Inhaltsverzeichnis

Development of
Algorithmic Constructions

15:24:05
Deutsch
29.Mar 2024

Polynom = x^2+103x+3

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) = 3 = 3
f(1) = 107 = 107
f(2) = 213 = 3*71
f(3) = 321 = 3*107
f(4) = 431 = 431
f(5) = 543 = 3*181
f(6) = 657 = 3*3*73
f(7) = 773 = 773
f(8) = 891 = 3*3*3*3*11
f(9) = 1011 = 3*337
f(10) = 1133 = 11*103
f(11) = 1257 = 3*419
f(12) = 1383 = 3*461
f(13) = 1511 = 1511
f(14) = 1641 = 3*547
f(15) = 1773 = 3*3*197
f(16) = 1907 = 1907
f(17) = 2043 = 3*3*227
f(18) = 2181 = 3*727
f(19) = 2321 = 11*211
f(20) = 2463 = 3*821
f(21) = 2607 = 3*11*79
f(22) = 2753 = 2753
f(23) = 2901 = 3*967
f(24) = 3051 = 3*3*3*113
f(25) = 3203 = 3203
f(26) = 3357 = 3*3*373
f(27) = 3513 = 3*1171
f(28) = 3671 = 3671
f(29) = 3831 = 3*1277
f(30) = 3993 = 3*11*11*11
f(31) = 4157 = 4157
f(32) = 4323 = 3*11*131
f(33) = 4491 = 3*3*499
f(34) = 4661 = 59*79
f(35) = 4833 = 3*3*3*179
f(36) = 5007 = 3*1669
f(37) = 5183 = 71*73
f(38) = 5361 = 3*1787
f(39) = 5541 = 3*1847
f(40) = 5723 = 59*97
f(41) = 5907 = 3*11*179
f(42) = 6093 = 3*3*677
f(43) = 6281 = 11*571
f(44) = 6471 = 3*3*719
f(45) = 6663 = 3*2221
f(46) = 6857 = 6857
f(47) = 7053 = 3*2351
f(48) = 7251 = 3*2417
f(49) = 7451 = 7451
f(50) = 7653 = 3*2551
f(51) = 7857 = 3*3*3*3*97
f(52) = 8063 = 11*733
f(53) = 8271 = 3*3*919
f(54) = 8481 = 3*11*257
f(55) = 8693 = 8693
f(56) = 8907 = 3*2969
f(57) = 9123 = 3*3041
f(58) = 9341 = 9341
f(59) = 9561 = 3*3187
f(60) = 9783 = 3*3*1087
f(61) = 10007 = 10007
f(62) = 10233 = 3*3*3*379
f(63) = 10461 = 3*11*317
f(64) = 10691 = 10691
f(65) = 10923 = 3*11*331
f(66) = 11157 = 3*3719
f(67) = 11393 = 11393
f(68) = 11631 = 3*3877
f(69) = 11871 = 3*3*1319
f(70) = 12113 = 12113
f(71) = 12357 = 3*3*1373
f(72) = 12603 = 3*4201
f(73) = 12851 = 71*181
f(74) = 13101 = 3*11*397
f(75) = 13353 = 3*4451
f(76) = 13607 = 11*1237
f(77) = 13863 = 3*4621
f(78) = 14121 = 3*3*3*523
f(79) = 14381 = 73*197
f(80) = 14643 = 3*3*1627
f(81) = 14907 = 3*4969
f(82) = 15173 = 15173
f(83) = 15441 = 3*5147
f(84) = 15711 = 3*5237
f(85) = 15983 = 11*1453
f(86) = 16257 = 3*5419
f(87) = 16533 = 3*3*11*167
f(88) = 16811 = 16811
f(89) = 17091 = 3*3*3*3*211
f(90) = 17373 = 3*5791
f(91) = 17657 = 17657
f(92) = 17943 = 3*5981
f(93) = 18231 = 3*59*103
f(94) = 18521 = 18521
f(95) = 18813 = 3*6271
f(96) = 19107 = 3*3*11*193
f(97) = 19403 = 19403
f(98) = 19701 = 3*3*11*199
f(99) = 20001 = 3*59*113
f(100) = 20303 = 79*257

