 5603 = 13*431
f(83) = 719 = 719
f(84) = 5903 = 5903
f(85) = 757 = 757
f(86) = 6211 = 6211
f(87) = 199 = 199
f(88) = 6527 = 61*107
f(89) = 209 = 11*19
f(90) = 6851 = 13*17*31
f(91) = 877 = 877
f(92) = 7183 = 11*653
f(93) = 919 = 919
f(94) = 7523 = 7523
f(95) = 481 = 13*37
f(96) = 7871 = 17*463
f(97) = 503 = 503
f(98) = 8227 = 19*433
f(99) = 1051 = 1051
f(100) = 8591 = 11*11*71

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^2-16x+191

f(0)=191
f(1)=11
f(2)=163
f(3)=19
f(4)=13
f(5)=17
f(6)=131
f(7)=1
f(8)=127
f(9)=1
f(10)=1
f(11)=1
f(12)=1
f(13)=1
f(14)=1
f(15)=1
f(16)=1
f(17)=1
f(18)=227
f(19)=31
f(20)=271
f(21)=37
f(22)=1
f(23)=1
f(24)=383
f(25)=1
f(26)=41
f(27)=61
f(28)=1
f(29)=71
f(30)=47
f(31)=1
f(32)=1
f(33)=1
f(34)=73
f(35)=107
f(36)=911
f(37)=1
f(38)=79
f(39)=1
f(40)=1151
f(41)=1
f(42)=1283
f(43)=1
f(44)=1423
f(45)=1
f(46)=1571
f(47)=103
f(48)=157
f(49)=113
f(50)=1
f(51)=1
f(52)=2063
f(53)=269
f(54)=2243
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=149
f(61)=367
f(62)=179
f(63)=197
f(64)=251
f(65)=211
f(66)=3491
f(67)=1
f(68)=3727
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=4483
f(75)=577
f(76)=4751
f(77)=1
f(78)=457
f(79)=1
f(80)=1
f(81)=1
f(82)=431
f(83)=719
f(84)=5903
f(85)=757
f(86)=6211
f(87)=199
f(88)=1
f(89)=1
f(90)=1
f(91)=877
f(92)=653
f(93)=919
f(94)=7523
f(95)=1
f(96)=463
f(97)=503
f(98)=433
f(99)=1051

b) Substitution of the polynom
The polynom f(x)=x^2-16x+191 could be written as f(y)= y^2+127 with x=y+8

