Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:18:17
Deutsch
29.Mar 2024

Polynom = x^2+67

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) = 67 = 67
f(1) = 17 = 17
f(2) = 71 = 71
f(3) = 19 = 19
f(4) = 83 = 83
f(5) = 23 = 23
f(6) = 103 = 103
f(7) = 29 = 29
f(8) = 131 = 131
f(9) = 37 = 37
f(10) = 167 = 167
f(11) = 47 = 47
f(12) = 211 = 211
f(13) = 59 = 59
f(14) = 263 = 263
f(15) = 73 = 73
f(16) = 323 = 17*19
f(17) = 89 = 89
f(18) = 391 = 17*23
f(19) = 107 = 107
f(20) = 467 = 467
f(21) = 127 = 127
f(22) = 551 = 19*29
f(23) = 149 = 149
f(24) = 643 = 643
f(25) = 173 = 173
f(26) = 743 = 743
f(27) = 199 = 199
f(28) = 851 = 23*37
f(29) = 227 = 227
f(30) = 967 = 967
f(31) = 257 = 257
f(32) = 1091 = 1091
f(33) = 289 = 17*17
f(34) = 1223 = 1223
f(35) = 323 = 17*19
f(36) = 1363 = 29*47
f(37) = 359 = 359
f(38) = 1511 = 1511
f(39) = 397 = 397
f(40) = 1667 = 1667
f(41) = 437 = 19*23
f(42) = 1831 = 1831
f(43) = 479 = 479
f(44) = 2003 = 2003
f(45) = 523 = 523
f(46) = 2183 = 37*59
f(47) = 569 = 569
f(48) = 2371 = 2371
f(49) = 617 = 617
f(50) = 2567 = 17*151
f(51) = 667 = 23*29
f(52) = 2771 = 17*163
f(53) = 719 = 719
f(54) = 2983 = 19*157
f(55) = 773 = 773
f(56) = 3203 = 3203
f(57) = 829 = 829
f(58) = 3431 = 47*73
f(59) = 887 = 887
f(60) = 3667 = 19*193
f(61) = 947 = 947
f(62) = 3911 = 3911
f(63) = 1009 = 1009
f(64) = 4163 = 23*181
f(65) = 1073 = 29*37
f(66) = 4423 = 4423
f(67) = 1139 = 17*67
f(68) = 4691 = 4691
f(69) = 1207 = 17*71
f(70) = 4967 = 4967
f(71) = 1277 = 1277
f(72) = 5251 = 59*89
f(73) = 1349 = 19*71
f(74) = 5543 = 23*241
f(75) = 1423 = 1423
f(76) = 5843 = 5843
f(77) = 1499 = 1499
f(78) = 6151 = 6151
f(79) = 1577 = 19*83
f(80) = 6467 = 29*223
f(81) = 1657 = 1657
f(82) = 6791 = 6791
f(83) = 1739 = 37*47
f(84) = 7123 = 17*419
f(85) = 1823 = 1823
f(86) = 7463 = 17*439
f(87) = 1909 = 23*83
f(88) = 7811 = 73*107
f(89) = 1997 = 1997
f(90) = 8167 = 8167
f(91) = 2087 = 2087
f(92) = 8531 = 19*449
f(93) = 2179 = 2179
f(94) = 8903 = 29*307
f(95) = 2273 = 2273
f(96) = 9283 = 9283
f(97) = 2369 = 23*103
f(98) = 9671 = 19*509
f(99) = 2467 = 2467
f(100) = 10067 = 10067

