Inhaltsverzeichnis

Development of
Algorithmic Constructions

09:50:18
Deutsch
28.Mar 2024

Polynom = x^2+54x-29

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) = 29 = 29
f(1) = 13 = 13
f(2) = 83 = 83
f(3) = 71 = 71
f(4) = 203 = 7*29
f(5) = 133 = 7*19
f(6) = 331 = 331
f(7) = 199 = 199
f(8) = 467 = 467
f(9) = 269 = 269
f(10) = 611 = 13*47
f(11) = 343 = 7*7*7
f(12) = 763 = 7*109
f(13) = 421 = 421
f(14) = 923 = 13*71
f(15) = 503 = 503
f(16) = 1091 = 1091
f(17) = 589 = 19*31
f(18) = 1267 = 7*181
f(19) = 679 = 7*97
f(20) = 1451 = 1451
f(21) = 773 = 773
f(22) = 1643 = 31*53
f(23) = 871 = 13*67
f(24) = 1843 = 19*97
f(25) = 973 = 7*139
f(26) = 2051 = 7*293
f(27) = 1079 = 13*83
f(28) = 2267 = 2267
f(29) = 1189 = 29*41
f(30) = 2491 = 47*53
f(31) = 1303 = 1303
f(32) = 2723 = 7*389
f(33) = 1421 = 7*7*29
f(34) = 2963 = 2963
f(35) = 1543 = 1543
f(36) = 3211 = 13*13*19
f(37) = 1669 = 1669
f(38) = 3467 = 3467
f(39) = 1799 = 7*257
f(40) = 3731 = 7*13*41
f(41) = 1933 = 1933
f(42) = 4003 = 4003
f(43) = 2071 = 19*109
f(44) = 4283 = 4283
f(45) = 2213 = 2213
f(46) = 4571 = 7*653
f(47) = 2359 = 7*337
f(48) = 4867 = 31*157
f(49) = 2509 = 13*193
f(50) = 5171 = 5171
f(51) = 2663 = 2663
f(52) = 5483 = 5483
f(53) = 2821 = 7*13*31
f(54) = 5803 = 7*829
f(55) = 2983 = 19*157
f(56) = 6131 = 6131
f(57) = 3149 = 47*67
f(58) = 6467 = 29*223
f(59) = 3319 = 3319
f(60) = 6811 = 7*7*139
f(61) = 3493 = 7*499
f(62) = 7163 = 13*19*29
f(63) = 3671 = 3671
f(64) = 7523 = 7523
f(65) = 3853 = 3853
f(66) = 7891 = 13*607
f(67) = 4039 = 7*577
f(68) = 8267 = 7*1181
f(69) = 4229 = 4229
f(70) = 8651 = 41*211
f(71) = 4423 = 4423
f(72) = 9043 = 9043
f(73) = 4621 = 4621
f(74) = 9443 = 7*19*71
f(75) = 4823 = 7*13*53
f(76) = 9851 = 9851
f(77) = 5029 = 47*107
f(78) = 10267 = 10267
f(79) = 5239 = 13*13*31
f(80) = 10691 = 10691
f(81) = 5453 = 7*19*41
f(82) = 11123 = 7*7*227
f(83) = 5671 = 53*107
f(84) = 11563 = 31*373
f(85) = 5893 = 71*83
f(86) = 12011 = 12011
f(87) = 6119 = 29*211
f(88) = 12467 = 7*13*137
f(89) = 6349 = 7*907
f(90) = 12931 = 67*193
f(91) = 6583 = 29*227
f(92) = 13403 = 13*1031
f(93) = 6821 = 19*359
f(94) = 13883 = 13883
f(95) = 7063 = 7*1009
f(96) = 14371 = 7*2053
f(97) = 7309 = 7309
f(98) = 14867 = 14867
f(99) = 7559 = 7559
f(100) = 15371 = 19*809

