Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:03:55
Deutsch
29.Mar 2024

Polynom = x^2+68x-61

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) = 61 = 61
f(1) = 1 = 1
f(2) = 79 = 79
f(3) = 19 = 19
f(4) = 227 = 227
f(5) = 19 = 19
f(6) = 383 = 383
f(7) = 29 = 29
f(8) = 547 = 547
f(9) = 79 = 79
f(10) = 719 = 719
f(11) = 101 = 101
f(12) = 899 = 29*31
f(13) = 31 = 31
f(14) = 1087 = 1087
f(15) = 37 = 37
f(16) = 1283 = 1283
f(17) = 173 = 173
f(18) = 1487 = 1487
f(19) = 199 = 199
f(20) = 1699 = 1699
f(21) = 113 = 113
f(22) = 1919 = 19*101
f(23) = 127 = 127
f(24) = 2147 = 19*113
f(25) = 283 = 283
f(26) = 2383 = 2383
f(27) = 313 = 313
f(28) = 2627 = 37*71
f(29) = 43 = 43
f(30) = 2879 = 2879
f(31) = 47 = 47
f(32) = 3139 = 43*73
f(33) = 409 = 409
f(34) = 3407 = 3407
f(35) = 443 = 443
f(36) = 3683 = 29*127
f(37) = 239 = 239
f(38) = 3967 = 3967
f(39) = 257 = 257
f(40) = 4259 = 4259
f(41) = 551 = 19*29
f(42) = 4559 = 47*97
f(43) = 589 = 19*31
f(44) = 4867 = 31*157
f(45) = 157 = 157
f(46) = 5183 = 71*73
f(47) = 167 = 167
f(48) = 5507 = 5507
f(49) = 709 = 709
f(50) = 5839 = 5839
f(51) = 751 = 751
f(52) = 6179 = 37*167
f(53) = 397 = 397
f(54) = 6527 = 61*107
f(55) = 419 = 419
f(56) = 6883 = 6883
f(57) = 883 = 883
f(58) = 7247 = 7247
f(59) = 929 = 929
f(60) = 7619 = 19*401
f(61) = 61 = 61
f(62) = 7999 = 19*421
f(63) = 1 = 1
f(64) = 8387 = 8387
f(65) = 1073 = 29*37
f(66) = 8783 = 8783
f(67) = 1123 = 1123
f(68) = 9187 = 9187
f(69) = 587 = 587
f(70) = 9599 = 29*331
f(71) = 613 = 613
f(72) = 10019 = 43*233
f(73) = 1279 = 1279
f(74) = 10447 = 31*337
f(75) = 1333 = 31*43
f(76) = 10883 = 10883
f(77) = 347 = 347
f(78) = 11327 = 47*241
f(79) = 361 = 19*19
f(80) = 11779 = 11779
f(81) = 1501 = 19*79
f(82) = 12239 = 12239
f(83) = 1559 = 1559
f(84) = 12707 = 97*131
f(85) = 809 = 809
f(86) = 13183 = 13183
f(87) = 839 = 839
f(88) = 13667 = 79*173
f(89) = 1739 = 37*47
f(90) = 14159 = 14159
f(91) = 1801 = 1801
f(92) = 14659 = 107*137
f(93) = 233 = 233
f(94) = 15167 = 29*523
f(95) = 241 = 241
f(96) = 15683 = 15683
f(97) = 1993 = 1993
f(98) = 16207 = 19*853
f(99) = 2059 = 29*71
f(100) = 16739 = 19*881