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+103x+3

f(0)=3
f(1)=107
f(2)=71
f(3)=1
f(4)=431
f(5)=181
f(6)=73
f(7)=773
f(8)=11
f(9)=337
f(10)=103
f(11)=419
f(12)=461
f(13)=1511
f(14)=547
f(15)=197
f(16)=1907
f(17)=227
f(18)=727
f(19)=211
f(20)=821
f(21)=79
f(22)=2753
f(23)=967
f(24)=113
f(25)=3203
f(26)=373
f(27)=1171
f(28)=3671
f(29)=1277
f(30)=1
f(31)=4157
f(32)=131
f(33)=499
f(34)=59
f(35)=179
f(36)=1669
f(37)=1
f(38)=1787
f(39)=1847
f(40)=97
f(41)=1
f(42)=677
f(43)=571
f(44)=719
f(45)=2221
f(46)=6857
f(47)=2351
f(48)=2417
f(49)=7451
f(50)=2551
f(51)=1
f(52)=733
f(53)=919
f(54)=257
f(55)=8693
f(56)=2969
f(57)=3041
f(58)=9341
f(59)=3187
f(60)=1087
f(61)=10007
f(62)=379
f(63)=317
f(64)=10691
f(65)=331
f(66)=3719
f(67)=11393
f(68)=3877
f(69)=1319
f(70)=12113
f(71)=1373
f(72)=4201
f(73)=1
f(74)=397
f(75)=4451
f(76)=1237
f(77)=4621
f(78)=523
f(79)=1
f(80)=1627
f(81)=4969
f(82)=15173
f(83)=5147
f(84)=5237
f(85)=1453
f(86)=5419
f(87)=167
f(88)=16811
f(89)=1
f(90)=5791
f(91)=17657
f(92)=5981
f(93)=1
f(94)=18521
f(95)=6271
f(96)=193
f(97)=19403
f(98)=199
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+103x+3 could be written as f(y)= y^2-2649.25 with x=y-51.5