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+54x-29

f(0)=29
f(1)=13
f(2)=83
f(3)=71
f(4)=7
f(5)=19
f(6)=331
f(7)=199
f(8)=467
f(9)=269
f(10)=47
f(11)=1
f(12)=109
f(13)=421
f(14)=1
f(15)=503
f(16)=1091
f(17)=31
f(18)=181
f(19)=97
f(20)=1451
f(21)=773
f(22)=53
f(23)=67
f(24)=1
f(25)=139
f(26)=293
f(27)=1
f(28)=2267
f(29)=41
f(30)=1
f(31)=1303
f(32)=389
f(33)=1
f(34)=2963
f(35)=1543
f(36)=1
f(37)=1669
f(38)=3467
f(39)=257
f(40)=1
f(41)=1933
f(42)=4003
f(43)=1
f(44)=4283
f(45)=2213
f(46)=653
f(47)=337
f(48)=157
f(49)=193
f(50)=5171
f(51)=2663
f(52)=5483
f(53)=1
f(54)=829
f(55)=1
f(56)=6131
f(57)=1
f(58)=223
f(59)=3319
f(60)=1
f(61)=499
f(62)=1
f(63)=3671
f(64)=7523
f(65)=3853
f(66)=607
f(67)=577
f(68)=1181
f(69)=4229
f(70)=211
f(71)=4423
f(72)=9043
f(73)=4621
f(74)=1
f(75)=1
f(76)=9851
f(77)=107
f(78)=10267
f(79)=1
f(80)=10691
f(81)=1
f(82)=227
f(83)=1
f(84)=373
f(85)=1
f(86)=12011
f(87)=1
f(88)=137
f(89)=907
f(90)=1
f(91)=1
f(92)=1031
f(93)=359
f(94)=13883
f(95)=1009
f(96)=2053
f(97)=7309
f(98)=14867
f(99)=7559

b) Substitution of the polynom
The polynom f(x)=x^2+54x-29 could be written as f(y)= y^2-758 with x=y-27