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-8
f'(x)>2x-17 with x > 11

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

191, 11, 163, 19, 13, 17, 131, 1, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 227, 31, 271, 37, 1, 1, 383, 1, 41, 61, 1, 71, 47, 1, 1, 1, 73, 107, 911, 1, 79, 1, 1151, 1, 1283, 1, 1423, 1, 1571, 103, 157, 113, 1, 1, 2063, 269, 2243, 1, 1, 1, 1, 1, 149, 367, 179, 197, 251, 211, 3491, 1, 3727, 1, 1, 1, 1, 1, 4483, 577, 4751, 1, 457, 1, 1, 1, 431, 719, 5903, 757, 6211, 199, 1, 1, 1, 877, 653, 919, 7523, 1, 463, 503, 433, 1051, 1, 1097, 8963, 1, 9343, 1, 263, 1, 1, 1291, 10531, 1, 353, 1, 1033, 1447, 907, 1, 12227, 389, 12671, 1, 1193, 1669, 1, 1, 14051, 1, 1, 1, 883, 1907, 419, 1, 1231, 1, 1, 1, 17027, 2161, 17551, 1, 1, 1, 1693, 1181, 1009, 1, 19727, 1, 1, 643, 673, 661, 523, 1, 22031, 2791, 1, 1433, 1787, 1471, 1, 3019, 1439, 1, 2281, 397, 1979, 1, 26371, 1, 443, 1, 1, 1, 28351, 1, 29027, 3671, 1, 1, 30403, 1, 1637, 983, 2447, 4021, 2957, 4111, 811, 1, 1999, 1, 2671, 1, 449, 4481, 2131, 1, 521, 1, 3433, 1, 38543, 1, 1063, 1, 40127, 1, 1, 5167, 1, 479, 1373, 1, 43391, 1, 941, 5581, 3467, 1, 2417, 2897, 4253, 1, 2803, 6011, 48527, 6121, 49411, 1, 1, 1, 51203, 587, 1, 6571, 4079, 3343, 53951, 1, 773, 1, 55823, 1, 1, 1789, 57727, 1, 3089, 569, 59663, 1, 1, 3821, 61631, 1, 62627, 607, 1, 8017, 509, 1, 5051, 1, 3923, 1, 1, 1, 68771, 1, 1, 4397, 971, 1, 1, 1, 709, 1, 2003, 2333, 1, 557, 76303, 739, 1, 1, 4133, 4943, 1, 1, 80783, 10169, 1, 1289, 83071, 1307, 1, 10601, 5023, 977, 809, 1, 87743, 5521, 787, 1, 90127, 1031, 91331, 1, 1, 1, 93763, 1, 7307, 1, 2347, 6053, 8861, 6131, 1, 1129, 7691, 12577, 1, 1, 1, 1, 103811, 1187, 1723, 13219, 1, 6691, 5669, 1, 3517, 13711, 677, 13877, 1, 3511, 113023, 1, 114371, 1, 115727, 14551, 9007, 1, 118463, 1, 1, 1, 1, 15241, 823, 1, 4001, 1949, 1, 1213, 1, 1, 128291, 733, 1, 1, 131171, 16487, 132623, 1, 7057, 1, 10427, 4259, 12457, 1013, 138511, 1, 1, 1, 1993, 1, 13001, 17971, 1279, 1, 146051, 1, 147583, 1, 11471, 1, 8863, 1721, 797, 1, 1, 9661, 1, 1, 156943, 19717, 3373, 1, 14557, 1, 161731, 1847, 8597, 1, 164963, 1, 2731, 10463, 168227, 1, 1, 1123, 1, 2693, 173183, 2719, 174851, 21961, 1, 22171, 953, 1, 179903, 1, 181603, 22807, 1, 23021, 185027, 1, 9829, 1, 188483, 23669, 17293, 23887, 14767, 1, 193727, 12163, 10289, 24547, 1, 1, 1, 1, 4273, 1, 202627, 1, 204431, 25667, 6653, 1, 12239, 1, 19081, 2027, 211727, 1399, 1, 6703, 1, 6761, 19753, 27277, 5923, 1, 11633, 13873, 1319, 1, 5483, 28219, 1, 1, 228611, 1, 1103, 3617, 232451, 1, 234383, 1, 236323, 14831, 21661, 1, 1087, 2741, 5153, 1, 6599, 1, 14479, 7723, 19087, 1, 1, 31391, 22921, 1217, 