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+51.5
f'(x)>2x+102

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

3, 107, 71, 1, 431, 181, 73, 773, 11, 337, 103, 419, 461, 1511, 547, 197, 1907, 227, 727, 211, 821, 79, 2753, 967, 113, 3203, 373, 1171, 3671, 1277, 1, 4157, 131, 499, 59, 179, 1669, 1, 1787, 1847, 97, 1, 677, 571, 719, 2221, 6857, 2351, 2417, 7451, 2551, 1, 733, 919, 257, 8693, 2969, 3041, 9341, 3187, 1087, 10007, 379, 317, 10691, 331, 3719, 11393, 3877, 1319, 12113, 1373, 4201, 1, 397, 4451, 1237, 4621, 523, 1, 1627, 4969, 15173, 5147, 5237, 1453, 5419, 167, 16811, 1, 5791, 17657, 5981, 1, 18521, 6271, 193, 19403, 199, 1, 1, 6869, 6971, 21221, 7177, 809, 22157, 1, 1, 191, 1, 7919, 24083, 1, 2749, 25073, 941, 8581, 2371, 8807, 811, 27107, 9151, 3089, 28151, 3167, 9619, 223, 9857, 907, 30293, 929, 383, 31391, 3529, 10711, 32507, 1, 11087, 33641, 1031, 3823, 3163, 1303, 1, 35963, 12119, 12251, 1, 12517, 4217, 1, 1, 1187, 39581, 13331, 13469, 40823, 233, 1543, 42083, 4723, 1301, 1, 1327, 14741, 44657, 15031, 5059, 45971, 1, 15619, 47303, 1447, 16067, 4423, 16369, 5507, 50021, 1, 16981, 51407, 17291, 239, 4801, 17761, 1, 281, 6079, 18397, 55673, 18719, 1, 57131, 19207, 587, 569, 1, 19867, 60101, 20201, 20369, 61613, 20707, 6959, 271, 643, 21391, 5881, 21737, 21911, 1123, 1, 277, 1, 1, 1, 1, 23327, 2137, 1, 23869, 8017, 72701, 2713, 1, 74357, 24971, 2287, 76031, 1, 8573, 77723, 8699, 1, 79433, 26669, 26861, 293, 2477, 3049, 7537, 9277, 28027, 1, 1, 28619, 86453, 29017, 9739, 1, 1097, 2711, 90071, 1, 30431, 1259, 30841, 1, 93761, 10487, 2879, 95633, 2917, 32297, 97523, 32719, 3659, 99431, 11119, 1, 1283, 1, 34217, 9391, 34651, 1, 105263, 3923, 35527, 107243, 35969, 36191, 9931, 36637, 1117, 1567, 12437, 37537, 1, 37991, 38219, 115343, 38677, 1, 117413, 1193, 1, 1637, 389, 1, 121607, 40771, 13669, 123731, 1, 41719, 11443, 42197, 42437, 128033, 1, 14387, 130211, 14549, 43891, 12037, 44381, 4057, 1307, 45121, 1, 1279, 15289, 1, 1231, 46619, 4261, 141371, 1, 15877, 1481, 1783, 48397, 1, 829, 49169, 1877, 4517, 16649, 13693, 16823, 1, 152981, 51257, 51521, 1, 52051, 5813, 14341, 1, 439, 160163, 53657, 53927, 162593, 54469, 1, 165041, 6143, 5051, 877, 5101, 1, 169991, 56941, 19073, 953, 19259, 58057, 175013, 1, 58901, 16141, 59467, 2213, 463, 20107, 60607, 182681, 1, 61469, 16843, 62047, 1889, 187883, 6991, 63211, 3229, 487, 64091, 1, 911, 1, 3319, 1987, 1, 198503, 66467, 1, 201203, 67369, 1, 203921, 2069, 68581, 18787, 69191, 69497, 1063, 70111, 23473, 212183, 1, 71347, 19543, 1, 6571, 217781, 72907, 24407, 1321, 1, 74167, 509, 1, 6829, 226313, 1, 1, 1, 1, 77041, 232091, 77687, 1, 235007, 7151, 1, 1, 8849, 1, 240893, 80627, 1109, 243863, 81619, 1, 22441, 27539, 7541, 249857, 83621, 1423, 252881, 84631, 1049, 255923, 28549, 7817, 258983, 7879, 87011, 3691, 87697, 29347, 265157, 9859, 1, 268271, 8161, 1, 2243, 1, 30389, 3761, 1, 92221, 1663, 92927, 93281, 25537, 1, 1, 284111, 31687, 95419, 287333, 96137, 96497, 1, 1, 2957, 293831, 1, 1249, 1493, 99401, 99767, 4231, 100501, 33623, 303713, 3079, 101977, 1, 1741, 103091, 3929, 