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
f'(x)>2x+53

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

29, 13, 83, 71, 7, 19, 331, 199, 467, 269, 47, 1, 109, 421, 1, 503, 1091, 31, 181, 97, 1451, 773, 53, 67, 1, 139, 293, 1, 2267, 41, 1, 1303, 389, 1, 2963, 1543, 1, 1669, 3467, 257, 1, 1933, 4003, 1, 4283, 2213, 653, 337, 157, 193, 5171, 2663, 5483, 1, 829, 1, 6131, 1, 223, 3319, 1, 499, 1, 3671, 7523, 3853, 607, 577, 1181, 4229, 211, 4423, 9043, 4621, 1, 1, 9851, 107, 10267, 1, 10691, 1, 227, 1, 373, 1, 12011, 1, 137, 907, 1, 1, 1031, 359, 13883, 1009, 2053, 7309, 14867, 7559, 809, 601, 2269, 1153, 349, 641, 16931, 8599, 17467, 1, 1, 1, 977, 9421, 1471, 313, 1, 1427, 1559, 541, 719, 1, 523, 1553, 1, 11173, 22651, 883, 439, 11789, 3413, 1, 179, 12421, 25163, 12743, 487, 1867, 1, 13399, 2087, 443, 27803, 14071, 1, 1, 941, 14759, 29867, 521, 1609, 1, 1, 1217, 32003, 16183, 461, 1, 683, 2417, 34211, 17293, 34963, 431, 1, 2579, 401, 18439, 449, 991, 2927, 409, 1, 2803, 1367, 20023, 2129, 659, 1423, 229, 859, 1, 42923, 1667, 1, 22093, 6373, 3217, 2393, 433, 1, 1, 3631, 1, 6869, 1277, 3767, 24709, 1061, 1, 7253, 3659, 51683, 1, 1283, 1, 1, 1, 251, 2113, 55411, 27943, 1, 1, 1, 4129, 821, 29389, 1, 29879, 60251, 4339, 673, 30871, 929, 1, 3329, 1, 9181, 661, 599, 2531, 2287, 1759, 1, 1, 1, 34469, 69467, 1129, 281, 35533, 1, 5153, 5591, 1, 73771, 37159, 1, 5387, 10853, 1, 4057, 38821, 78203, 39383, 1619, 1, 1201, 40519, 81611, 1, 82763, 5953, 631, 1, 85091, 42839, 307, 317, 1, 1, 88643, 44621, 6911, 1103, 1097, 6547, 1, 46439, 93491, 1, 94723, 1, 13709, 6899, 2371, 1, 98467, 49549, 1, 1, 1, 50821, 3527, 971, 1, 1, 1, 7537, 8167, 1723, 5657, 1, 108803, 1117, 15733, 55399, 111467, 1, 1163, 1, 347, 1, 115523, 1, 2851, 58789, 1, 1, 17093, 3167, 9311, 2099, 122443, 919, 1361, 1, 1, 62989, 126691, 63703, 4133, 9203, 1, 5011, 131011, 1, 953, 1, 1, 9619, 135403, 68071, 136883, 68813, 138371, 1, 1, 70309, 141371, 397, 379, 71821, 1, 10369, 145931, 1789, 1, 1, 149011, 823, 1, 2441, 152123, 5881, 8089, 77239, 1, 1, 1439, 78823, 158443, 79621, 947, 11489, 3299, 2801, 1, 82039, 911, 2857, 23789, 11953, 1, 4447, 3613, 6563, 171467, 1, 24733, 6691, 2609, 1657, 5693, 88663, 25453, 1, 1, 90359, 13967, 1721, 183283, 1879, 1, 1, 1, 967, 1, 2309, 937, 13649, 1979, 7417, 193723, 97303, 10289, 1, 28181, 99079, 6421, 1, 200843, 5309, 28949, 1, 15727, 102679, 1, 1459, 16007, 14929, 29989, 2243, 1, 1, 213611, 107269, 30781, 1, 3061, 109133, 1, 8467, 221083, 15859, 1, 111959, 2063, 112909, 5531, 3673, 1, 1, 1, 115783, 1, 1, 234467, 1, 33773, 6247, 238363, 119671, 8287, 9281, 34613, 17377, 8423, 9433, 246251, 1, 248243, 1, 5107, 1, 1307, 1, 19559, 1171, 1, 18379, 1, 129671, 260363, 130693, 262411, 1, 37781, 132749, 1489, 1, 1, 4649, 38669, 1493, 272771, 4721, 274867, 1, 1, 2837, 39869, 140071, 1, 1319, 1, 2683, 3137, 