Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:23:32
Deutsch
28.Mar 2024

Polynom = x^2-144x+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) = 3 = 3
f(2) = 93 = 3*31
f(3) = 29 = 29
f(4) = 369 = 3*3*41
f(5) = 63 = 3*3*7
f(6) = 637 = 7*7*13
f(7) = 3 = 3
f(8) = 897 = 3*13*23
f(9) = 1 = 1
f(10) = 1149 = 3*383
f(11) = 159 = 3*53
f(12) = 1393 = 7*199
f(13) = 189 = 3*3*3*7
f(14) = 1629 = 3*3*181
f(15) = 109 = 109
f(16) = 1857 = 3*619
f(17) = 123 = 3*41
f(18) = 2077 = 31*67
f(19) = 273 = 3*7*13
f(20) = 2289 = 3*7*109
f(21) = 299 = 13*23
f(22) = 2493 = 3*3*277
f(23) = 81 = 3*3*3*3
f(24) = 2689 = 2689
f(25) = 87 = 3*29
f(26) = 2877 = 3*7*137
f(27) = 371 = 7*53
f(28) = 3057 = 3*1019
f(29) = 393 = 3*131
f(30) = 3229 = 3229
f(31) = 207 = 3*3*23
f(32) = 3393 = 3*3*13*29
f(33) = 217 = 7*31
f(34) = 3549 = 3*7*13*13
f(35) = 453 = 3*151
f(36) = 3697 = 3697
f(37) = 471 = 3*157
f(38) = 3837 = 3*1279
f(39) = 61 = 61
f(40) = 3969 = 3*3*3*3*7*7
f(41) = 63 = 3*3*7
f(42) = 4093 = 4093
f(43) = 519 = 3*173
f(44) = 4209 = 3*23*61
f(45) = 533 = 13*41
f(46) = 4317 = 3*1439
f(47) = 273 = 3*7*13
f(48) = 4417 = 7*631
f(49) = 279 = 3*3*31
f(50) = 4509 = 3*3*3*167
f(51) = 569 = 569
f(52) = 4593 = 3*1531
f(53) = 579 = 3*193
f(54) = 4669 = 7*23*29
f(55) = 147 = 3*7*7
f(56) = 4737 = 3*1579
f(57) = 149 = 149
f(58) = 4797 = 3*3*13*41
f(59) = 603 = 3*3*67
f(60) = 4849 = 13*373
f(61) = 609 = 3*7*29
f(62) = 4893 = 3*7*233
f(63) = 307 = 307
f(64) = 4929 = 3*31*53
f(65) = 309 = 3*103
f(66) = 4957 = 4957
f(67) = 621 = 3*3*3*23
f(68) = 4977 = 3*3*7*79
f(69) = 623 = 7*89
f(70) = 4989 = 3*1663
f(71) = 39 = 3*13
f(72) = 4993 = 4993
f(73) = 39 = 3*13
f(74) = 4989 = 3*1663
f(75) = 623 = 7*89
f(76) = 4977 = 3*3*7*79
f(77) = 621 = 3*3*3*23
f(78) = 4957 = 4957
f(79) = 309 = 3*103
f(80) = 4929 = 3*31*53
f(81) = 307 = 307
f(82) = 4893 = 3*7*233
f(83) = 609 = 3*7*29
f(84) = 4849 = 13*373
f(85) = 603 = 3*3*67
f(86) = 4797 = 3*3*13*41
f(87) = 149 = 149
f(88) = 4737 = 3*1579
f(89) = 147 = 3*7*7
f(90) = 4669 = 7*23*29
f(91) = 579 = 3*193
f(92) = 4593 = 3*1531
f(93) = 569 = 569
f(94) = 4509 = 3*3*3*167
f(95) = 279 = 3*3*31
f(96) = 4417 = 7*631
f(97) = 273 = 3*7*13
f(98) = 4317 = 3*1439
f(99) = 533 = 13*41
f(100) = 4209 = 3*23*61

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^2-144x+191

f(0)=191
f(1)=3
f(2)=31
f(3)=29
f(4)=41
f(5)=7
f(6)=13
f(7)=1
f(8)=23
f(9)=1
f(10)=383
f(11)=53
f(12)=199
f(13)=1
f(14)=181
f(15)=109
f(16)=619
f(17)=1
f(18)=67
f(19)=1
f(20)=1
f(21)=1
f(22)=277
f(23)=1
f(24)=2689
f(25)=1
f(26)=137
f(27)=1
f(28)=1019
f(29)=131
f(30)=3229
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=151
f(36)=3697