3217, 1, 256163, 1, 1, 32401, 1, 1, 262271, 1, 264323, 1, 1, 33427, 1, 16843, 270527, 1543, 2081, 1, 1, 34469, 2113, 1, 278911, 1, 1, 1, 25741, 35527, 285283, 1627, 1459, 1, 9341, 1, 291727, 2153, 1, 1, 1987, 4643, 1427, 1, 300431, 37691, 23279, 1, 304831, 19121, 1, 2963, 1, 3527, 1, 9769, 313727, 9839, 2953, 3049, 18719, 1, 320483, 20101, 1, 1, 325027, 40771, 327311, 41057, 329603, 1, 1, 1301, 10781, 1, 336527, 1, 338851, 1, 17957, 21397, 7309, 3917, 2203, 1, 31657, 1, 1, 10993, 1429, 44269, 355343, 1, 1913, 22433, 11617, 2053, 1, 2393, 364943, 1237, 3251, 1, 369791, 1, 5099, 46681, 34061, 2473, 377123, 1, 10259, 23801, 5381, 47911, 2689, 48221, 20369, 1, 1481, 12211, 23059, 49157, 30347, 1, 397027, 1, 1, 1, 3323, 1, 8609, 1, 9931, 3191, 13217, 1, 1, 3041, 1, 4729, 1, 26171, 2819, 1549, 422627, 2789, 6971, 1, 32911, 13411, 39133, 1, 2657, 4177, 1, 54631, 1, 27481, 40093, 27647, 26099, 1, 2843, 1, 23633, 1, 34747, 7079, 11083, 5179, 2297, 3371, 41801, 1, 1, 1, 4517, 1, 1, 4513, 42793, 14753, 473471, 1, 1, 1, 478991, 1, 1667, 30197, 484543, 1, 1973, 61091, 2621, 61441, 15901, 1931, 6791, 1, 38351, 62497, 2399, 62851, 1, 1, 507071, 1, 1, 1559, 13859, 1, 515651, 1, 12647, 16249, 1, 65357, 1, 65719, 527203, 1, 31183, 33223, 1, 1, 41227, 1, 31699, 8443, 28517, 1, 544771, 68281, 547727, 1, 550691, 34511, 50333, 1, 29297, 69767, 559631, 70141, 1, 1, 51421, 1, 568643, 1, 1, 71647, 44207, 36013, 1, 1, 1, 1, 1, 3851, 53353, 4597, 34703, 4621, 3313, 5717, 3121, 74707, 1, 37547, 1, 1, 605411, 4463, 608527, 76261, 1597, 19163, 1, 1, 1, 1, 1, 5987, 16871, 39113, 627391, 2069, 1, 2549, 3389, 1, 636931, 1, 640127, 1, 1, 4243, 646543, 81019, 1, 3701, 652991, 1, 1, 82231, 1, 4861, 1, 1, 665983, 1, 5531, 4933, 8513, 1, 35569, 3257, 14449, 1, 22013, 2311, 52747, 1, 40531, 1, 1, 2711, 1, 1, 3163, 87587, 1, 1, 64157, 1, 4759, 1, 1, 89269, 715843, 1, 23201, 1733, 1, 8231, 2699, 5351, 5101, 45697, 732863, 1, 1, 92251, 38933, 92681, 5197, 1, 1, 1, 750083, 7229, 16033, 4969, 44531, 1, 760511, 1, 764003, 1, 6343, 96157, 771011, 1, 4583, 1, 778051, 1, 1, 97919, 785123, 1, 1, 1, 792227, 99251, 795791, 7669, 799363, 1, 13163, 6287, 1, 1, 1, 1, 1, 50971, 48079, 51197, 5741, 5413, 824591, 9391, 26717, 25939, 22483, 26053, 64271, 1, 4217, 1, 1, 1, 1877, 3119, 850211, 1, 853903, 2609, 45137, 1, 78301, 13487, 4391, 9851, 3931, 108827, 14303, 54647, 1, 54881, 1, 1, 67979, 3571, 80681, 27793, 12553, 1, 6007, 1, 3581, 1, 2647, 56533, 906431, 1, 3461, 1, 2111, 1, 70607, 3593, 1, 1, 1, 1, 4447, 2477, 1, 58451, 937151, 58693, 941027, 9067, 1, 1, 49937, 1, 8431, 29833, 12109, 1, 960527, 3881, 6143, 1, 1, 60647, 2389, 121787, 976271, 7193, 9161, 1,