103837, 11579, 1487, 593, 105337, 2621, 1, 9679, 4391, 107227, 35869, 2473, 12041, 108751, 327401, 109517, 1, 330857, 10061, 37019, 334331, 37277, 1, 1, 112997, 1597, 341333, 1, 4243, 1, 38449, 115741, 348407, 116531, 116927, 351971, 117721, 39373, 32323, 1, 10847, 359153, 1, 631, 1291, 121327, 40577, 2797, 40847, 11177, 370061, 11251, 1, 1, 124987, 13933, 6397, 42073, 126631, 1483, 11587, 1801, 1, 1, 1, 4919, 4813, 130369, 392363, 1, 131627, 36013, 132469, 4027, 399941, 44579, 134161, 403757, 1709, 1, 3607, 1867, 1381, 411443, 1, 138007, 2087, 138869, 139301, 419201, 140167, 46867, 1, 1429, 141907, 38821, 2011, 1, 683, 1399, 48179, 434933, 48473, 145861, 39901, 1, 13381, 2447, 1, 5501, 446921, 49807, 2053, 4649, 150767, 1, 455003, 13829, 50857, 4457, 17053, 2609, 463157, 154841, 1601, 467261, 1, 52223, 42853, 52529, 158047, 475523, 2239, 159431, 4483, 160357, 1, 43987, 53917, 14747, 488051, 163151, 1, 2333, 2083, 55009, 496493, 1, 15131, 500741, 15217, 167861, 2113, 757, 1, 4507, 56747, 2161, 513593, 15607, 172157, 1, 743, 19289, 522251, 58189, 1, 7417, 176021, 176507, 48271, 177481, 5393, 2803, 1, 179437, 539783, 180419, 1, 1, 3083, 5527, 548657, 5557, 183877, 7577, 1, 185369, 1759, 1, 769, 562103, 5693, 1667, 51511, 189377, 189881, 571157, 190891, 63799, 575711, 21379, 1873, 1, 193937, 1607, 584873, 195469, 65327, 589481, 1, 197521, 594107, 1, 18097, 1, 18191, 22291, 1823, 67219, 1, 6269, 1973, 2579, 612791, 18617, 68437, 1, 7643, 206887, 622241, 207941, 208469, 10627, 1, 70019, 1, 70373, 1, 10789, 1, 1993, 3583, 2003, 23873, 646181, 1, 1789, 3637, 19777, 218087, 655883, 219169, 73237, 1, 24533, 1, 3449, 1, 222977, 60961, 224071, 74873, 1811, 75239, 226267, 680453, 839, 2017, 62311, 229027, 1, 690407, 1, 3257, 695411, 1, 232919, 700433, 1, 7109, 1, 2381, 853, 710531, 1, 237971, 3209, 1, 79889, 859, 7297, 4091, 65983, 1, 243077, 730943, 244219, 1, 736091, 1, 246511, 1, 1, 22567, 7247, 2207, 1, 751643, 1, 251707, 1997, 252869, 23041, 762101, 1, 1, 767357, 881, 256957, 772631, 258131, 883, 1, 23627, 1, 6473, 87223, 262261, 788561, 1, 3617, 1, 265231, 88609, 72661, 29669, 24329, 804653, 268817, 1, 810053, 270619, 90407, 815471, 1, 24821, 820907, 1, 274847, 7723, 4679, 10247, 1, 1, 278497, 1, 1, 1, 1, 281557, 94057, 3301, 31489, 1, 853901, 285251, 1, 7103, 287107, 8719, 865043, 96323, 3967, 870641, 290837, 1, 876257, 292711, 2963, 12421, 1, 295219, 887543, 296477, 1, 893213, 298369, 99667, 2347, 1, 1, 82237, 302171, 3833, 3907, 304081, 101573, 916073, 101999, 1, 1, 307919, 28051, 1, 1, 34499, 933407, 1009, 312427, 15919, 313721, 28579, 5659, 28697, 1, 1, 35291, 318271, 8467, 319577, 1, 1, 29231, 1, 1, 107837, 1, 9461, 325487, 326147, 980423, 327469, 4051, 89671, 109819, 30011, 992357, 331451, 332117, 7621, 333451, 111373, 1004363, 37273, 30557, 13841, 2789, 338141, 1016441, 1, 1, 1022507, 113837, 342187, 9613, 31231, 3217, 94063, 345577, 1, 1040813, 115873, 1, 1046951, 349667, 350351, 95737, 1, 1, 6343, 13103, 354469, 13487, 355847, 6043, 1071683, 4903, 10867, 1077911, 10909, 1, 1084157, 362081, 1, 1090421, 364171, 1, 1096703, 1, 366967,