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+68x-61

f(0)=61
f(1)=1
f(2)=79
f(3)=19
f(4)=227
f(5)=1
f(6)=383
f(7)=29
f(8)=547
f(9)=1
f(10)=719
f(11)=101
f(12)=31
f(13)=1
f(14)=1087
f(15)=37
f(16)=1283
f(17)=173
f(18)=1487
f(19)=199
f(20)=1699
f(21)=113
f(22)=1
f(23)=127
f(24)=1
f(25)=283
f(26)=2383
f(27)=313
f(28)=71
f(29)=43
f(30)=2879
f(31)=47
f(32)=73
f(33)=409
f(34)=3407
f(35)=443
f(36)=1
f(37)=239
f(38)=3967
f(39)=257
f(40)=4259
f(41)=1
f(42)=97
f(43)=1
f(44)=157
f(45)=1
f(46)=1
f(47)=167
f(48)=5507
f(49)=709
f(50)=5839
f(51)=751
f(52)=1
f(53)=397
f(54)=107
f(55)=419
f(56)=6883
f(57)=883
f(58)=7247
f(59)=929
f(60)=401
f(61)=1
f(62)=421
f(63)=1
f(64)=8387
f(65)=1
f(66)=8783
f(67)=1123
f(68)=9187
f(69)=587
f(70)=331
f(71)=613
f(72)=233
f(73)=1279
f(74)=337
f(75)=1
f(76)=10883
f(77)=347
f(78)=241
f(79)=1
f(80)=11779
f(81)=1
f(82)=12239
f(83)=1559
f(84)=131
f(85)=809
f(86)=13183
f(87)=839
f(88)=1
f(89)=1
f(90)=14159
f(91)=1801
f(92)=137
f(93)=1
f(94)=523
f(95)=1
f(96)=15683
f(97)=1993
f(98)=853
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+68x-61 could be written as f(y)= y^2-1217 with x=y-34