1, 287611, 1, 289763, 4691, 291923, 20929, 42013, 11353, 1, 1, 298451, 1, 1481, 3079, 302843, 1831, 1, 3257, 307267, 22027, 1, 155303, 7603, 156421, 24151, 157543, 1, 1193, 318467, 159799, 1, 160933, 323003, 1, 1499, 1, 327571, 1, 329867, 1, 6779, 1, 6311, 1, 336803, 5827, 1373, 1, 48781, 2557, 1, 1, 8443, 173671, 49789, 24979, 350891, 1, 18593, 13633, 355651, 1, 7307, 1, 7669, 1, 362851, 182029, 52181, 26177, 1, 1327, 1, 2237, 1, 26699, 1, 9901, 3463, 189349, 379931, 190583, 54629, 1, 1, 14851, 387371, 194309, 389867, 1, 56053, 2029, 1, 4831, 1997, 3761, 1, 28657, 1, 6961, 405011, 6553, 1, 1, 1429, 205703, 2969, 1, 3881, 208279, 59693, 1, 1, 1, 423043, 1, 6353, 30497, 1, 214789, 430891, 216103, 22817, 217421, 4793, 31249, 3203, 1, 1, 221399, 444131, 677, 1, 224071, 1, 1, 452171, 17443, 9283, 32587, 457571, 1, 2543, 3251, 463003, 1, 66533, 233549, 8839, 1, 1, 236293, 67709, 1787, 36671, 239053, 1, 8291, 1, 1, 1, 8387, 487843, 1, 490643, 709, 1, 2719, 1931, 248839, 3643, 250253, 26417, 1, 72109, 6173, 39047, 254519, 10861, 1, 5641, 1, 1, 3863, 519083, 260263, 1, 1, 1, 1, 18199, 733, 1, 1, 2459, 38219, 536531, 268999, 28393, 1511, 2161, 1, 1, 3851, 548323, 5849, 42407, 276389, 79181, 1, 1, 1, 560243, 280871, 7933, 1, 1973, 1, 569267, 757, 18461, 1, 82189, 41203, 578363, 1, 581411, 7109, 44959, 2203, 83933, 1, 1, 296071, 2207, 297613, 1, 42737, 1, 3623, 602971, 23251, 606083, 43403, 3001, 1, 612331, 306949, 1, 308519, 1, 1, 13229, 1, 1, 16487, 20261, 1, 1, 316429, 2027, 318023, 1, 2333, 1, 1, 644051, 1, 15787, 324439, 2917, 3583, 1987, 11299, 1, 329293, 1663, 11411, 94781, 47507, 51287, 334183, 21613, 335821, 1, 1, 5087, 10939, 679867, 1, 7043, 342413, 1, 1, 689803, 345733, 6359, 26723, 16987, 1, 1, 1, 24247, 352421, 706523, 18637, 1, 1, 713267, 8719, 55127, 1861, 37897, 1, 103349, 6841, 6793, 5437, 23557, 1, 1, 52529, 1, 1, 38977, 371143, 1, 53267, 106781, 1, 4783, 1, 58031, 378071, 15467, 1871, 58567, 381559, 4273, 9349, 768371, 55009, 110269, 20359, 25013, 1, 16573, 4703, 111781, 1, 1, 5879, 4091, 395671, 793123, 56779, 1, 399239, 61559, 401029, 1, 402823, 1, 57803, 27967, 406423, 17333, 408229, 15439, 3083, 117413, 2437, 2689, 1, 7607, 1, 118973, 59617, 4673, 1, 1, 22157, 8699, 8629, 1, 1, 851171, 4397, 3461, 14771, 122653, 1, 862283, 1, 866003, 1, 2833, 4789, 124781, 1, 1, 33811, 3847, 441421, 1, 1, 28661, 23431, 892267, 447079, 68927, 2069, 128549, 1, 5347, 452773, 12781, 454679, 1, 3433, 2549, 1, 31687, 1, 4637, 1, 6967, 1, 930467, 4357, 11257, 468133, 1, 67153, 1, 1, 72767, 15289, 49993, 67987, 1, 477863, 30893, 479821, 7019, 25357, 137933, 2383, 969467, 37363, 973411, 1, 977363, 5381, 2861, 491653, 985291, 25981, 2467, 495629, 3019, 71089, 1, 499621, 52697, 501623, 1, 71947, 4651, 505639, 1013291, 1, 1017323, 1, 2753, 5623, 1, 513719, 1029467, 1, 1, 10567, 148229, 1, 10739, 7789, 1, 1, 1, 1,