6. Sequence of the polynom (only primes)

3, 107, 71, 431, 181, 73, 773, 11, 337, 103, 419, 461, 1511, 547, 197, 1907, 227, 727, 211, 821, 79, 2753, 967, 113, 3203, 373, 1171, 3671, 1277, 4157, 131, 499, 59, 179, 1669, 1787, 1847, 97, 677, 571, 719, 2221, 6857, 2351, 2417, 7451, 2551, 733, 919, 257, 8693, 2969, 3041, 9341, 3187, 1087, 10007, 379, 317, 10691, 331, 3719, 11393, 3877, 1319, 12113, 1373, 4201, 397, 4451, 1237, 4621, 523, 1627, 4969, 15173, 5147, 5237, 1453, 5419, 167, 16811, 5791, 17657, 5981, 18521, 6271, 193, 19403, 199, 6869, 6971, 21221, 7177, 809, 22157, 191, 7919, 24083, 2749, 25073, 941, 8581, 2371, 8807, 811, 27107, 9151, 3089, 28151, 3167, 9619, 223, 9857, 907, 30293, 929, 383, 31391, 3529, 10711, 32507, 11087, 33641, 1031, 3823, 3163, 1303, 35963, 12119, 12251, 12517, 4217, 1187, 39581, 13331, 13469, 40823, 233, 1543, 42083, 4723, 1301, 1327, 14741, 44657, 15031, 5059, 45971, 15619, 47303, 1447, 16067, 4423, 16369, 5507, 50021, 16981, 51407, 17291, 239, 4801, 17761, 281, 6079, 18397, 55673, 18719, 57131, 19207, 587, 569, 19867, 60101, 20201, 20369, 61613, 20707, 6959, 271, 643, 21391, 5881, 21737, 21911, 1123, 277, 23327, 2137, 23869, 8017, 72701, 2713, 74357, 24971, 2287, 76031, 8573, 77723, 8699, 79433, 26669, 26861, 293, 2477, 3049, 7537, 9277, 28027, 28619, 86453, 29017, 9739, 1097, 2711, 90071, 30431, 1259, 30841, 93761, 10487, 2879, 95633, 2917, 32297, 97523, 32719, 3659, 99431, 11119, 1283, 34217, 9391, 34651, 105263, 3923, 35527, 107243, 35969, 36191, 9931, 36637, 1117, 1567, 12437, 37537, 37991, 38219, 115343, 38677, 117413, 1193, 1637, 389, 121607, 40771, 13669, 123731, 41719, 11443, 42197, 42437, 128033, 14387, 130211, 14549, 43891, 12037, 44381, 4057, 1307, 45121, 1279, 15289, 1231, 46619, 4261, 141371, 15877, 1481, 1783, 48397, 829, 49169, 1877, 4517, 16649, 13693, 16823, 152981, 51257, 51521, 52051, 5813, 14341, 439, 160163, 53657, 53927, 162593, 54469, 165041, 6143, 5051, 877, 5101, 169991, 56941, 19073, 953, 19259, 58057, 175013, 58901, 16141, 59467, 2213, 463, 20107, 60607, 182681, 61469, 16843, 62047, 1889, 187883, 6991, 63211, 3229, 487, 64091, 911, 3319, 1987, 198503, 66467, 201203, 67369, 203921, 2069, 68581, 18787, 69191, 69497, 1063, 70111, 23473, 212183, 71347, 19543, 6571, 217781, 72907, 24407, 1321, 74167, 509, 6829, 226313, 77041, 232091, 77687, 235007, 7151, 8849, 240893, 80627, 1109, 243863, 81619, 22441, 27539, 7541, 249857, 83621, 1423, 252881, 84631, 1049, 255923, 28549, 7817, 258983, 7879, 87011, 3691, 87697, 29347, 265157, 9859, 268271, 8161, 2243, 30389, 3761, 92221, 1663, 92927, 93281, 25537, 284111, 31687, 95419, 287333, 96137, 96497, 2957, 293831, 1249, 1493, 99401, 99767, 4231, 100501, 33623, 303713, 3079, 101977, 1741, 103091, 3929, 103837, 11579, 1487, 593, 105337, 2621, 9679, 4391, 107227, 35869, 2473, 12041, 108751, 327401, 