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+34
f'(x)>2x+67

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

61, 1, 79, 19, 227, 1, 383, 29, 547, 1, 719, 101, 31, 1, 1087, 37, 1283, 173, 1487, 199, 1699, 113, 1, 127, 1, 283, 2383, 313, 71, 43, 2879, 47, 73, 409, 3407, 443, 1, 239, 3967, 257, 4259, 1, 97, 1, 157, 1, 1, 167, 5507, 709, 5839, 751, 1, 397, 107, 419, 6883, 883, 7247, 929, 401, 1, 421, 1, 8387, 1, 8783, 1123, 9187, 587, 331, 613, 233, 1279, 337, 1, 10883, 347, 241, 1, 11779, 1, 12239, 1559, 131, 809, 13183, 839, 1, 1, 14159, 1801, 137, 1, 523, 1, 15683, 1993, 853, 1, 881, 1063, 467, 1097, 17827, 1, 593, 2333, 18947, 601, 149, 619, 1, 2549, 151, 1, 21283, 1, 509, 1, 1, 2851, 379, 1, 23747, 1, 659, 193, 863, 3169, 25679, 3251, 26339, 1667, 1, 1709, 1, 1, 1493, 1, 29059, 919, 29759, 941, 30467, 3853, 31183, 3943, 31907, 2017, 1, 2063, 1151, 4219, 34127, 1, 34883, 1, 829, 563, 461, 1, 37199, 1, 37987, 2399, 38783, 1, 1277, 4999, 569, 5101, 877, 1301, 2213, 1327, 1, 5413, 599, 5519, 44579, 1, 45439, 1, 46307, 5843, 1627, 5953, 677, 1, 1, 1, 683, 1, 50767, 1, 51683, 3259, 1697, 1, 1447, 1, 1, 6869, 1289, 1747, 56383, 1777, 57347, 7229, 2011, 7351, 3121, 1, 1, 1, 61283, 7723, 1, 1, 63299, 997, 64319, 1013, 647, 8233, 66383, 8363, 67427, 1, 1, 1, 69539, 1, 70607, 8893, 739, 1, 72767, 1, 73859, 1, 74959, 9439, 1, 4789, 977, 1, 78307, 9859, 1, 1, 4241, 317, 81727, 643, 82883, 10433, 84047, 1, 2749, 1, 86399, 5437, 87587, 1, 1889, 11173, 1, 1, 91199, 1, 92419, 1, 2531, 11783, 1, 1, 991, 6047, 97379, 12251, 98639, 12409, 99907, 1571, 101183, 1, 5393, 12889, 1, 1, 3389, 6607, 106367, 6689, 1, 1, 1787, 13709, 110339, 3469, 3851, 3511, 113027, 1, 1069, 757, 1, 1, 117119, 1, 1669, 1, 1187, 15073, 121283, 953, 1553, 1, 1, 15601, 4049, 1, 1, 1, 1, 8069, 6833, 16319, 1, 1, 132739, 1, 134207, 4217, 1, 17053, 1879, 17239, 1, 8713, 140159, 8807, 141667, 937, 1093, 947, 3911, 2273, 146239, 2297, 147779, 1, 4817, 1, 150883, 9479, 971, 1, 1, 1, 155599, 1, 8273, 4937, 1, 4987, 160387, 20149, 161999, 433, 1249, 1, 165247, 1, 3881, 20963, 168527, 21169, 170179, 1, 1, 1, 1, 1, 1049, 22003, 1291, 1, 178559, 11213, 180259, 22639, 181967, 22853, 183683, 1, 5011, 5821, 187139, 1, 9941, 23719, 1, 11969, 192383, 1, 1303, 24379, 195919, 1, 197699, 1, 199487, 1, 1, 1, 1, 1, 3359, 1, 1, 1, 1, 26183, 2083, 1, 212227, 6661, 2207, 6719, 215939, 27109, 1259, 1, 219683, 13789, 221567, 13907, 1, 28051, 1, 28289, 227267, 1783, 7393, 1, 231107, 29009, 1, 29251, 234979, 14747, 1, 14869, 238883, 29983, 1061, 1, 242819, 1, 5693, 7681, 246787, 1, 248783, 31223, 250787, 15737, 3463, 1, 1, 1, 1, 32233, 1, 1, 1, 4093, 13841, 541, 2477, 1, 3659, 16759, 269183, 16889, 4447, 34039, 273359, 34301, 275459, 8641, 277567, 8707, 7559, 1847, 281807, 1861, 9791, 1, 6653, 1, 288227, 1, 290383, 36433, 