f(37)=157
f(38)=1279
f(39)=61
f(40)=1
f(41)=1
f(42)=4093
f(43)=173
f(44)=1
f(45)=1
f(46)=1439
f(47)=1
f(48)=631
f(49)=1
f(50)=167
f(51)=569
f(52)=1531
f(53)=193
f(54)=1
f(55)=1
f(56)=1579
f(57)=149
f(58)=1
f(59)=1
f(60)=373
f(61)=1
f(62)=233
f(63)=307
f(64)=1
f(65)=103
f(66)=4957
f(67)=1
f(68)=79
f(69)=89
f(70)=1663
f(71)=1
f(72)=4993
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-144x+191 could be written as f(y)= y^2-4993 with x=y+72

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-72
f'(x)>2x-145 with x > 71

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, 3, 31, 29, 41, 7, 13, 1, 23, 1, 383, 53, 199, 1, 181, 109, 619, 1, 67, 1, 1, 1, 277, 1, 2689, 1, 137, 1, 1019, 131, 3229, 1, 1, 1, 1, 151, 3697, 157, 1279, 61, 1, 1, 4093, 173, 1, 1, 1439, 1, 631, 1, 167, 569, 1531, 193, 1, 1, 1579, 149, 1, 1, 373, 1, 233, 307, 1, 103, 4957, 1, 79, 89, 1663, 1, 4993, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1091, 1, 1, 1, 577, 1, 2063, 1, 1, 1, 1, 1, 239, 1, 1, 457, 1, 1, 1, 1, 1, 601, 1669, 1, 773, 1, 647, 1, 2081, 269, 953, 1, 1, 229, 839, 1, 1, 1, 1, 1087, 1, 1, 409, 1, 1, 1, 3461, 443, 10883, 1, 3797, 1, 1, 1, 401, 1, 1, 827, 643, 1, 14051, 1, 541, 1861, 389, 1, 1, 1, 5441, 1, 1879, 1, 1, 1, 6037, 1151, 1, 1, 1, 1, 317, 1, 6869, 1, 1, 1, 1, 397, 1, 1, 1787, 491, 613, 433, 1171, 1039, 1, 1, 2887, 823, 1, 1, 27407, 1, 9377, 1, 1069, 1, 4229, 1249, 1, 1, 10369, 1, 1, 1, 1, 4129, 857, 1409, 34211, 1, 1667, 2213, 1, 503, 691, 1543, 1783, 1, 12757, 1, 39107, 1, 1, 1, 1, 859, 683, 877, 14177, 1, 1, 1, 3407, 1, 15061, 1, 15361, 1, 1, 659, 5323, 3023, 1, 1, 7109, 1, 16901, 6397, 1913, 1, 1697, 1, 2551, 1, 18181, 2293, 4271, 1, 1, 509, 661, 2417, 58511, 2459, 19841, 1, 1, 1, 2677, 1, 1, 607, 1, 1, 727, 1, 7307, 8287, 22277, 1, 1, 1, 1, 1, 599, 1, 10169, 1, 24097, 1, 24469, 1, 74531, 1, 1201, 733, 25601, 1, 77951, 1, 3767, 1423, 991, 1123, 3541, 1709, 27541, 743, 1, 3517, 1, 1, 1, 2713, 1, 1, 88643, 3719, 1033, 11311, 1, 1, 1, 1, 761, 11779, 1, 1, 1, 1, 1, 1, 1, 4139, 7691, 1, 1, 1, 11399, 1, 3583, 4357, 5011, 1, 35521, 1117, 107903, 1, 1, 1, 1, 4637, 2731, 2347, 37781, 7127, 1, 1, 1733, 1, 1, 1847, 3049, 1, 17189, 1, 1, 15307, 41057, 1, 2543, 1, 42017, 1, 1, 1, 4447, 1, 6211, 4099, 43969, 5527, 133391, 1, 2141, 1, 1, 2857, 10607, 1, 1499, 2503, 2237, 1, 1307, 1, 2087, 18097, 1, 1, 147107, 1, 16519, 9341, 821, 1, 1, 6359, 51137, 1, 5741, 1, 1721, 937, 4057, 1, 53281, 3347, 161471, 1, 863, 1, 54917, 6899, 166403, 1, 1, 1, 18859, 1, 947, 7177, 57697, 1553, 1, 3659, 5693, 1, 1523, 22381, 1, 1, 1, 1901, 61121, 23029, 1, 1, 26693, 3911, 62869, 911, 1, 1, 27449, 1, 1, 