109517, 330857, 10061, 37019, 334331, 37277, 112997, 1597, 341333, 4243, 38449, 115741, 348407, 116531, 116927, 351971, 117721, 39373, 32323, 10847, 359153, 631, 1291, 121327, 40577, 2797, 40847, 11177, 370061, 11251, 124987, 13933, 6397, 42073, 126631, 1483, 11587, 1801, 4919, 4813, 130369, 392363, 131627, 36013, 132469, 4027, 399941, 44579, 134161, 403757, 1709, 3607, 1867, 1381, 411443, 138007, 2087, 138869, 139301, 419201, 140167, 46867, 1429, 141907, 38821, 2011, 683, 1399, 48179, 434933, 48473, 145861, 39901, 13381, 2447, 5501, 446921, 49807, 2053, 4649, 150767, 455003, 13829, 50857, 4457, 17053, 2609, 463157, 154841, 1601, 467261, 52223, 42853, 52529, 158047, 475523, 2239, 159431, 4483, 160357, 43987, 53917, 14747, 488051, 163151, 2333, 2083, 55009, 496493, 15131, 500741, 15217, 167861, 2113, 757, 4507, 56747, 2161, 513593, 15607, 172157, 743, 19289, 522251, 58189, 7417, 176021, 176507, 48271, 177481, 5393, 2803, 179437, 539783, 180419, 3083, 5527, 548657, 5557, 183877, 7577, 185369, 1759, 769, 562103, 5693, 1667, 51511, 189377, 189881, 571157, 190891, 63799, 575711, 21379, 1873, 193937, 1607, 584873, 195469, 65327, 589481, 197521, 594107, 18097, 18191, 22291, 1823, 67219, 6269, 1973, 2579, 612791, 18617, 68437, 7643, 206887, 622241, 207941, 208469, 10627, 70019, 70373, 10789, 1993, 3583, 2003, 23873, 646181, 1789, 3637, 19777, 218087, 655883, 219169, 73237, 24533, 3449, 222977, 60961, 224071, 74873, 1811, 75239, 226267, 680453, 839, 2017, 62311, 229027, 690407, 3257, 695411, 232919, 700433, 7109, 2381, 853, 710531, 237971, 3209, 79889, 859, 7297, 4091, 65983, 243077, 730943, 244219, 736091, 246511, 22567, 7247, 2207, 751643, 251707, 1997, 252869, 23041, 762101, 767357, 881, 256957, 772631, 258131, 883, 23627, 6473, 87223, 262261, 788561, 3617, 265231, 88609, 72661, 29669, 24329, 804653, 268817, 810053, 270619, 90407, 815471, 24821, 820907, 274847, 7723, 4679, 10247, 278497, 281557, 94057, 3301, 31489, 853901, 285251, 7103, 287107, 8719, 865043, 96323, 3967, 870641, 290837, 876257, 292711, 2963, 12421, 295219, 887543, 296477, 893213, 298369, 99667, 2347, 82237, 302171, 3833, 3907, 304081, 101573, 916073, 101999, 307919, 28051, 34499, 933407, 1009, 312427, 15919, 313721, 28579, 5659, 28697, 35291, 318271, 8467, 319577, 29231, 107837, 9461, 325487, 326147, 980423, 327469, 4051, 89671, 109819, 30011, 992357, 331451, 332117, 7621, 333451, 111373, 1004363, 37273, 30557, 13841, 2789, 338141, 1016441, 1022507, 113837, 342187, 9613, 31231, 3217, 94063, 345577, 1040813, 115873, 1046951, 349667, 350351, 95737, 6343, 13103, 354469, 13487, 355847, 6043, 1071683, 4903, 10867, 1077911, 10909, 1084157, 362081, 1090421, 364171, 1096703, 366967,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+103x+3 and
the reducible primes which appear as divisor for the first time
p | x^2+103x+3 and p < x^2+103x+3