9437, 1, 1481, 2311, 6317, 1, 299087, 1, 1, 18899, 15973, 19037, 2857, 38351, 307919, 38629, 1, 1, 312383, 1, 314627, 1361, 316879, 1, 1, 1, 11083, 1, 1871, 2137, 1399, 1319, 10589, 5147, 5419, 1, 2621, 1, 9059, 42043, 1, 1, 339839, 21313, 342179, 1, 18133, 1, 18257, 1, 12043, 1, 1459, 44101, 4481, 1531, 356387, 22349, 358783, 1, 1, 1, 1, 1471, 366019, 1, 368447, 1, 370883, 1, 373327, 46819, 375779, 23563, 378239, 641, 380707, 47743, 3391, 1657, 8969, 1, 388159, 1, 1, 48989, 20693, 1, 5573, 24809, 398207, 24967, 400739, 1621, 13009, 1, 405827, 6361, 8689, 1, 3137, 1, 6779, 2729, 1741, 1373, 5897, 26249, 1, 1, 423887, 53149, 11527, 1, 429119, 13451, 4451, 54133, 434383, 1, 4327, 27397, 1, 1, 1, 1789, 1, 55793, 447683, 1, 450367, 3529, 3307, 1, 9697, 57139, 12391, 1, 461183, 28909, 463907, 3061, 16091, 3079, 469379, 1, 472127, 14797, 1, 1609, 3761, 59879, 480419, 30113, 3527, 1, 1, 60923, 5039, 1, 25873, 7703, 26021, 1, 3167, 1, 1, 62683, 502883, 733, 505727, 1, 508579, 1723, 511439, 64109, 1, 1, 517183, 1, 520067, 1, 522959, 65551, 1, 32957, 1, 1, 4969, 66643, 534607, 1, 11437, 4211, 4783, 1, 543427, 68113, 1, 68483, 1, 1, 1, 34613, 1, 1, 558287, 1, 3767, 1, 1, 17681, 1, 1, 1, 71479, 3797, 1, 18593, 1901, 579427, 1, 1, 1973, 1, 9173, 5501, 9221, 12589, 2557, 8377, 1, 597859, 37463, 600959, 37657, 31793, 75703, 31957, 1619, 1811, 19121, 613439, 19219, 616579, 77269, 2593, 2099, 1, 1, 1, 39227, 14633, 2719, 8663, 1, 1, 1, 22027, 5003, 17351, 80449, 3343, 1, 1693, 40627, 651647, 40829, 8971, 1, 658127, 82469, 3823, 20719, 34981, 1, 35153, 1, 11003, 2713, 21757, 42257, 23371, 42463, 1, 1, 14561, 2957, 687683, 10771, 5441, 1, 694339, 1, 1, 1, 1, 1, 16381, 44129, 707747, 1, 2767, 89101, 6323, 22381, 717887, 1, 721283, 90373, 1, 1, 38321, 45613, 38501, 45827, 25343, 92083, 738383, 1, 741827, 1, 1, 1, 1, 93809, 752207, 1, 3329, 47339, 759167, 2503, 762659, 1, 20707, 95989, 2459, 24107, 17981, 1, 26783, 1, 25169, 1, 783779, 1, 12907, 1, 790883, 99083, 41813, 99529, 1, 12497, 1999, 12553, 805187, 1, 21859, 101323, 812387, 1, 6229, 1, 819619, 102679, 2909, 1, 826883, 1, 10513, 1, 1, 3371, 1, 104959, 1, 52709, 845183, 1, 18061, 2473, 852559, 106801, 19913, 1, 1, 6733, 1, 108193, 1, 1, 1, 54563, 874879, 1, 2087, 110063, 30427, 110533, 1, 27751, 8317, 1, 1, 111949, 6551, 1, 24359, 2971, 905087, 56687, 8999, 1, 12503, 114329, 19501, 1, 1, 14411, 8179, 115769, 928079, 1, 21673, 58367, 49253, 1, 1, 117703, 943567, 118189, 1, 29669, 30689, 1, 955267, 119653, 959183, 1, 9001, 60317, 12241, 1, 1, 1, 974927, 6427, 978883, 1, 982847, 3847, 986819, 123601, 990799, 1, 34303, 62299, 3191, 62549, 1002787, 1, 1006799, 2683, 53201, 1021, 1, 1, 16703, 3449, 23789, 128119, 1026979, 1, 21937, 64567, 1035107, 129643, 6619, 1, 1, 16333, 28307, 1, 1051459, 1, 1055567, 1, 1059683, 66359, 36683, 66617,