6. Sequence of the polynom (only primes)

29, 13, 83, 71, 7, 19, 331, 199, 467, 269, 47, 109, 421, 503, 1091, 31, 181, 97, 1451, 773, 53, 67, 139, 293, 2267, 41, 1303, 389, 2963, 1543, 1669, 3467, 257, 1933, 4003, 4283, 2213, 653, 337, 157, 193, 5171, 2663, 5483, 829, 6131, 223, 3319, 499, 3671, 7523, 3853, 607, 577, 1181, 4229, 211, 4423, 9043, 4621, 9851, 107, 10267, 10691, 227, 373, 12011, 137, 907, 1031, 359, 13883, 1009, 2053, 7309, 14867, 7559, 809, 601, 2269, 1153, 349, 641, 16931, 8599, 17467, 977, 9421, 1471, 313, 1427, 1559, 541, 719, 523, 1553, 11173, 22651, 883, 439, 11789, 3413, 179, 12421, 25163, 12743, 487, 1867, 13399, 2087, 443, 27803, 14071, 941, 14759, 29867, 521, 1609, 1217, 32003, 16183, 461, 683, 2417, 34211, 17293, 34963, 431, 2579, 401, 18439, 449, 991, 2927, 409, 2803, 1367, 20023, 2129, 659, 1423, 229, 859, 42923, 1667, 22093, 6373, 3217, 2393, 433, 3631, 6869, 1277, 3767, 24709, 1061, 7253, 3659, 51683, 1283, 251, 2113, 55411, 27943, 4129, 821, 29389, 29879, 60251, 4339, 673, 30871, 929, 3329, 9181, 661, 599, 2531, 2287, 1759, 34469, 69467, 1129, 281, 35533, 5153, 5591, 73771, 37159, 5387, 10853, 4057, 38821, 78203, 39383, 1619, 1201, 40519, 81611, 82763, 5953, 631, 85091, 42839, 307, 317, 88643, 44621, 6911, 1103, 1097, 6547, 46439, 93491, 94723, 13709, 6899, 2371, 98467, 49549, 50821, 3527, 971, 7537, 8167, 1723, 5657, 108803, 1117, 15733, 55399, 111467, 1163, 347, 115523, 2851, 58789, 17093, 3167, 9311, 2099, 122443, 919, 1361, 62989, 126691, 63703, 4133, 9203, 5011, 131011, 953, 9619, 135403, 68071, 136883, 68813, 138371, 70309, 141371, 397, 379, 71821, 10369, 145931, 1789, 149011, 823, 2441, 152123, 5881, 8089, 77239, 1439, 78823, 158443, 79621, 947, 11489, 3299, 2801, 82039, 911, 2857, 23789, 11953, 4447, 3613, 6563, 171467, 24733, 6691, 2609, 1657, 5693, 88663, 25453, 90359, 13967, 1721, 183283, 1879, 967, 2309, 937, 13649, 1979, 7417, 193723, 97303, 10289, 28181, 99079, 6421, 200843, 5309, 28949, 15727, 102679, 1459, 16007, 14929, 29989, 2243, 213611, 107269, 30781, 3061, 109133, 8467, 221083, 15859, 111959, 2063, 112909, 5531, 3673, 115783, 234467, 33773, 6247, 238363, 119671, 8287, 9281, 34613, 17377, 8423, 9433, 246251, 248243, 5107, 1307, 19559, 1171, 18379, 129671, 260363, 130693, 262411, 37781, 132749, 1489, 4649, 38669, 1493, 272771, 4721, 274867, 2837, 39869, 140071, 1319, 2683, 3137, 287611, 289763, 4691, 291923, 20929, 42013, 11353, 298451, 1481, 3079, 302843, 1831, 3257, 