1, 1, 1, 197507, 1181, 9491, 1, 1, 1, 1, 1, 1, 3673, 1129, 8647, 1399, 1, 7789, 1, 10103, 1, 214031, 1, 1, 13553, 3457, 1, 219683, 1, 1, 27817, 1, 1, 2477, 1, 1, 28537, 76421, 1, 33029, 1, 1, 14633, 1, 1093, 237071, 1, 11383, 1, 80341, 2521, 243011, 3389, 3889, 1, 1, 5167, 1, 5209, 1, 1, 1, 3529, 255107, 1, 85717, 2017, 12343, 1549, 4283, 3643, 3251, 1, 88469, 1, 1, 1, 3907, 33829, 30187, 1, 39113, 1, 91969, 1, 7129, 1, 1, 1, 4481, 1, 1789, 11897, 9883, 11987, 13751, 1, 32327, 1, 293123, 1, 1, 1, 1, 6221, 299711, 2089, 33547, 37879, 2069, 1, 23567, 3203, 1, 9679, 1, 1, 44729, 13093, 3391, 1, 1, 1, 1987, 1, 11933, 3109, 108161, 1, 1427, 1, 15671, 41281, 1601, 1, 5471, 6977, 1231, 3011, 8677, 14149, 8311, 1583, 1, 1, 16451, 1, 2539, 14543, 116741, 1, 1, 1, 355007, 1, 1, 1949, 1, 1, 51749, 1, 40519, 5717, 9413, 1, 1, 2207, 124001, 23327, 1, 2609, 376931, 2251, 1, 47581, 127297, 1, 384383, 1, 1, 1, 1, 1, 391907, 8191, 131477, 3533, 6301, 5531, 1, 16699, 1, 1, 19267, 1, 9931, 1, 1, 1, 1, 1, 59273, 1, 139169, 4027, 46679, 1951, 8627, 1, 4889, 1, 142657, 1, 1, 1, 1, 1, 11177, 9109, 2423, 18329, 21011, 1, 1, 1, 19417, 2333, 4831, 1, 21523, 1453, 15679, 3167, 5647, 1, 21911, 1, 462863, 1, 155201, 14593, 4003, 1, 1, 19687, 1, 59407, 158881, 1, 68489, 1, 1, 1, 1, 1559, 487811, 1, 1, 1, 1, 1, 21577, 1, 23767, 1, 167317, 10487, 1, 1, 4339, 9091, 1, 5333, 513407, 1, 172097, 64717, 1, 1, 522083, 1, 1699, 2531, 1, 1, 75833, 7393, 1, 8363, 1, 2803, 77093, 3221, 13913, 2957, 4663, 1, 548543, 1637, 26263, 69127, 184837, 23167, 1, 1, 1, 1, 8167, 1811, 566543, 23669, 1, 5099, 1, 1, 575651, 24049, 8387, 1, 2131, 1, 44987, 1, 65323, 73681, 6793, 3527, 84869, 12409, 1, 1627, 7411, 1, 86201, 1, 1, 1, 203221, 6367, 7757, 1, 3259, 77191, 206369, 1, 1, 1, 1, 11197, 69847, 8753, 1, 3299, 9203, 1, 30391, 1, 7207, 1, 1747, 1, 30851, 1933, 28309, 27197, 218117, 1, 2357, 1, 13487, 1, 17029, 83221, 1, 1, 95813, 1, 74887, 42227, 3701, 28289, 1777, 1, 1, 1, 25453, 1, 16843, 28843, 1, 12421, 232417, 14561, 13219, 4877, 2897, 12601, 2591, 1, 2377, 1, 238037, 22369, 1, 1, 24859, 30109, 241441, 45377, 1, 1, 1, 1, 2633, 92041, 246017, 3853, 105929, 1, 1, 93337, 1, 1, 57839, 2243, 2767, 1, 1931, 1, 24593, 10613, 12157, 1, 256469, 1, 12671, 1, 3863, 1, 4127, 1, 1, 16361, 262369, 3181, 1, 1, 4591, 1, 2273, 6247, 20549, 1, 115001, 33617, 11719, 50651, 90247, 5653, 116549, 1, 273157, 1, 9461, 1, 2693, 1, 1, 4523, 6781, 1, 837731, 17491, 1, 7529, 1, 1, 1, 1, 284161, 1907,

6. Sequence of the polynom (only primes)