6. Sequence of the polynom (only primes)

191, 11, 163, 19, 13, 17, 131, 127, 227, 31, 271, 37, 383, 41, 61, 71, 47, 73, 107, 911, 79, 1151, 1283, 1423, 1571, 103, 157, 113, 2063, 269, 2243, 149, 367, 179, 197, 251, 211, 3491, 3727, 4483, 577, 4751, 457, 431, 719, 5903, 757, 6211, 199, 877, 653, 919, 7523, 463, 503, 433, 1051, 1097, 8963, 9343, 263, 1291, 10531, 353, 1033, 1447, 907, 12227, 389, 12671, 1193, 1669, 14051, 883, 1907, 419, 1231, 17027, 2161, 17551, 1693, 1181, 1009, 19727, 643, 673, 661, 523, 22031, 2791, 1433, 1787, 1471, 3019, 1439, 2281, 397, 1979, 26371, 443, 28351, 29027, 3671, 30403, 1637, 983, 2447, 4021, 2957, 4111, 811, 1999, 2671, 449, 4481, 2131, 521, 3433, 38543, 1063, 40127, 5167, 479, 1373, 43391, 941, 5581, 3467, 2417, 2897, 4253, 2803, 6011, 48527, 6121, 49411, 51203, 587, 6571, 4079, 3343, 53951, 773, 55823, 1789, 57727, 3089, 569, 59663, 3821, 61631, 62627, 607, 8017, 509, 5051, 3923, 68771, 4397, 971, 709, 2003, 2333, 557, 76303, 739, 4133, 4943, 80783, 10169, 1289, 83071, 1307, 10601, 5023, 977, 809, 87743, 5521, 787, 90127, 1031, 91331, 93763, 7307, 2347, 6053, 8861, 6131, 1129, 7691, 12577, 103811, 1187, 1723, 13219, 6691, 5669, 3517, 13711, 677, 13877, 3511, 113023, 114371, 115727, 14551, 9007, 118463, 15241, 823, 4001, 1949, 1213, 128291, 733, 131171, 16487, 132623, 7057, 10427, 4259, 12457, 1013, 138511, 1993, 13001, 17971, 1279, 146051, 147583, 11471, 8863, 1721, 797, 9661, 156943, 19717, 3373, 14557, 161731, 1847, 8597, 164963, 2731, 10463, 168227, 1123, 2693, 173183, 2719, 174851, 21961, 22171, 953, 179903, 181603, 22807, 23021, 185027, 9829, 188483, 23669, 17293, 23887, 14767, 193727, 12163, 10289, 24547, 4273, 202627, 204431, 25667, 6653, 12239, 19081, 2027, 211727, 1399, 6703, 6761, 19753, 27277, 5923, 11633, 13873, 1319, 5483, 28219, 228611, 1103, 3617, 232451, 234383, 236323, 14831, 21661, 1087, 2741, 5153, 6599, 14479, 7723, 19087, 31391, 22921, 1217, 3217, 256163, 32401, 262271, 264323, 33427, 16843, 270527, 1543, 2081, 34469, 2113, 278911, 25741, 35527, 285283, 1627, 1459, 9341, 291727, 2153, 1987, 4643, 1427, 300431, 37691, 23279, 304831, 19121, 2963, 3527, 9769, 313727, 9839, 2953, 3049, 18719, 320483, 20101, 325027, 40771, 327311, 41057, 329603, 1301, 10781, 336527, 338851, 17957, 21397, 7309, 3917, 2203, 31657, 10993, 1429, 44269, 355343, 1913, 22433, 11617, 2053, 2393, 364943, 1237, 3251, 369791, 5099, 46681, 34061, 2473, 377123, 10259, 23801, 5381, 47911, 2689, 48221, 20369, 1481, 12211, 23059, 49157, 30347, 397027, 3323, 8609, 9931, 3191, 13217, 3041, 4729, 26171, 2819, 1549, 422627, 2789, 6971, 32911, 13411, 39133, 2657, 4177, 54631, 27481, 40093, 27647, 26099, 2843, 23633, 34747, 7079, 11083, 5179, 2297, 3371, 41801, 4517, 4513, 42793, 14753, 473471, 478991, 1667, 30197, 484543, 1973, 61091, 2621, 61441, 15901, 1931, 6791, 38351, 62497, 2399, 62851, 507071, 1559, 13859, 515651, 12647, 16249, 65357, 65719, 527203, 31183, 33223, 41227, 31699, 8443, 28517, 544771, 68281, 547727, 550691, 34511, 50333, 29297, 69767, 559631, 70141, 51421, 568643, 71647, 44207, 36013, 3851, 53353, 4597, 34703, 4621, 3313, 5717, 3121, 74707, 37547, 605411, 4463, 608527, 76261, 1597, 19163, 5987, 16871, 39113, 627391, 2069, 2549, 3389, 636931, 640127, 4243, 646543, 81019, 3701, 652991, 82231, 4861, 665983, 5531, 4933, 8513, 35569, 3257, 14449, 22013, 2311, 52747, 40531, 2711, 3163, 87587, 64157, 4759, 89269, 715843, 23201, 1733, 8231, 2699, 5351, 5101, 45697, 732863, 92251, 38933, 92681, 5197, 750083, 7229, 16033, 4969, 44531, 760511, 764003, 6343, 96157, 771011, 4583, 778051, 97919, 785123, 792227, 99251, 795791, 7669, 799363, 13163, 6287, 50971, 48079, 51197, 5741, 5413, 824591, 9391, 26717, 25939, 22483, 26053, 64271, 4217, 1877, 3119, 850211, 853903, 2609, 45137, 78301, 13487, 4391, 9851, 3931, 108827, 14303, 54647, 54881, 67979, 3571, 80681, 27793, 12553, 6007, 3581, 2647, 56533, 906431, 3461, 2111, 70607, 3593, 4447, 2477, 58451, 937151, 58693, 941027, 9067, 49937, 8431, 29833, 12109, 960527, 3881, 6143, 60647, 2389, 121787, 976271, 7193, 9161,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-16x+191 and
the reducible primes which appear as divisor for the first time
p | x^2-16x+191 and p < x^2-16x+191