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+67

f(0)=67
f(1)=17
f(2)=71
f(3)=19
f(4)=83
f(5)=23
f(6)=103
f(7)=29
f(8)=131
f(9)=37
f(10)=167
f(11)=47
f(12)=211
f(13)=59
f(14)=263
f(15)=73
f(16)=1
f(17)=89
f(18)=1
f(19)=107
f(20)=467
f(21)=127
f(22)=1
f(23)=149
f(24)=643
f(25)=173
f(26)=743
f(27)=199
f(28)=1
f(29)=227
f(30)=967
f(31)=257
f(32)=1091
f(33)=1
f(34)=1223
f(35)=1
f(36)=1
f(37)=359
f(38)=1511
f(39)=397
f(40)=1667
f(41)=1
f(42)=1831
f(43)=479
f(44)=2003
f(45)=523
f(46)=1
f(47)=569
f(48)=2371
f(49)=617
f(50)=151
f(51)=1
f(52)=163
f(53)=719
f(54)=157
f(55)=773
f(56)=3203
f(57)=829
f(58)=1
f(59)=887
f(60)=193
f(61)=947
f(62)=3911
f(63)=1009
f(64)=181
f(65)=1
f(66)=4423
f(67)=1
f(68)=4691
f(69)=1
f(70)=4967
f(71)=1277
f(72)=1
f(73)=1
f(74)=241
f(75)=1423
f(76)=5843
f(77)=1499
f(78)=6151
f(79)=1
f(80)=223
f(81)=1657
f(82)=6791
f(83)=1
f(84)=419
f(85)=1823
f(86)=439
f(87)=1
f(88)=1
f(89)=1997
f(90)=8167
f(91)=2087
f(92)=449
f(93)=2179
f(94)=307
f(95)=2273
f(96)=9283
f(97)=1
f(98)=509
f(99)=2467

b) Substitution of the polynom
The polynom f(x)=x^2+67 could be written as f(y)= y^2+67 with x=y+0