191, 3, 31, 29, 41, 7, 13, 23, 383, 53, 199, 181, 109, 619, 67, 277, 2689, 137, 1019, 131, 3229, 151, 3697, 157, 1279, 61, 4093, 173, 1439, 631, 167, 569, 1531, 193, 1579, 149, 373, 233, 307, 103, 4957, 79, 89, 1663, 4993, 1091, 577, 2063, 239, 457, 601, 1669, 773, 647, 2081, 269, 953, 229, 839, 1087, 409, 3461, 443, 10883, 3797, 401, 827, 643, 14051, 541, 1861, 389, 5441, 1879, 6037, 1151, 317, 6869, 397, 1787, 491, 613, 433, 1171, 1039, 2887, 823, 27407, 9377, 1069, 4229, 1249, 10369, 4129, 857, 1409, 34211, 1667, 2213, 503, 691, 1543, 1783, 12757, 39107, 859, 683, 877, 14177, 3407, 15061, 15361, 659, 5323, 3023, 7109, 16901, 6397, 1913, 1697, 2551, 18181, 2293, 4271, 509, 661, 2417, 58511, 2459, 19841, 2677, 607, 727, 7307, 8287, 22277, 599, 10169, 24097, 24469, 74531, 1201, 733, 25601, 77951, 3767, 1423, 991, 1123, 3541, 1709, 27541, 743, 3517, 2713, 88643, 3719, 1033, 11311, 761, 11779, 4139, 7691, 11399, 3583, 4357, 5011, 35521, 1117, 107903, 4637, 2731, 2347, 37781, 7127, 1733, 1847, 3049, 17189, 15307, 41057, 2543, 42017, 4447, 6211, 4099, 43969, 5527, 133391, 2141, 2857, 10607, 1499, 2503, 2237, 1307, 2087, 18097, 147107, 16519, 9341, 821, 6359, 51137, 5741, 1721, 937, 4057, 53281, 3347, 161471, 863, 54917, 6899, 166403, 18859, 947, 7177, 57697, 1553, 3659, 5693, 1523, 22381, 1901, 61121, 23029, 26693, 3911, 62869, 911, 27449, 197507, 1181, 9491, 3673, 1129, 8647, 1399, 7789, 10103, 214031, 13553, 3457, 219683, 27817, 2477, 28537, 76421, 33029, 14633, 1093, 237071, 11383, 80341, 2521, 243011, 3389, 3889, 5167, 5209, 3529, 255107, 85717, 2017, 12343, 1549, 4283, 3643, 3251, 88469, 3907, 33829, 30187, 39113, 91969, 7129, 4481, 1789, 11897, 9883, 11987, 13751, 32327, 293123, 6221, 299711, 2089, 33547, 37879, 2069, 23567, 3203, 9679, 44729, 13093, 3391, 1987, 11933, 3109, 108161, 1427, 15671, 41281, 1601, 5471, 6977, 1231, 3011, 8677, 14149, 8311, 1583, 16451, 2539, 14543, 116741, 355007, 1949, 51749, 40519, 5717, 9413, 2207, 124001, 23327, 2609, 376931, 2251, 47581, 127297, 384383, 391907, 8191, 131477, 3533, 6301, 5531, 16699, 19267, 9931, 59273, 139169, 4027, 46679, 1951, 8627, 4889, 142657, 11177, 9109, 2423, 18329, 21011, 19417, 2333, 4831, 21523, 1453, 15679, 3167, 5647, 21911, 462863, 155201, 14593, 4003, 19687, 59407, 158881, 68489, 1559, 487811, 21577, 23767, 167317, 10487, 4339, 9091, 5333, 513407, 172097, 64717, 522083, 1699, 2531, 75833, 7393, 8363, 2803, 77093, 3221, 13913, 2957, 4663, 548543, 1637, 26263, 69127, 184837, 23167, 8167, 1811, 566543, 23669, 5099, 575651, 24049, 8387, 2131, 44987, 65323, 73681, 6793, 3527, 84869, 12409, 1627, 7411, 86201, 203221, 6367, 7757, 3259, 77191, 206369, 11197, 69847, 8753, 3299, 9203, 30391, 7207, 1747, 30851, 1933, 28309, 27197, 