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)
1108710.8000000.7000000.1000000.0000000.0000000.000000
21005341120.5300000.4100000.1200006.6250005.85714312.000000
31.0006133232900.6130000.3230000.29000011.5660387.87804924.166666
410.0006.4842.2794.2050.6484000.2279000.42050010.5774887.05572714.500000
5100.00065.81717.37248.4450.6581700.1737200.48445010.1506797.62264211.520808
61.000.000665.225140.124525.1010.6652250.1401240.52510110.1071918.06608310.839116
710.000.0006.692.6491.172.9195.519.7300.6692650.1172920.55197310.0607308.37057910.511749
8100.000.00067.232.05010.092.28657.139.7640.6723200.1009230.57139810.0456568.60441910.351912
91.000.000.000674.657.84488.587.669586.070.1750.6746580.0885880.58607010.0347668.77776110.256783
1010.000.000.0006.765.197.079789.404.2495.975.792.8300.6765200.0789400.59757910.0275978.91099510.196378


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
245411.2500001.0000000.2500001.6666671.333333inf
388711.0000000.8750000.1250001.6000001.7500001.000000
4168710.5000000.4375000.0625001.0000001.0000001.000000
532131210.4062500.3750000.0312501.6250001.7142861.000000
664322570.5000000.3906250.1093752.4615392.0833337.000000
71286952170.5390620.4062500.1328122.1562502.0800002.428571
825614795520.5742190.3710940.2031252.1304351.8269233.058824
95123011791220.5878910.3496090.2382812.0476191.8842102.346154
101.0246273273000.6123050.3193360.2929692.0830561.8268162.459016
112.0481.2955807150.6323240.2832030.3491212.0653911.7737002.383333
124.0962.6281.0351.5930.6416020.2526860.3889162.0293441.7844832.227972
138.1925.3151.9253.3900.6488040.2349850.4138182.0224501.8599032.128060
1416.38410.6553.5137.1420.6503300.2144170.4359132.0047041.8249352.106785
1532.76821.4206.49314.9270.6536870.1981510.4555362.0103241.8482782.090031
1665.53643.05211.92531.1270.6569210.1819610.4749602.0098971.8365932.085282
17131.07286.45722.14764.3100.6596150.1689680.4906462.0081991.8571912.066052
18262.144173.51741.388132.1290.6619150.1578830.5040322.0069741.8687862.054564
19524.288348.12477.398270.7260.6639940.1476250.5163692.0062821.8700592.048952
201.048.576697.488146.332551.1560.6651760.1395530.5256232.0035621.8906432.035844
212.097.1521.398.008276.3501.121.6580.6666220.1317740.5348482.0043471.8885142.035101
224.194.3042.801.560523.8552.277.7050.6679440.1248970.5430472.0039661.8956222.030659
238.388.6085.611.975996.2304.615.7450.6690000.1187600.5502402.0031611.9017292.026489
2416.777.21611.241.4041.898.2279.343.1770.6700400.1131430.5568972.0031101.9054102.024197
2533.554.43222.515.1423.625.43618.889.7060.6710040.1080460.5629572.0028761.9099072.021765
2667.108.86445.087.8506.939.80138.148.0490.6718610.1034110.5684502.0025571.9141982.019515
27134.217.72890.281.83613.308.01576.973.8210.6726520.0991520.5735002.0023541.9176362.017766