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+0
f'(x)>2x-1 with x > 8

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

67, 17, 71, 19, 83, 23, 103, 29, 131, 37, 167, 47, 211, 59, 263, 73, 1, 89, 1, 107, 467, 127, 1, 149, 643, 173, 743, 199, 1, 227, 967, 257, 1091, 1, 1223, 1, 1, 359, 1511, 397, 1667, 1, 1831, 479, 2003, 523, 1, 569, 2371, 617, 151, 1, 163, 719, 157, 773, 3203, 829, 1, 887, 193, 947, 3911, 1009, 181, 1, 4423, 1, 4691, 1, 4967, 1277, 1, 1, 241, 1423, 5843, 1499, 6151, 1, 223, 1657, 6791, 1, 419, 1823, 439, 1, 1, 1997, 8167, 2087, 449, 2179, 307, 2273, 9283, 1, 509, 2467, 10067, 1, 283, 1, 10883, 1, 1, 2879, 11731, 1, 1, 1, 12611, 3209, 13063, 3323, 13523, 1, 823, 3557, 1, 3677, 14951, 1, 15443, 3923, 1, 4049, 16451, 4177, 1, 1, 17491, 1, 269, 1, 977, 277, 659, 1, 1, 4987, 20231, 1, 293, 5273, 21383, 5419, 1, 1, 22567, 5717, 1, 5869, 1399, 317, 1061, 1, 25031, 6337, 25667, 1, 1, 6659, 457, 6823, 1201, 1, 1489, 421, 349, 431, 1, 7499, 1597, 7673, 839, 1, 31751, 1, 32467, 1, 33191, 8389, 33923, 8573, 2039, 461, 2083, 389, 613, 9137, 36931, 491, 1019, 1, 1327, 9719, 1, 1, 1, 1, 1777, 607, 571, 619, 2237, 10729, 43331, 10937, 1523, 1, 1, 1, 45863, 1, 46723, 11789, 47591, 12007, 2851, 12227, 2903, 1, 1069, 1, 1, 12899, 52051, 13127, 52967, 1, 53891, 1, 751, 601, 55763, 827, 56711, 1, 57667, 14537, 58631, 14779, 3137, 1, 1289, 15269, 2677, 1, 1, 15767, 1, 1, 1, 16273, 1, 16529, 2897, 16787, 1, 17047, 68711, 911, 69763, 17573, 70823, 17839, 1, 953, 557, 1, 74051, 1097, 1, 1, 76243, 1, 77351, 19477, 78467, 859, 1, 691, 907, 20323, 1153, 1, 1, 20897, 4951, 21187, 85331, 1, 3761, 21773, 87683, 761, 1, 22367, 90067, 1193, 853, 1, 4021, 1, 93703, 1, 1609, 23887, 96167, 24197, 3359, 24509, 98663, 1, 99923, 1093, 2153, 25457, 5393, 1, 1, 26099, 1, 26423, 1, 1163, 1297, 27077, 108967, 27407, 110291, 27739, 111623, 1, 1361, 28409, 114311, 1, 1, 1, 3163, 29429, 709, 1567, 119783, 30119, 121171, 30467, 1, 30817, 821, 1, 125383, 1087, 7459, 1, 1, 32237, 1021, 881, 131111, 1433, 6977, 1, 1, 1, 863, 34057, 4723, 1, 138451, 1, 139943, 2069, 141443, 1871, 877, 971, 1979, 36307, 145991, 1931, 5087, 1, 6481, 797, 150611, 37847, 8951, 38237, 9043, 38629, 155303, 39023, 1, 39419, 4283, 1373, 2713, 1, 1, 1, 1, 41023, 1601, 2437, 1, 1, 1, 1, 3613, 1471, 1, 2267, 4679, 1, 1697, 1, 176467, 2333, 178151, 1, 1, 1, 1, 45599, 1, 46027, 184967, 46457, 8117, 46889, 1, 1279, 1, 1, 191911, 48197, 10193, 2861, 1, 2887, 197203, 49523, 1031, 1, 200771, 50417, 202567, 50867, 204371, 1, 206183, 2251, 208003, 1801, 12343, 1, 12451, 53147, 2399, 53609, 1, 2351, 217223, 54539, 219091, 1, 1483, 1913, 1, 55949, 224743, 3319, 226643, 3347, 1, 1, 230467, 1231, 232391, 1, 234323, 997, 8147, 1, 10357, 59797, 6491, 1, 14243, 60779, 1, 1, 3371, 3251, 248071, 1, 8623, 2729, 3037, 1, 254083, 63773, 5449, 64279, 258131, 1, 13693, 1, 1, 65809, 