218117, 2357, 13487, 17029, 83221, 95813, 74887, 42227, 3701, 28289, 1777, 25453, 16843, 28843, 12421, 232417, 14561, 13219, 4877, 2897, 12601, 2591, 2377, 238037, 22369, 24859, 30109, 241441, 45377, 2633, 92041, 246017, 3853, 105929, 93337, 57839, 2243, 2767, 1931, 24593, 10613, 12157, 256469, 12671, 3863, 4127, 16361, 262369, 3181, 4591, 2273, 6247, 20549, 115001, 33617, 11719, 50651, 90247, 5653, 116549, 273157, 9461, 2693, 4523, 6781, 837731, 17491, 7529, 284161, 1907,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-144x+191 and
the reducible primes which appear as divisor for the first time
p | x^2-144x+191 and p < x^2-144x+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)
1108350.8000000.3000000.5000000.0000000.0000000.000000
21004013270.4000000.1300000.2700005.0000004.3333335.400000
31.000479993800.4790000.0990000.38000011.9750007.61538514.074074
410.0005.8607605.1000.5860000.0760000.51000012.2338217.67676813.421053
5100.00061.5365.81655.7200.6153600.0581600.55720010.5010247.65263210.925490
61.000.000630.52046.828583.6920.6305200.0468280.58369210.2463608.05158110.475449
710.000.0006.399.201391.4856.007.7160.6399200.0391480.60077210.1490858.36006310.292613
8100.000.00064.685.6083.366.74761.318.8610.6468560.0336670.61318910.1083888.59993810.206684
91.000.000.000652.192.48829.558.273622.634.2150.6521930.0295580.62263410.0824988.77947510.154041
1010.000.000.0006.564.048.041263.411.6006.300.636.4410.6564050.0263410.63006410.0645878.91160410.119323


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
387340.8750000.3750000.5000001.4000001.0000002.000000
41613490.8125000.2500000.5625001.8571431.3333332.250000
532206140.6250000.1875000.4375001.5384621.5000001.555556
6643711260.5781250.1718750.4062501.8500001.8333331.857143
71284013270.3125000.1015620.2109381.0810811.1818181.038462
82567725520.3007810.0976560.2031251.9250001.9230771.925926
9512208521560.4062500.1015620.3046882.7012992.0800003.000000
101.0244911023890.4794920.0996090.3798832.3605771.9615382.493590
112.0481.0791918880.5268550.0932620.4335942.1975561.8725492.282776
124.0962.2913541.9370.5593260.0864260.4729002.1232621.8534032.181306
138.1924.7536384.1150.5802000.0778810.5023192.0746401.8022602.124419
1416.3849.7401.1728.5680.5944820.0715330.5229492.0492321.8369912.082139
1532.76819.8042.14217.6620.6043700.0653690.5390012.0332651.8276452.061391
1665.53640.0583.98636.0720.6112370.0608220.5504152.0227231.8608782.042351
17131.07280.9687.40673.5620.6177370.0565030.5612342.0212691.8580032.039310
18262.144163.27713.874149.4030.6228520.0529250.5699272.0165621.8733462.030981
19524.288328.63526.004302.6310.6268220.0495990.5772232.0127451.8742972.025602