28268.435.456180.763.67425.561.156155.202.5180.6733970.0952230.5781742.0022151.9207342.016303
29536.870.912361.895.30149.180.206312.715.0950.6740830.0916050.5824772.0020351.9240212.014884
301.073.741.824724.474.94494.761.116629.713.8280.6747200.0882530.5864672.0018911.9268142.013698
312.147.483.6481.450.239.690182.823.2191.267.416.4710.6753210.0851340.5901872.0017811.9293062.012686
324.294.967.2962.902.890.624353.157.4782.549.733.1460.6758820.0822260.5936562.0016621.9316882.011756
338.589.934.5925.810.301.898683.005.6855.127.296.2130.6764080.0795120.5968962.0015571.9339982.010915
3417.179.869.18411.629.117.4961.322.381.24210.306.736.2540.6769040.0769730.5999312.0014651.9361202.010170
3534.359.738.36823.274.285.8212.562.928.71420.711.357.1070.6773710.0745910.6027802.0013801.9381162.009497
3668.719.476.73646.578.882.3894.972.074.78841.606.807.6010.6778120.0723530.6054592.0013021.9399972.008888


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
123120201
244220301
387341402
4167341402
53212571515
6642581729311
7128522329420721
82569544519311441
9512179809920632571
101.0243271451824211446125
112.0485802653158220978211
124.0961.035458577152378132373
138.1921.9258411.084269707264685
1416.3843.5131.5381.9754701.3094741.260
1532.7686.4932.8863.6078762.3948772.346
1665.53611.9255.3416.5841.5994.3841.6094.333
17131.07222.1479.94412.2032.9898.0942.9448.120
18262.14441.38818.64022.7485.54215.1375.48415.225
19524.28877.39834.87442.52410.29428.39310.21028.501
201.048.576146.33265.88980.44319.44353.83719.22353.829
212.097.152276.350124.250152.10036.552101.75636.157101.885
224.194.304523.855235.275288.58069.086193.11668.542193.111
238.388.608996.230447.179549.051130.727367.585130.276367.642
2416.777.2161.898.227851.7621.046.465248.435700.732248.019701.041
2533.554.4323.625.4361.626.8121.998.624473.4451.338.817473.4941.339.680
2667.108.8646.939.8013.112.9903.826.811904.1562.565.110904.5242.566.011
27134.217.72813.308.0155.964.6037.343.4121.732.3294.920.6561.731.9714.923.059
28268.435.45625.561.15611.454.08814.107.0683.322.7189.457.2103.321.6509.459.578
29536.870.91249.180.20622.031.25727.148.9496.383.79018.205.0266.384.92718.206.463
301.073.741.82494.761.11642.437.42852.323.68812.282.00335.096.91012.285.69735.096.506
312.147.483.648182.823.21981.854.373100.968.84623.664.19267.748.31623.668.68467.742.027
324.294.967.296353.157.478158.073.400195.084.07845.657.905130.922.09045.663.939130.913.544
338.589.934.592683.005.685305.629.223377.376.46288.208.264253.305.96688.209.388253.282.067
3417.179.869.1841.322.381.242591.611.241730.770.001170.592.968490.591.738170.609.944490.586.592
3534.359.738.3682.562.928.7141.146.369.2801.416.559.434330.321.424951.151.510330.342.211951.113.569
3668.719.476.7364.972.074.7882.223.517.5092.748.557.279640.228.4231.845.827.526640.277.1981.845.741.641