264263, 2287, 1, 1, 4549, 1, 1, 1, 1, 68399, 11941, 1, 1, 1877, 1, 1, 280967, 70507, 2161, 71039, 12401, 3767, 4289, 72109, 289511, 72647, 1109, 1, 293831, 4337, 1, 1, 8059, 3253, 15809, 75367, 302567, 1, 304771, 1627, 1, 77023, 1283, 3373, 10739, 78137, 18451, 78697, 18583, 79259, 8599, 79823, 4513, 4231, 1, 1, 1, 81527, 327251, 1, 2099, 1759, 331843, 1, 1, 4931, 14629, 84407, 338791, 2297, 341123, 1, 18077, 2971, 345811, 1, 1, 87337, 1, 3823, 20759, 88523, 20899, 89119, 357671, 1229, 1, 2441, 1, 1, 364883, 4817, 4127, 1, 369731, 92737, 372167, 1, 3637, 5527, 2969, 94573, 1, 95189, 381991, 1, 10391, 1, 8233, 1, 1, 97673, 17041, 98299, 23203, 98927, 1, 3433, 399491, 100189, 402023, 100823, 6857, 1429, 11003, 1, 4603, 1447, 5647, 5441, 414803, 1, 417383, 1, 1, 1, 422567, 105967, 1873, 106619, 2833, 107273, 3389, 2917, 2593, 108587, 1, 1, 1, 1, 25939, 110573, 1, 1, 1, 2381, 6701, 112577, 1, 1, 15667, 3079, 457043, 114599, 2753, 6781, 462467, 1, 2963, 116639, 1, 5101, 470663, 6211, 473411, 4093, 476167, 1, 478931, 120079, 1, 1, 28499, 121469, 28663, 122167, 25793, 122867, 492871, 4261, 3041, 1, 26237, 124979, 1, 125687, 2221, 126397, 1, 7477, 509863, 1, 10909, 1, 1, 6803, 518467, 2203, 3499, 130699, 524243, 6917, 527143, 1, 530051, 1493, 1, 1, 1, 134339, 538823, 1, 541763, 2027, 28669, 136547, 2131, 1, 550631, 1663, 29137, 1901, 556583, 1, 559571, 1, 1, 1699, 565571, 141769, 1, 1, 571603, 7541, 574631, 144037, 577667, 4993, 580711, 1, 1, 146323, 34519, 147089, 1, 1, 592967, 148627, 1, 149399, 599143, 1, 1, 6563, 4621, 2137, 10313, 8971, 32189, 1, 6907, 154073, 16699, 6733, 1, 1, 21523, 156437, 1, 157229, 4813, 8317, 1999, 1, 37463, 159617, 1637, 8443, 2503, 1, 9649, 1, 1, 162829, 652931, 163637, 1, 164447, 2957, 2801, 34877, 9769, 665923, 9817, 1, 5783, 35393, 168527, 675751, 1, 1, 1, 8221, 1, 1, 171827, 6689, 172657, 1, 1, 1, 3709, 1, 1, 702311, 1, 4079, 1, 19163, 177679, 1, 1, 31121, 179369, 24799, 10601, 10177, 10651, 2011, 181919, 10273, 182773, 1, 3907, 38749, 184487, 19991, 2539, 743111, 6421, 746563, 187073, 44119, 1, 1, 8209, 756967, 1, 12889, 2141, 763943, 1789, 767443, 1, 770951, 1, 774467, 194057, 16553, 11467, 781523, 11519, 1, 196709, 5023, 197597, 1, 2719, 1, 199379, 799303, 200273, 42257, 5437, 13669, 202067, 2803, 202967, 2081, 203869, 17389, 1, 28307, 2311, 824531, 1, 6521, 207497, 831811, 208409, 1, 1, 22679, 1, 842791, 12421, 1, 1, 2029, 212999, 853843, 1, 5261, 214849, 45329, 4591, 864967, 3673, 868691, 2113, 1, 7537, 51539, 1, 1, 220447, 883667, 1, 3299, 222329, 8329, 223273, 1, 11801, 1, 225167, 31123, 1, 906371, 1, 1949, 228023, 6053, 3881, 917831, 229937, 921667, 10039, 10399, 231859, 1, 232823, 49117, 1, 1, 1, 55351, 1, 1, 236699, 1, 237673, 20269, 238649, 2459, 8263, 960467, 240607, 9013, 241589, 42101, 1, 972263, 14327, 976211, 244547, 1, 12923, 1, 8501, 42961, 1, 1, 1, 21193, 249517,