201.048.576661.38148.864612.5170.6307420.0466000.5841422.0125091.8790962.023973
212.097.1521.329.24492.5871.236.6570.6338330.0441490.5896842.0098011.8947902.018976
224.194.3042.670.957175.2232.495.7340.6368060.0417760.5950292.0093811.8925232.018130
238.388.6085.362.869332.6345.030.2350.6393040.0396530.5996512.0078451.8983472.015533
2416.777.21610.765.466633.46510.132.0010.6416720.0377570.6039142.0074081.9043902.014220
2533.554.43221.602.4381.209.88620.392.5520.6438030.0360570.6077452.0066421.9099492.012687
2667.108.86443.337.4042.314.72641.022.6780.6457780.0344920.6112862.0061351.9131772.011650
27134.217.72886.923.9784.440.55282.483.4260.6476340.0330850.6145492.0057491.9183922.010679
28268.435.456174.302.8108.528.366165.774.4440.6493290.0317710.6175582.0052331.9205642.009791
29536.870.912349.443.81116.411.234333.032.5770.6508900.0305680.6203222.0048091.9243122.008950
301.073.741.824700.442.65431.617.104668.825.5500.6523380.0294460.6228922.0044501.9265532.008289
312.147.483.6481.403.792.40161.002.5721.342.789.8290.6536920.0284070.6252852.0041501.9294172.007683
324.294.967.2962.813.031.291117.843.0332.695.188.2580.6549600.0274370.6275222.0038801.9317722.007156
338.589.934.5925.636.303.595227.903.0045.408.400.5910.6561520.0265310.6296212.0036411.9339542.006687
3417.179.869.18411.291.842.444441.276.96710.850.565.4770.6572720.0256860.6315862.0034131.9362492.006243


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
122010101
243020111
383020111
4164120121
5326321131
66411643251
712813844261
82562513116676
95125224279151216
101.024102455621301734
112.0481917511532642768
124.0963541302235612250126
138.1926382274109323583227
1416.3841.172431740157434162419
1532.7682.1427791.362296803284759
1665.5363.9861.4342.5515571.4545361.439
17131.0727.4062.6134.7921.0202.7049712.711
18262.14413.8744.8669.0071.8725.0611.8495.092
19524.28826.0049.12916.8743.4479.5783.4229.557
201.048.57648.86417.25731.6066.47217.9726.51217.908
212.097.15292.58732.43260.15412.22633.93112.23134.199
224.194.304175.22360.948114.27423.01164.45623.15964.597
238.388.608332.634115.413217.22043.681122.43043.752122.771
2416.777.216633.465219.302414.16282.729233.58883.158233.990
2533.554.4321.209.886418.227791.658157.619446.875158.369447.023
2667.108.8642.314.726798.8781.515.847301.331855.196302.180856.019
27134.217.7284.440.5521.530.5892.909.962577.4161.642.030578.7571.642.349
28268.435.4568.528.3662.936.9195.591.4461.107.8443.156.5501.108.5703.155.402
29536.870.91216.411.2345.641.74610.769.4872.130.6796.075.9872.129.0236.075.545
301.073.741.82431.617.10410.856.66020.760.4434.099.85511.710.6494.096.72311.709.877