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)
1109450.9000000.4000000.5000000.0000000.0000000.000000
21008723640.8700000.2300000.6400009.6666675.75000012.800000
31.0007801666140.7800000.1660000.6140008.9655177.2173919.593750
410.0007.6461.1956.4510.7646000.1195000.6451009.8025647.19879510.506515
5100.00074.7259.41665.3090.7472500.0941600.6530909.7730847.87949810.123857
61.000.000736.93277.078659.8540.7369320.0770780.6598549.8619208.18585410.103569
710.000.0007.299.861652.3646.647.4970.7299860.0652360.6647509.9057468.46368610.074194
8100.000.00072.497.2195.651.55266.845.6670.7249720.0565160.6684579.9313148.66318810.055765
91.000.000.000721.190.54949.868.856671.321.6930.7211910.0498690.6713229.9478388.82392210.042860
1010.000.000.0007.182.119.754446.229.2106.735.890.5440.7182120.0446230.6735899.9587008.94805410.033775


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
244311.0000000.7500000.2500001.3333331.5000001.000000
387430.8750000.5000000.3750001.7500001.3333333.000000
41615690.9375000.3750000.5625002.1428571.5000003.000000
5323010200.9375000.3125000.6250002.0000001.6666672.222222
6645816420.9062500.2500000.6562501.9333331.6000002.100000
712810929800.8515620.2265620.6250001.8793101.8125001.904762
8256207531540.8085940.2070310.6015621.8990831.8275861.925000
9512408943140.7968750.1835940.6132811.9710141.7735852.038961
101.0248011696320.7822270.1650390.6171881.9632351.7978722.012739
112.0481.5933081.2850.7778320.1503910.6274411.9887641.8224852.033228
124.0963.1625612.6010.7719730.1369630.6350101.9849341.8214292.024125
138.1926.2861.0085.2780.7673340.1230470.6442871.9879821.7967912.029220
1416.38412.4251.86010.5650.7583620.1135250.6448361.9766151.8452382.001705
1532.76824.6923.44721.2450.7535400.1051940.6483461.9872841.8532262.010885
1665.53649.1286.44542.6830.7496340.0983430.6512911.9896321.8697422.009084
17131.07297.75812.05085.7080.7458340.0919340.6539001.9898631.8696662.008013
18262.144194.62722.558172.0690.7424430.0860520.6563911.9909061.8720332.007619
19524.288387.73042.462345.2680.7395360.0809900.6585461.9921701.8823482.006567
201.048.576772.48080.525691.9550.7366940.0767950.6599001.9923141.8964022.004110
212.097.1521.539.995152.7661.387.2290.7343270.0728450.6614821.9935731.8971252.004797
224.194.3043.071.466290.3572.781.1090.7322950.0692270.6630681.9944651.9006652.004794
238.388.6086.127.374553.4295.573.9450.7304400.0659740.6644661.9949351.9060292.004217
2416.777.21612.225.7051.057.57811.168.1270.7287090.0630370.6656721.9952601.9109552.003631
2533.554.43224.398.7632.024.31422.374.4490.7271400.0603290.6668111.9956941.9141042.003420
2667.108.86448.702.4523.881.56944.820.8830.7257230.0578400.6678831.9961031.9174742.003217
27134.217.72897.232.2197.457.14089.775.0790.7244360.0555600.6688761.9964541.9211662.002974
28268.435.456194.144.11814.348.825179.795.2930.7232430.0534540.6697901.9967061.9241732.002731
29536.870.912387.687.81127.646.257360.041.5540.7221250.0514950.6706301.9969071.9267262.002508
301.073.741.824774.264.76053.352.289720.912.4710.7210900.0496880.6714021.9971351.9298202.002303
312.147.483.6481.546.463.981103.069.6191.443.394.3620.7201280.0479960.6721331.9973321.9318692.002177
324.294.967.2963.089.072.038199.338.9112.889.733.1270.7192310.0464120.6728181.9975071.9340222.002040
338.589.934.5926.170.921.894385.980.3575.784.941.5370.7183900.0449340.6734561.9976621.9363022.001895
3417.179.869.18412.328.308.016748.146.70811.580.161.3080.7176020.0435480.6740541.9978061.9383032.001777