6. Sequence of the polynom (only primes)

67, 17, 71, 19, 83, 23, 103, 29, 131, 37, 167, 47, 211, 59, 263, 73, 89, 107, 467, 127, 149, 643, 173, 743, 199, 227, 967, 257, 1091, 1223, 359, 1511, 397, 1667, 1831, 479, 2003, 523, 569, 2371, 617, 151, 163, 719, 157, 773, 3203, 829, 887, 193, 947, 3911, 1009, 181, 4423, 4691, 4967, 1277, 241, 1423, 5843, 1499, 6151, 223, 1657, 6791, 419, 1823, 439, 1997, 8167, 2087, 449, 2179, 307, 2273, 9283, 509, 2467, 10067, 283, 10883, 2879, 11731, 12611, 3209, 13063, 3323, 13523, 823, 3557, 3677, 14951, 15443, 3923, 4049, 16451, 4177, 17491, 269, 977, 277, 659, 4987, 20231, 293, 5273, 21383, 5419, 22567, 5717, 5869, 1399, 317, 1061, 25031, 6337, 25667, 6659, 457, 6823, 1201, 1489, 421, 349, 431, 7499, 1597, 7673, 839, 31751, 32467, 33191, 8389, 33923, 8573, 2039, 461, 2083, 389, 613, 9137, 36931, 491, 1019, 1327, 9719, 1777, 607, 571, 619, 2237, 10729, 43331, 10937, 1523, 45863, 46723, 11789, 47591, 12007, 2851, 12227, 2903, 1069, 12899, 52051, 13127, 52967, 53891, 751, 601, 55763, 827, 56711, 57667, 14537, 58631, 14779, 3137, 1289, 15269, 2677, 15767, 16273, 16529, 2897, 16787, 17047, 68711, 911, 69763, 17573, 70823, 17839, 953, 557, 74051, 1097, 76243, 77351, 19477, 78467, 859, 691, 907, 20323, 1153, 20897, 4951, 21187, 85331, 3761, 21773, 87683, 761, 22367, 90067, 1193, 853, 4021, 93703, 1609, 23887, 96167, 24197, 3359, 24509, 98663, 99923, 1093, 2153, 25457, 5393, 26099, 26423, 1163, 1297, 27077, 108967, 27407, 110291, 27739, 111623, 1361, 28409, 114311, 3163, 29429, 709, 1567, 119783, 30119, 121171, 30467, 30817, 821, 125383, 1087, 7459, 32237, 1021, 881, 131111, 1433, 6977, 863, 34057, 4723, 138451, 139943, 2069, 141443, 1871, 877, 971, 1979, 36307, 145991, 1931, 5087, 6481, 797, 150611, 37847, 8951, 38237, 9043, 38629, 155303, 39023, 39419, 4283, 1373, 2713, 41023, 1601, 2437, 3613, 1471, 2267, 4679, 1697, 176467, 2333, 178151, 45599, 46027, 184967, 46457, 8117, 46889, 1279, 191911, 48197, 10193, 2861, 2887, 197203, 49523, 1031, 200771, 50417, 202567, 50867, 204371, 206183, 2251, 208003, 1801, 12343, 12451, 53147, 2399, 53609, 2351, 217223, 54539, 219091, 1483, 1913, 55949, 224743, 3319, 226643, 3347, 230467, 1231, 232391, 234323, 997, 8147, 10357, 59797, 6491, 14243, 60779, 3371, 3251, 248071, 8623, 2729, 3037, 254083, 63773, 5449, 64279, 258131, 13693, 65809, 264263, 2287, 4549, 68399, 11941, 1877, 280967, 70507, 2161, 71039, 12401, 3767, 4289, 72109, 289511, 72647, 1109, 293831, 