307267, 22027, 155303, 7603, 156421, 24151, 157543, 1193, 318467, 159799, 160933, 323003, 1499, 327571, 329867, 6779, 6311, 336803, 5827, 1373, 48781, 2557, 8443, 173671, 49789, 24979, 350891, 18593, 13633, 355651, 7307, 7669, 362851, 182029, 52181, 26177, 1327, 2237, 26699, 9901, 3463, 189349, 379931, 190583, 54629, 14851, 387371, 194309, 389867, 56053, 2029, 4831, 1997, 3761, 28657, 6961, 405011, 6553, 1429, 205703, 2969, 3881, 208279, 59693, 423043, 6353, 30497, 214789, 430891, 216103, 22817, 217421, 4793, 31249, 3203, 221399, 444131, 677, 224071, 452171, 17443, 9283, 32587, 457571, 2543, 3251, 463003, 66533, 233549, 8839, 236293, 67709, 1787, 36671, 239053, 8291, 8387, 487843, 490643, 709, 2719, 1931, 248839, 3643, 250253, 26417, 72109, 6173, 39047, 254519, 10861, 5641, 3863, 519083, 260263, 18199, 733, 2459, 38219, 536531, 268999, 28393, 1511, 2161, 3851, 548323, 5849, 42407, 276389, 79181, 560243, 280871, 7933, 1973, 569267, 757, 18461, 82189, 41203, 578363, 581411, 7109, 44959, 2203, 83933, 296071, 2207, 297613, 42737, 3623, 602971, 23251, 606083, 43403, 3001, 612331, 306949, 308519, 13229, 16487, 20261, 316429, 2027, 318023, 2333, 644051, 15787, 324439, 2917, 3583, 1987, 11299, 329293, 1663, 11411, 94781, 47507, 51287, 334183, 21613, 335821, 5087, 10939, 679867, 7043, 342413, 689803, 345733, 6359, 26723, 16987, 24247, 352421, 706523, 18637, 713267, 8719, 55127, 1861, 37897, 103349, 6841, 6793, 5437, 23557, 52529, 38977, 371143, 53267, 106781, 4783, 58031, 378071, 15467, 1871, 58567, 381559, 4273, 9349, 768371, 55009, 110269, 20359, 25013, 16573, 4703, 111781, 5879, 4091, 395671, 793123, 56779, 399239, 61559, 401029, 402823, 57803, 27967, 406423, 17333, 408229, 15439, 3083, 117413, 2437, 2689, 7607, 118973, 59617, 4673, 22157, 8699, 8629, 851171, 4397, 3461, 14771, 122653, 862283, 866003, 2833, 4789, 124781, 33811, 3847, 441421, 28661, 23431, 892267, 447079, 68927, 2069, 128549, 5347, 452773, 12781, 454679, 3433, 2549, 31687, 4637, 6967, 930467, 4357, 11257, 468133, 67153, 72767, 15289, 49993, 67987, 477863, 30893, 479821, 7019, 25357, 137933, 2383, 969467, 37363, 973411, 977363, 5381, 2861, 491653, 985291, 25981, 2467, 495629, 3019, 71089, 499621, 52697, 501623, 71947, 4651, 505639, 1013291, 1017323, 2753, 5623, 513719, 1029467, 10567, 148229, 10739, 7789,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+54x-29 and
the reducible primes which appear as divisor for the first time
p | x^2+54x-29 and p < x^2+54x-29