312.147.483.64861.002.57220.926.44840.076.1237.897.24122.606.1397.894.83722.604.355
324.294.967.296117.843.03340.391.20577.451.82715.237.81243.686.24915.233.25443.685.718
338.589.934.592227.903.00478.043.779149.859.22429.434.95484.516.66829.429.13784.522.245
3417.179.869.184441.276.967150.978.568290.298.39856.934.458163.708.43356.926.771163.707.305


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
121100001
242111001
384221012
4169541134
53214772444
6642615114679
712827161146710
825652282410111615
9512156777939334737
101.0243891892001018611290
112.048888435453245192247204
124.0961.937947990512436554435
138.1924.1152.0082.1071.0719571.102985
1416.3848.5684.1994.3692.1942.0372.3032.034
1532.76817.6628.6149.0484.6124.1454.7014.204
1665.53636.07217.63318.4399.3798.5039.6298.561
17131.07273.56236.22137.34119.18517.41619.35017.611
18262.144149.40373.68675.71739.06435.47539.04835.816
19524.288302.631149.472153.15978.58772.49878.73472.812
201.048.576612.517302.173310.344158.470147.551159.067147.429
212.097.1521.236.657610.677625.980319.533298.438320.763297.923
224.194.3042.495.7341.233.9851.261.749645.119603.020645.204602.391
238.388.6085.030.2352.489.2162.541.0191.297.7151.217.1291.299.7551.215.636
2416.777.21610.132.0015.015.7135.116.2882.609.6922.455.1442.613.7632.453.402
2533.554.43220.392.55210.100.77610.291.7765.245.7594.950.4535.248.5294.947.811
2667.108.86441.022.67820.328.60620.694.07210.539.0949.974.03010.538.8939.970.661
27134.217.72882.483.42640.892.18041.591.24621.159.82120.080.77921.165.68920.077.137
28268.435.456165.774.44482.210.81383.563.63142.480.80240.404.83742.491.14940.397.656
29536.870.912333.032.577165.210.517167.822.06085.258.57181.263.51085.269.14181.241.355
301.073.741.824668.825.550331.893.442336.932.108171.079.153163.347.938171.085.731163.312.728
312.147.483.6481.342.789.829666.541.151676.248.678343.177.820328.239.932343.175.415328.196.662
324.294.967.2962.695.188.2581.338.218.2081.356.970.050688.246.887659.358.995688.250.994659.331.382
338.589.934.5925.408.400.5912.686.076.7772.722.323.8141.380.081.8791.324.110.4141.380.099.3181.324.108.980
3417.179.869.18410.850.565.4775.390.148.2925.460.417.1852.766.796.4552.658.439.8262.766.919.1892.658.410.007


8. Check for existing Integer Sequences by OEIS

Found in Database : 191, 3, 31, 29, 41, 7, 13, 1, 23, 1, 383, 53, 199, 1, 181, 109, 619, 1, 67, 1,
Found in Database : 191, 3, 31, 29, 41, 7, 13, 23, 383, 53, 199, 181, 109, 619, 67, 277, 2689, 137, 1019, 131, 3229, 151, 3697, 157, 1279, 61,
Found in Database : 3, 7, 13, 23, 29, 31, 41, 53, 61, 67, 79, 89, 103, 109, 131, 137, 149,