4337, 8059, 3253, 15809, 75367, 302567, 304771, 1627, 77023, 1283, 3373, 10739, 78137, 18451, 78697, 18583, 79259, 8599, 79823, 4513, 4231, 81527, 327251, 2099, 1759, 331843, 4931, 14629, 84407, 338791, 2297, 341123, 18077, 2971, 345811, 87337, 3823, 20759, 88523, 20899, 89119, 357671, 1229, 2441, 364883, 4817, 4127, 369731, 92737, 372167, 3637, 5527, 2969, 94573, 95189, 381991, 10391, 8233, 97673, 17041, 98299, 23203, 98927, 3433, 399491, 100189, 402023, 100823, 6857, 1429, 11003, 4603, 1447, 5647, 5441, 414803, 417383, 422567, 105967, 1873, 106619, 2833, 107273, 3389, 2917, 2593, 108587, 25939, 110573, 2381, 6701, 112577, 15667, 3079, 457043, 114599, 2753, 6781, 462467, 2963, 116639, 5101, 470663, 6211, 473411, 4093, 476167, 478931, 120079, 28499, 121469, 28663, 122167, 25793, 122867, 492871, 4261, 3041, 26237, 124979, 125687, 2221, 126397, 7477, 509863, 10909, 6803, 518467, 2203, 3499, 130699, 524243, 6917, 527143, 530051, 1493, 134339, 538823, 541763, 2027, 28669, 136547, 2131, 550631, 1663, 29137, 1901, 556583, 559571, 1699, 565571, 141769, 571603, 7541, 574631, 144037, 577667, 4993, 580711, 146323, 34519, 147089, 592967, 148627, 149399, 599143, 6563, 4621, 2137, 10313, 8971, 32189, 6907, 154073, 16699, 6733, 21523, 156437, 157229, 4813, 8317, 1999, 37463, 159617, 1637, 8443, 2503, 9649, 162829, 652931, 163637, 164447, 2957, 2801, 34877, 9769, 665923, 9817, 5783, 35393, 168527, 675751, 8221, 171827, 6689, 172657, 3709, 702311, 4079, 19163, 177679, 31121, 179369, 24799, 10601, 10177, 10651, 2011, 181919, 10273, 182773, 3907, 38749, 184487, 19991, 2539, 743111, 6421, 746563, 187073, 44119, 8209, 756967, 12889, 2141, 763943, 1789, 767443, 770951, 774467, 194057, 16553, 11467, 781523, 11519, 196709, 5023, 197597, 2719, 199379, 799303, 200273, 42257, 5437, 13669, 202067, 2803, 202967, 2081, 203869, 17389, 28307, 2311, 824531, 6521, 207497, 831811, 208409, 22679, 842791, 12421, 2029, 212999, 853843, 5261, 214849, 45329, 4591, 864967, 3673, 868691, 2113, 7537, 51539, 220447, 883667, 3299, 222329, 8329, 223273, 11801, 225167, 31123, 906371, 1949, 228023, 6053, 3881, 917831, 229937, 921667, 10039, 10399, 231859, 232823, 49117, 55351, 236699, 237673, 20269, 238649, 2459, 8263, 960467, 240607, 9013, 241589, 42101, 972263, 14327, 976211, 244547, 12923, 8501, 42961, 21193, 249517,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+67 and
the reducible primes which appear as divisor for the first time
p | x^2x+67 and p < x^2x+67