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)
11010821.0000000.8000001.0000000.0000000.0000000.000000
21007343300.7300000.4300000.7300007.3000005.37500015.000000
31.0006982694290.6980000.2690000.6980009.5616446.25581414.300000
410.0006.9701.9315.0390.6970000.1931000.6970009.9856737.17843911.745921
5100.00069.84014.98054.8600.6984000.1498000.69840010.0200867.75763810.887081
61.000.000697.720121.799575.9210.6977200.1217990.6977209.9902638.13077410.498013
710.000.0006.968.6531.029.2835.939.3700.6968650.1029280.6968659.9877508.45066810.312820
8100.000.00069.623.9458.904.39260.719.5530.6962390.0890440.6962399.9910198.65106310.223231
91.000.000.000695.799.34078.472.590617.326.7500.6957990.0784730.6957999.9936798.81279710.166852
1010.000.000.0006.954.566.847701.712.4506.252.854.3970.6954570.0701710.6954579.9950758.94213410.128922


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
244401.0000001.0000000.0000001.3333331.333333-nan
388711.0000000.8750000.1250002.0000001.750000inf
416141130.8750000.6875000.1875001.7500001.5714293.000000
5322515100.7812500.4687500.3125001.7857141.3636363.333333
6644830180.7500000.4687500.2812501.9200002.0000001.800000
71289549460.7421880.3828120.3593751.9791671.6333332.555556
8256187831040.7304690.3242190.4062501.9684211.6938782.260870
95123631532100.7089840.2988280.4101561.9411761.8433742.019231
101.0247142744400.6972660.2675780.4296881.9669421.7908502.095238
112.0481.4184959230.6923830.2416990.4506841.9859941.8065692.097727
124.0962.8528801.9720.6962890.2148440.4814452.0112831.7777782.136511
138.1925.7091.6294.0800.6968990.1988530.4980472.0017531.8511362.068965
1416.38411.4372.9958.4420.6980590.1828000.5152592.0033281.8385512.069118
1532.76822.8475.56517.2820.6972350.1698300.5274051.9976391.8580972.047145
1665.53645.77610.23435.5420.6984860.1561580.5423282.0035891.8389942.056591
17131.07291.52019.17772.3430.6982420.1463090.5519331.9993011.8738522.035423
18262.144183.01635.879147.1370.6981510.1368680.5612831.9997381.8709392.033880
19524.288365.99567.516298.4790.6980800.1287770.5693041.9997981.8817692.028579
201.048.576731.615127.143604.4720.6977220.1212530.5764691.9989751.8831542.025174
212.097.1521.462.856241.2841.221.5720.6975440.1150530.5824911.9994891.8977372.020891
224.194.3042.924.499458.2382.466.2610.6972550.1092520.5880021.9991711.8991642.018924
238.388.6085.846.429873.3734.973.0560.6969490.1041140.5928341.9991221.9059382.016435
2416.777.21611.687.8071.668.70210.019.1050.6966480.0994620.5971851.9991361.9106412.014678
2533.554.43223.370.2733.192.93520.177.3380.6964880.0951570.6013321.9995431.9134242.013886
2667.108.86446.729.1956.118.95340.610.2420.6963190.0911800.6051401.9995141.9164042.012666
27134.217.72893.440.39711.746.63281.693.7650.6961850.0875190.6086661.9996151.9197132.011654
28268.435.456186.839.87422.594.139164.245.7350.6960330.0841700.6118631.9995621.9234572.010505
29536.870.912373.607.40243.526.641330.080.7610.6958980.0810750.6148231.9996131.9264572.009676
301.073.741.824747.093.94883.951.369663.142.5790.6957850.0781860.6176001.9996771.9287352.009031
312.147.483.6481.493.945.227162.161.6511.331.783.5760.6956720.0755120.6201601.9996751.9316142.008291
324.294.967.2962.987.452.199313.578.9252.673.873.2740.6955700.0730110.6225601.9997071.9337432.007739
338.589.934.5925.974.104.058607.004.3905.367.099.6680.6954770.0706650.6248131.9997321.9357312.007238
3417.179.869.18411.946.703.3261.176.244.15110.770.459.1750.6953900.0684660.6269231.9997481.9377852.006756