6. Sequence of the polynom (only primes)

61, 79, 19, 227, 383, 29, 547, 719, 101, 31, 1087, 37, 1283, 173, 1487, 199, 1699, 113, 127, 283, 2383, 313, 71, 43, 2879, 47, 73, 409, 3407, 443, 239, 3967, 257, 4259, 97, 157, 167, 5507, 709, 5839, 751, 397, 107, 419, 6883, 883, 7247, 929, 401, 421, 8387, 8783, 1123, 9187, 587, 331, 613, 233, 1279, 337, 10883, 347, 241, 11779, 12239, 1559, 131, 809, 13183, 839, 14159, 1801, 137, 523, 15683, 1993, 853, 881, 1063, 467, 1097, 17827, 593, 2333, 18947, 601, 149, 619, 2549, 151, 21283, 509, 2851, 379, 23747, 659, 193, 863, 3169, 25679, 3251, 26339, 1667, 1709, 1493, 29059, 919, 29759, 941, 30467, 3853, 31183, 3943, 31907, 2017, 2063, 1151, 4219, 34127, 34883, 829, 563, 461, 37199, 37987, 2399, 38783, 1277, 4999, 569, 5101, 877, 1301, 2213, 1327, 5413, 599, 5519, 44579, 45439, 46307, 5843, 1627, 5953, 677, 683, 50767, 51683, 3259, 1697, 1447, 6869, 1289, 1747, 56383, 1777, 57347, 7229, 2011, 7351, 3121, 61283, 7723, 63299, 997, 64319, 1013, 647, 8233, 66383, 8363, 67427, 69539, 70607, 8893, 739, 72767, 73859, 74959, 9439, 4789, 977, 78307, 9859, 4241, 317, 81727, 643, 82883, 10433, 84047, 2749, 86399, 5437, 87587, 1889, 11173, 91199, 92419, 2531, 11783, 991, 6047, 97379, 12251, 98639, 12409, 99907, 1571, 101183, 5393, 12889, 3389, 6607, 106367, 6689, 1787, 13709, 110339, 3469, 3851, 3511, 113027, 1069, 757, 117119, 1669, 1187, 15073, 121283, 953, 1553, 15601, 4049, 8069, 6833, 16319, 132739, 134207, 4217, 17053, 1879, 17239, 8713, 140159, 8807, 141667, 937, 1093, 947, 3911, 2273, 146239, 2297, 147779, 4817, 150883, 9479, 971, 155599, 8273, 4937, 4987, 160387, 20149, 161999, 433, 1249, 165247, 3881, 20963, 168527, 21169, 170179, 1049, 22003, 1291, 178559, 11213, 180259, 22639, 181967, 22853, 183683, 5011, 5821, 187139, 9941, 23719, 11969, 192383, 1303, 24379, 195919, 197699, 199487, 3359, 26183, 2083, 212227, 6661, 2207, 6719, 215939, 27109, 1259, 219683, 13789, 221567, 13907, 28051, 28289, 227267, 1783, 7393, 231107, 29009, 29251, 234979, 14747, 14869, 238883, 29983, 1061, 242819, 5693, 7681, 246787, 248783, 31223, 250787, 15737, 3463, 32233, 4093, 13841, 541, 2477, 3659, 16759, 269183, 16889, 4447, 34039, 273359, 34301, 275459, 8641, 277567, 8707, 7559, 1847, 281807, 1861, 9791, 6653, 288227, 290383, 36433, 9437, 1481, 2311, 6317, 299087, 18899, 15973, 19037, 2857, 38351, 307919, 38629, 312383, 314627, 1361, 316879, 