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)
110111101.1000001.1000001.1000000.0000000.0000000.000000
21008068120.8000000.6800000.8000007.2727276.181818inf
31.0007163933230.7160000.3930000.7160008.9500005.77941226.916666
410.0007.0342.7254.3090.7034000.2725000.7034009.8240226.93384213.340557
5100.00070.41820.86949.5490.7041800.2086900.70418010.0110897.65834911.498956
61.000.000702.573169.158533.4150.7025730.1691580.7025739.9771808.10570710.765404
710.000.0007.010.5411.421.8905.588.6510.7010540.1421890.7010549.9783818.40569210.477117
8100.000.00069.986.87712.278.84957.708.0280.6998690.1227880.6998699.9830928.63558310.325932
91.000.000.000699.029.551108.086.061590.943.4900.6990300.1080860.6990309.9880088.80262210.240231
1010.000.000.0006.983.656.729965.350.0356.018.306.6940.6983660.0965350.6983669.9905038.93130910.184234


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
389901.1250001.1250000.0000001.8000001.800000-nan
416161601.0000001.0000000.0000001.7777781.777778-nan
532292900.9062500.9062500.0000001.8125001.812500-nan
664544950.8437500.7656250.0781251.8620691.689655inf
71289783140.7578120.6484380.1093751.7962961.6938782.800000
8256185132530.7226560.5156250.2070311.9072171.5903613.785714
95123692301390.7207030.4492190.2714841.9945951.7424242.622642
101.0247323983340.7148440.3886720.3261721.9837401.7304352.402878
112.0481.4477127350.7065430.3476560.3588871.9767761.7889452.200599
124.0962.8841.2571.6270.7041020.3068850.3972171.9930891.7654492.213605
138.1925.7842.3083.4760.7060550.2817380.4243162.0055481.8361182.136447
1416.38411.5134.2147.2990.7026980.2572020.4454961.9904911.8258232.099827
1532.76823.0917.67615.4150.7046810.2342530.4704282.0056461.8215472.111933
1665.53646.11714.27931.8380.7036900.2178800.4858091.9971851.8602142.065391
17131.07292.25426.65265.6020.7038420.2033390.5005042.0004341.8665172.060494
18262.144184.42249.879134.5430.7035140.1902730.5132411.9990681.8714922.050898
19524.288368.49993.969274.5300.7028560.1792320.5236241.9981291.8839392.040463
201.048.576736.671176.730559.9410.7025440.1685430.5340011.9991131.8807272.039635
212.097.1521.472.273334.1001.138.1730.7020340.1593110.5427231.9985491.8904542.032666
224.194.3042.942.638634.6922.307.9460.7015800.1513220.5502571.9987041.8997072.027764
238.388.6085.881.4301.207.5494.673.8810.7011210.1439510.5571701.9986931.9025752.025126
2416.777.21611.756.2382.303.5419.452.6970.7007260.1373020.5634251.9988741.9076172.022451
2533.554.43223.500.6854.404.02019.096.6650.7003750.1312500.5691251.9989971.9118482.020234
2667.108.86446.978.9768.439.23638.539.7400.7000410.1257540.5742871.9990471.9162572.018140
27134.217.72893.919.06316.199.03777.720.0260.6997520.1206920.5790591.9991721.9194912.016620
28268.435.456187.764.97031.142.370156.622.6000.6994790.1160140.5834651.9992211.9224832.015215
29536.870.912375.402.17059.965.174315.436.9960.6992410.1116940.5875471.9993191.9255172.013994
301.073.741.824750.554.623115.628.351634.926.2720.6990080.1076870.5913211.9993351.9282582.012846
312.147.483.6481.500.643.590223.238.5361.277.405.0540.6987920.1039540.5948381.9993801.9306562.011895
324.294.967.2963.000.423.126431.553.1232.568.870.0030.6985900.1004790.5981121.9994241.9331482.011007
338.589.934.5925.999.242.887835.121.4805.164.121.4070.6984040.0972210.6011831.9994661.9351532.010270
3417.179.869.18411.995.557.4151.617.826.97510.377.730.4400.6982330.0941700.6040631.9995121.9372352.009583
3534.359.738.36823.985.455.9543.137.347.29420.848.108.6600.6980690.0913090.6067601.9995281.9392352.008928
3668.719.476.73647.960.352.6776.089.659.79341.870.692.8840.6979150.0886160.6092991.9995601.9410222.008369


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
123121101
245231301
389361413
416166102626
532291019411410
664491633717718
712883265711321129
8256132448820501745
95122307715330883181
101.0243981352635614349150
112.0487122434699325886275
124.0961.257422835169461153474
138.1922.3087791.529327853283845
1416.3844.2141.4172.7975671.5745191.554
1532.7687.6762.5545.1221.0162.8799652.816
1665.53614.2794.7819.4981.8695.3301.8095.271
17131.07226.6528.86917.7833.4749.8493.4239.906
18262.14449.87916.63333.2466.45218.5626.46618.399
19524.28893.96931.41762.55212.11834.93012.17434.747
201.048.576176.73059.028117.70222.78865.65022.70265.590
212.097.152334.100111.479222.62143.002124.24442.806124.048
224.194.304634.692211.498423.19481.641235.79081.258236.003
238.388.6081.207.549402.393805.156154.838449.085154.374449.252
2416.777.2162.303.541768.3501.535.191294.836856.889294.518857.298
2533.554.4324.404.0201.468.6132.935.407562.8251.639.376562.3841.639.435
2667.108.8648.439.2362.813.6155.625.6211.077.6083.142.1221.076.8013.142.705
27134.217.72816.199.0375.399.70210.799.3352.065.7756.032.5262.066.5786.034.158
28268.435.45631.142.37010.381.61220.760.7583.969.82111.602.5603.966.80411.603.185
29536.870.91259.965.17419.985.30539.979.8697.636.04222.350.3157.633.06922.345.748
301.073.741.824115.628.35138.538.61077.089.74114.710.65943.104.55914.712.54243.100.591
312.147.483.648223.238.53674.407.201148.831.33528.391.09583.225.29128.390.28583.231.865
324.294.967.296431.553.123143.836.265287.716.85854.850.140160.922.07154.848.352160.932.560
338.589.934.592835.121.480278.356.681556.764.799106.077.366311.476.754106.090.636311.476.724
3417.179.869.1841.617.826.975539.265.2271.078.561.748205.406.774603.513.649205.412.440603.494.112
3534.359.738.3683.137.347.2941.045.786.7612.091.560.533398.139.9631.170.545.004398.149.9281.170.512.399
3668.719.476.7366.089.659.7932.029.912.1974.059.747.596772.458.7902.272.397.482772.450.3742.272.353.147