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
123120120
244130121
387340322
41611470443
532155100654
66430102001488
71284921280241411
82568339440372323
951215378750743742
101.02427413713701316776
112.0484952482470235126134
124.0968804384420420225235
138.1921.6298138160793406430
1416.3842.9951.4891.50601.467769759
1532.7685.5652.7702.79502.7331.4201.412
1665.53610.2345.1435.09105.0082.6232.603
17131.07219.1779.6769.50109.4374.8504.890
18262.14435.87918.03217.847017.6959.0929.092
19524.28867.51633.93533.581033.36517.04517.106
201.048.576127.14363.81663.327062.97032.10432.069
212.097.152241.284120.962120.3220119.32860.77861.178
224.194.304458.238229.923228.3150226.503115.362116.373
238.388.608873.373438.117435.2560432.053220.286221.034
2416.777.2161.668.702837.293831.4090825.133421.635421.934
2533.554.4323.192.9351.602.4041.590.53101.578.811806.501807.623
2667.108.8646.118.9533.069.0443.049.90903.026.8851.544.8981.547.170
27134.217.72811.746.6325.892.7775.853.85505.813.2042.966.3892.967.039
28268.435.45622.594.13911.332.32311.261.816011.186.5595.703.3605.704.220
29536.870.91243.526.64121.827.83021.698.811021.560.67810.982.69710.983.266
301.073.741.82483.951.36942.096.29641.855.073041.598.20021.176.64421.176.525
312.147.483.648162.161.65181.308.89780.852.754080.380.21140.888.04340.893.397
324.294.967.296313.578.925157.217.326156.361.5990155.491.72579.034.86279.052.338
338.589.934.592607.004.390304.304.506302.699.8840301.071.943152.958.778152.973.669
3417.179.869.1841.176.244.151589.647.401586.596.7500583.551.689296.337.670296.354.792


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
381100100
4163210111
53210641351
664181264482
71284629171310149
8256104515332242622
951221010110963505245
101.02444021822211110912694
112.048923482441243215251214
124.0961.972986986499471525477
138.1924.0802.0182.0621.0379901.058995
1416.3848.4424.2214.2212.1492.0552.1682.070
1532.76817.2828.6098.6734.4564.1954.4724.159
1665.53635.54217.74817.7949.2008.5359.1808.627
17131.07272.34336.23436.10918.69517.42818.65117.569
18262.144147.13773.69173.44637.85335.86737.72035.697
19524.288298.479149.536148.94376.97572.63076.22672.648
201.048.576604.472302.604301.868155.862147.228154.491146.891
212.097.1521.221.572611.388610.184313.662298.498311.971297.441
224.194.3042.466.2611.234.3291.231.932632.251602.567629.542601.901
238.388.6084.973.0562.487.9862.485.0701.274.0201.215.2961.268.7431.214.997
2416.777.21610.019.1055.011.5935.007.5122.563.0612.450.6542.553.6252.451.765
2533.554.43220.177.33810.093.05710.084.2815.153.1554.941.2885.139.5874.943.308
2667.108.86440.610.24220.313.19620.297.04610.362.2869.955.50910.336.2969.956.151
27134.217.72881.693.76540.862.88140.830.88420.828.93120.046.60620.774.17220.044.056
28268.435.456164.245.73582.155.75882.089.97741.832.52840.340.40441.743.55740.329.246
29536.870.912330.080.761165.099.425164.981.33684.006.19081.126.32883.846.84181.101.402
301.073.741.824663.142.579331.677.570331.465.009168.634.483163.097.614168.353.151163.057.331
312.147.483.6481.331.783.576666.066.021665.717.555338.430.160327.748.108337.928.254327.677.054
324.294.967.2962.673.873.2741.337.246.1891.336.627.085679.017.823658.451.508678.122.363658.281.580
338.589.934.5925.367.099.6682.684.114.3582.682.985.3101.362.147.4331.322.394.8881.360.504.1161.322.053.231
3417.179.869.18410.770.459.1755.386.290.2465.384.168.9292.731.967.6792.655.229.3232.728.938.9772.654.323.196


8. Check for existing Integer Sequences by OEIS

Found in Database : 29, 13, 83, 71, 7, 19, 331, 199, 467, 269, 47, 1, 109, 421, 1, 503, 1091, 31, 181, 97,
Found in Database : 29, 13, 83, 71, 7, 19, 331, 199, 467, 269, 47, 109, 421, 503, 1091, 31, 181, 97, 1451, 773, 53, 67, 139, 293, 2267, 41, 1303, 389, 2963, 1543, 1669, 3467, 257,
Found in Database : 7, 13, 19, 29, 31, 41, 47, 53, 67, 71, 83, 97, 107, 109, 137, 139,