11083, 1871, 2137, 1399, 1319, 10589, 5147, 5419, 2621, 9059, 42043, 339839, 21313, 342179, 18133, 18257, 12043, 1459, 44101, 4481, 1531, 356387, 22349, 358783, 1471, 366019, 368447, 370883, 373327, 46819, 375779, 23563, 378239, 641, 380707, 47743, 3391, 1657, 8969, 388159, 48989, 20693, 5573, 24809, 398207, 24967, 400739, 1621, 13009, 405827, 6361, 8689, 3137, 6779, 2729, 1741, 1373, 5897, 26249, 423887, 53149, 11527, 429119, 13451, 4451, 54133, 434383, 4327, 27397, 1789, 55793, 447683, 450367, 3529, 3307, 9697, 57139, 12391, 461183, 28909, 463907, 3061, 16091, 3079, 469379, 472127, 14797, 1609, 3761, 59879, 480419, 30113, 3527, 60923, 5039, 25873, 7703, 26021, 3167, 62683, 502883, 733, 505727, 508579, 1723, 511439, 64109, 517183, 520067, 522959, 65551, 32957, 4969, 66643, 534607, 11437, 4211, 4783, 543427, 68113, 68483, 34613, 558287, 3767, 17681, 71479, 3797, 18593, 1901, 579427, 1973, 9173, 5501, 9221, 12589, 2557, 8377, 597859, 37463, 600959, 37657, 31793, 75703, 31957, 1619, 1811, 19121, 613439, 19219, 616579, 77269, 2593, 2099, 39227, 14633, 2719, 8663, 22027, 5003, 17351, 80449, 3343, 1693, 40627, 651647, 40829, 8971, 658127, 82469, 3823, 20719, 34981, 35153, 11003, 2713, 21757, 42257, 23371, 42463, 14561, 2957, 687683, 10771, 5441, 694339, 16381, 44129, 707747, 2767, 89101, 6323, 22381, 717887, 721283, 90373, 38321, 45613, 38501, 45827, 25343, 92083, 738383, 741827, 93809, 752207, 3329, 47339, 759167, 2503, 762659, 20707, 95989, 2459, 24107, 17981, 26783, 25169, 783779, 12907, 790883, 99083, 41813, 99529, 12497, 1999, 12553, 805187, 21859, 101323, 812387, 6229, 819619, 102679, 2909, 826883, 10513, 3371, 104959, 52709, 845183, 18061, 2473, 852559, 106801, 19913, 6733, 108193, 54563, 874879, 2087, 110063, 30427, 110533, 27751, 8317, 111949, 6551, 24359, 2971, 905087, 56687, 8999, 12503, 114329, 19501, 14411, 8179, 115769, 928079, 21673, 58367, 49253, 117703, 943567, 118189, 29669, 30689, 955267, 119653, 959183, 9001, 60317, 12241, 974927, 6427, 978883, 982847, 3847, 986819, 123601, 990799, 34303, 62299, 3191, 62549, 1002787, 1006799, 2683, 53201, 1021, 16703, 3449, 23789, 128119, 1026979, 21937, 64567, 1035107, 129643, 6619, 16333, 28307, 1051459, 1055567, 1059683, 66359, 36683, 66617,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+68x-61 and
the reducible primes which appear as divisor for the first time
p | x^2+68x-61 and p < x^2+68x-61