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
380000000
4160000000
5320000000
6645501121
7128141133434
825653312211131712
9512139706935353732
101.02433418514989799571
112.048735392343180175201179
124.0961.627872755406394435392
138.1923.4761.8131.663897824922833
1416.3847.2993.7913.5081.8571.7511.9331.758
1532.76815.4158.0547.3613.9353.7543.9753.751
1665.53631.83816.53015.3088.1617.7298.2257.723
17131.07265.60233.90431.69816.76716.02816.79216.015
18262.144134.54369.37365.17034.35032.80334.45032.940
19524.288274.530140.939133.59170.32667.10670.17966.919
201.048.576559.941286.674273.267142.736137.194143.123136.888
212.097.1521.138.173581.124557.049290.000279.319289.831279.023
224.194.3042.307.9461.177.3201.130.626587.151566.630586.867567.298
238.388.6084.673.8812.382.2832.291.5981.187.2881.149.5681.187.7801.149.245
2416.777.2169.452.6974.812.4474.640.2502.398.4942.328.8182.398.4342.326.951
2533.554.43219.096.6659.712.6809.383.9854.839.2744.708.9024.841.5344.706.955
2667.108.86438.539.74019.582.73118.957.0099.760.6429.508.1309.764.9609.506.008
27134.217.72877.720.02639.458.59438.261.43219.672.90819.186.09319.674.24519.186.780
28268.435.456156.622.60079.453.09177.169.50939.621.79638.687.86939.629.51438.683.421
29536.870.912315.436.996159.903.827155.533.16979.758.53277.958.41979.770.40077.949.645
301.073.741.824634.926.272321.671.313313.254.959160.461.222157.001.549160.481.129156.982.372
312.147.483.6481.277.405.054646.803.334630.601.720322.674.391316.016.593322.709.247316.004.823
324.294.967.2962.568.870.0031.300.046.4761.268.823.527648.605.127635.797.714648.675.390635.791.772
338.589.934.5925.164.121.4072.612.245.5962.551.875.8111.303.360.6411.278.664.4811.303.470.7921.278.625.493
3417.179.869.18410.377.730.4405.247.158.2745.130.572.1662.618.344.6732.570.496.4722.618.425.7002.570.463.595
3534.359.738.36820.848.108.66010.536.816.42510.311.292.2355.258.285.3875.165.779.5205.258.361.9715.165.681.782
3668.719.476.73641.870.692.88421.153.754.71820.716.938.16610.557.192.99910.378.068.27710.557.391.79410.378.039.814


8. Check for existing Integer Sequences by OEIS

Found in Database : 67, 17, 71, 19, 83, 23, 103, 29, 131, 37, 167, 47, 211, 59, 263, 73, 1, 89, 1, 107,
Found in Database : 67, 17, 71, 19, 83, 23, 103, 29, 131, 37, 167, 47, 211, 59, 263, 73, 89, 107, 467, 127, 149, 643, 173, 743, 199, 227, 967, 257, 1091, 1223, 359, 1511, 397,
Found in Database : 17, 19, 23, 29, 37, 47, 59, 67, 71, 73, 83, 89, 103, 107, 127, 131, 149,