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
241100010
381100010
4161100010
5321100010
6647431231
712817986344
825652232915141112
9512122586434312730
101.02430013516581747867
112.048715342373181170197167
124.0961.593795798407393421372
138.1923.3901.7011.689877828881804
1416.3847.1423.5583.5841.7891.7551.8421.756
1532.76814.9277.4837.4443.8033.6513.8403.633
1665.53631.12715.65315.4747.9537.5697.9607.645
17131.07264.31032.17332.13716.46415.72216.28715.837
18262.144132.12966.02966.10033.67232.45333.52932.475
19524.288270.726135.478135.24868.80266.56768.90366.454
201.048.576551.156275.732275.424139.992135.538140.013135.613
212.097.1521.121.658561.243560.415284.784275.873284.176276.825
224.194.3042.277.7051.139.2171.138.488577.328561.730577.083561.564
238.388.6084.615.7452.310.4392.305.3061.167.1961.140.6071.168.7981.139.144
2416.777.2169.343.1774.677.5064.665.6712.361.0272.310.3972.363.2862.308.467
2533.554.43218.889.7069.456.3989.433.3084.773.5094.673.8574.771.8724.670.468
2667.108.86438.148.04919.092.84719.055.2029.633.4079.441.1509.634.5029.438.990
27134.217.72876.973.82138.524.61838.449.20319.428.65619.061.25419.424.85119.059.060
28268.435.456155.202.51877.671.78277.530.73639.151.90138.453.09939.153.20538.444.313
29536.870.912312.715.095156.481.473156.233.62278.847.99077.510.66678.850.56877.505.871
301.073.741.824629.713.828315.097.883314.615.945158.723.997156.126.483158.737.547156.125.801
312.147.483.6481.267.416.471634.155.783633.260.688319.351.878314.345.913319.368.339314.350.341
324.294.967.2962.549.733.1461.275.752.8541.273.980.292642.251.934632.575.273642.274.021632.631.918
338.589.934.5925.127.296.2132.565.390.0732.561.906.1401.291.130.2281.272.495.5811.291.141.7251.272.528.679
3417.179.869.18410.306.736.2545.156.700.7875.150.035.4672.594.626.4822.558.737.9532.594.628.4372.558.743.382
3534.359.738.36820.711.357.10710.362.045.43310.349.311.6745.212.528.1325.143.158.8985.212.506.2745.143.163.803
3668.719.476.73641.606.807.60120.815.727.52020.791.080.08110.468.834.86110.334.519.20410.468.805.42410.334.648.112


8. Check for existing Integer Sequences by OEIS

Found in Database : 191, 11, 163, 19, 13, 17, 131, 1, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 227, 31,
Found in Database : 191, 11, 163, 19, 13, 17, 131, 127, 227, 31, 271, 37, 383, 41, 61, 71, 47, 73, 107, 911, 79,
Found in Database : 11, 13, 17, 19, 31, 37, 41, 47, 61, 71, 73, 79, 103, 107, 113, 127, 131, 149,