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)
1108800.8000000.8000000.0000000.0000000.0000000.000000
21007359140.7300000.5900000.1400009.1250007.375000inf
31.0006804512290.6800000.4510000.2290009.3150687.64406816.357143
410.0006.8493.2343.6150.6849000.3234000.36150010.0720597.17073215.786026
5100.00068.85124.70744.1440.6885100.2470700.44144010.0527097.63976512.211342
61.000.000690.121198.180491.9410.6901210.1981800.49194110.0233988.02120911.144006
710.000.0006.902.7951.658.0985.244.6970.6902790.1658100.52447010.0022968.36662710.661232
8100.000.00069.054.18414.267.49554.786.6890.6905420.1426750.54786710.0038008.60473510.446111
91.000.000.000690.784.421125.208.698565.575.7230.6907840.1252090.56557610.0035138.77580110.323233
1010.000.000.0006.909.762.5341.115.755.1425.794.007.3920.6909760.1115760.57940110.0027778.91116310.244441


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
244401.0000001.0000000.0000002.0000002.000000-nan
387700.8750000.8750000.0000001.7500001.750000-nan
416131210.8125000.7500000.0625001.8571431.714286inf
532252230.7812500.6875000.0937501.9230771.8333333.000000
664484170.7500000.6406250.1093751.9200001.8636362.333333
71289070200.7031250.5468750.1562501.8750001.7073172.857143
8256180139410.7031250.5429690.1601562.0000001.9857142.050000
9512352261910.6875000.5097660.1777341.9555561.8776982.219512
101.0246944632310.6777340.4521480.2255861.9715911.7739462.538461
112.0481.4038036000.6850590.3920900.2929692.0216141.7343412.597403
124.0962.8121.4831.3290.6865230.3620610.3244632.0042771.8468242.215000
138.1925.6032.7282.8750.6839600.3330080.3509521.9925321.8395142.163281
1416.38411.2335.0126.2210.6856080.3059080.3797002.0048191.8372432.163826
1532.76822.5309.16413.3660.6875610.2796630.4078982.0056971.8284122.148529
1665.53645.09716.93328.1640.6881260.2583770.4297492.0016421.8477742.107137
17131.07290.29431.46658.8280.6888890.2400670.4488222.0022181.8582652.088766
18262.144180.74458.575122.1690.6894840.2234460.4660382.0017281.8615332.076715
19524.288361.613109.959251.6540.6897220.2097300.4799922.0006921.8772342.059884
201.048.576723.610206.873516.7370.6900880.1972890.4927992.0010621.8813652.053363
212.097.1521.447.327390.7861.056.5410.6901390.1863410.5037982.0001481.8890142.044640
224.194.3042.894.763740.9552.153.8080.6901650.1766570.5135082.0000751.8960632.038547
238.388.6085.790.1831.408.4944.381.6890.6902440.1679060.5223382.0002271.9009172.034392
2416.777.21611.581.9102.684.4898.897.4210.6903360.1600080.5303282.0002671.9059292.030592
2533.554.43223.166.6085.126.62618.039.9820.6904190.1527850.5376332.0002411.9097212.027552
2667.108.86446.338.5049.810.93036.527.5740.6904980.1461940.5443032.0002281.9137212.024812
27134.217.72892.688.23718.813.55173.874.6860.6905810.1401720.5504092.0002421.9176112.022436
28268.435.456185.396.95936.135.394149.261.5650.6906570.1346150.5560432.0002211.9207112.020470
29536.870.912370.831.29269.520.702301.310.5900.6907270.1294920.5612352.0002021.9238952.018675
301.073.741.824741.730.251133.937.302607.792.9490.6907900.1247390.5660512.0001821.9265822.017164
312.147.483.6481.483.594.077258.400.4331.225.193.6440.6908520.1203270.5705252.0001801.9292642.015807
324.294.967.2962.967.434.950499.149.2502.468.285.7000.6909100.1162170.5746932.0001661.9316892.014609
338.589.934.5925.935.347.546965.351.2774.969.996.2690.6909650.1123820.5785842.0001611.9339932.013542
3417.179.869.18411.871.603.3071.869.073.19610.002.530.1110.6910180.1087940.5822242.0001531.9361592.012583
3534.359.738.36823.744.888.8103.622.516.39720.122.372.4130.6910670.1054290.5856382.0001421.9381352.011728
3668.719.476.73647.493.090.3007.027.632.61040.465.457.6900.6911150.1022660.5888502.0001391.9399862.010968


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
122200011
244310211
387430322
41612660444
5322212102659
664412021513716
712870343610261024
8256139667315542248
951226112014137943892
101.0244632092546216774160
112.048803368435110292122279
124.0961.483676807206528219530
138.1922.7281.2281.500361979397991
1416.3845.0122.2742.7386701.8207061.816
1532.7689.1644.1295.0351.2083.3751.2753.306
1665.53616.9337.6809.2532.2516.2132.3296.140
17131.07231.46614.17317.2934.15611.5504.23611.524
18262.14458.57526.39032.1857.81221.4517.85221.460
19524.288109.95949.42360.53614.64240.27114.72740.319
201.048.576206.87392.962113.91127.24376.05927.49576.076
212.097.152390.786175.526215.26051.533143.78251.489143.982
224.194.304740.955332.633408.32297.507272.83097.498273.120
238.388.6081.408.494631.830776.664184.911519.620184.374519.589
2416.777.2162.684.4891.203.6751.480.814352.008990.635350.852990.994
2533.554.4325.126.6262.298.0172.828.609669.8681.893.220668.9091.894.629
2667.108.8649.810.9304.397.4485.413.4821.280.0103.626.6891.278.0553.626.176
27134.217.72818.813.5518.430.71710.382.8342.449.2876.958.2102.448.0496.958.005
28268.435.45636.135.39416.190.06419.945.3304.697.55113.372.3774.694.75513.370.711
29536.870.91269.520.70231.139.77338.380.9299.023.82225.739.0929.021.02025.736.768
301.073.741.824133.937.30259.976.40473.960.89817.360.02249.604.94317.361.24449.611.093
312.147.483.648258.400.433115.678.683142.721.75033.450.85895.748.50933.448.90195.752.165
324.294.967.296499.149.250223.405.324275.743.92664.536.512185.045.00864.533.005185.034.725
338.589.934.592965.351.277431.967.360533.383.917124.671.376357.999.990124.678.631358.001.280
3417.179.869.1841.869.073.196836.185.0121.032.888.184241.139.559693.393.984241.150.383693.389.270
3534.359.738.3683.622.516.3971.620.313.1972.002.203.200466.907.6741.344.347.533466.919.8731.344.341.317
3668.719.476.7367.027.632.6103.142.780.9323.884.851.678904.950.8402.608.843.695904.992.5362.608.845.539


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
4161100001
5323211002
6647523022
7128201198543
8256411823149117
951291355628202518
101.02423110612567526250
112.048600283317155152159134
124.0961.329632697340328345316
138.1922.8751.3761.499722712733708
1416.3846.2213.0363.1851.5761.5031.6111.531
1532.76813.3666.5676.7993.3963.2123.4423.316
1665.53628.16413.93714.2277.1386.8577.2376.932
17131.07258.82829.08829.74015.04114.30915.10414.374
18262.144122.16960.46961.70031.27229.75431.30729.836
19524.288251.654124.644127.01063.82461.64464.42761.759
201.048.576516.737256.215260.522131.104126.813131.910126.910
212.097.1521.056.541524.130532.411268.151259.441269.146259.803
224.194.3042.153.8081.069.8291.083.979546.007530.350547.543529.908
238.388.6084.381.6892.176.4962.205.1931.110.8431.080.1341.111.5541.079.158
2416.777.2168.897.4214.422.7844.474.6372.254.4572.193.9722.254.6552.194.337
2533.554.43218.039.9828.969.0279.070.9554.569.4464.451.2514.568.1124.451.173
2667.108.86436.527.57418.167.34018.360.2349.245.2619.018.7389.244.2449.019.331
27134.217.72873.874.68636.752.89637.121.79018.687.75118.253.55618.682.06918.251.310
28268.435.456149.261.56574.287.02274.974.54337.729.45136.901.62337.730.48536.900.006
29536.870.912301.310.590150.000.601151.309.98976.121.10474.535.24076.129.48674.524.760
301.073.741.824607.792.949302.627.699305.165.250153.477.312150.416.624153.490.526150.408.487
312.147.483.6481.225.193.644610.147.928615.045.716309.253.981303.340.364309.271.710303.327.589
324.294.967.2962.468.285.7001.229.415.0521.238.870.648622.777.577611.364.676622.791.188611.352.259
338.589.934.5924.969.996.2692.475.844.7492.494.151.5201.253.542.4171.231.478.8311.253.536.0601.231.438.961
3417.179.869.18410.002.530.1114.983.531.1025.018.999.0092.521.996.0802.479.292.3382.522.009.4752.479.232.218
3534.359.738.36820.122.372.41310.026.877.38510.095.495.0285.071.997.5634.989.209.9125.071.929.8254.989.235.113
3668.719.476.73640.465.457.69020.166.138.73520.299.318.95510.196.607.24010.036.213.23110.196.370.68310.036.266.536


8. Check for existing Integer Sequences by OEIS

Found in Database : 61, 1, 79, 19, 227, 1, 383, 29, 547, 1, 719, 101, 31, 1, 1087, 37, 1283, 173, 1487, 199,
Found in Database : 61, 79, 19, 227, 383, 29, 547, 719, 101, 31, 1087, 37, 1283, 173, 1487, 199, 1699, 113, 127, 283, 2383, 313, 71, 43, 2879, 47, 73, 409, 3407, 443, 239, 3967, 257,
Found in Database : 19, 29, 31, 37, 43, 47, 61, 71, 73, 79, 97, 101, 107, 113, 127, 131, 137, 149,