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
122010200
243020201
384030211
4166050312
53210091423
664160152644
71282902871075
82565305211161511
95129409322262224
101.024169016839464440
112.048308030779827572
124.0965610560143144145129
138.1921.00801.006265246248249
1416.3841.86001.858486454461459
1532.7683.44703.445874876867830
1665.5366.44506.4431.6161.6621.5831.584
17131.07212.050012.0482.9563.0853.0372.972
18262.14422.558022.5565.5465.7715.6565.585
19524.28842.462042.46010.51110.71210.58210.657
201.048.57680.525080.52320.01120.19820.12620.190
212.097.152152.7660152.76437.94938.27538.28938.253
224.194.304290.3570290.35572.25972.81472.58772.697
238.388.608553.4290553.427137.916138.810138.311138.392
2416.777.2161.057.57801.057.576264.072264.890264.124264.492
2533.554.4322.024.31402.024.312505.917506.227505.709506.461
2667.108.8643.881.56903.881.567970.245970.744970.315970.265
27134.217.7287.457.14007.457.1381.864.9361.863.8341.864.6801.863.690
28268.435.45614.348.825014.348.8233.586.5903.586.7453.589.6633.585.827
29536.870.91227.646.257027.646.2556.910.4596.912.9986.912.8446.909.956
301.073.741.82453.352.289053.352.28713.338.93913.339.49713.337.44113.336.412
312.147.483.648103.069.6190103.069.61725.769.52225.765.31125.766.28325.768.503
324.294.967.296199.338.9110199.338.90949.838.66949.832.16449.834.73149.833.347
338.589.934.592385.980.3570385.980.35596.496.05896.498.58596.488.25296.497.462
3417.179.869.184748.146.7080748.146.706187.033.249187.042.588187.030.937187.039.934


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
121010001
241010001
383211011
4169542232
532201193665
6644222208121111
712880443615222320
8256154837133443542
951231416814672817982
101.024632341291150160164158
112.0481.285688597306334324321
124.0962.6011.3741.227659639668635
138.1925.2782.7672.5111.3021.3071.3671.302
1416.38410.5655.5505.0152.6152.6122.6852.653
1532.76821.24510.99110.2545.2565.3095.3275.353
1665.53642.68322.06720.61610.60110.66810.71610.698
17131.07285.70844.11641.59221.33621.40821.52721.437
18262.144172.06988.53083.53943.16242.87543.13442.898
19524.288345.268176.907168.36186.33286.08886.52286.326
201.048.576691.955354.251337.704173.098172.806172.927173.124
212.097.1521.387.229709.534677.695347.253346.319346.460347.197
224.194.3042.781.1091.421.9301.359.179695.614694.326695.041696.128
238.388.6085.573.9452.847.2082.726.7371.394.0431.392.7981.393.2081.393.896
2416.777.21611.168.1275.700.4645.467.6632.792.2372.791.6572.790.4422.793.791
2533.554.43222.374.44911.407.84410.966.6055.593.9045.592.4105.591.7035.596.432
2667.108.86444.820.88322.830.20321.990.68011.205.54811.206.60711.205.79711.202.931
27134.217.72889.775.07945.694.61644.080.46322.441.99922.445.89722.447.11322.440.070
28268.435.456179.795.29391.450.33488.344.95944.945.64944.952.37044.951.88544.945.389
29536.870.912360.041.554182.991.012177.050.54290.000.57590.012.46490.016.11790.012.398
301.073.741.824720.912.471366.186.435354.726.036180.212.872180.233.744180.242.980180.222.875
312.147.483.6481.443.394.362732.746.172710.648.190360.833.007360.851.168360.861.036360.849.151
324.294.967.2962.889.733.1271.466.204.9381.423.528.189722.449.732722.430.063722.422.238722.431.094
338.589.934.5925.784.941.5372.933.720.0692.851.221.4681.446.249.1481.446.214.1871.446.226.2491.446.251.953
3417.179.869.18411.580.161.3085.869.842.3575.710.318.9512.895.058.5122.894.996.0982.895.033.4132.895.073.285


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 107, 71, 1, 431, 181, 73, 773, 11, 337, 103, 419, 461, 1511, 547, 197, 1907, 227, 727, 211,
Found in Database : 3, 107, 71, 431, 181, 73, 773, 11, 337, 103, 419, 461, 1511, 547, 197, 1907, 227, 727, 211, 821, 79, 2753, 967, 113, 3203, 373, 1171, 3671, 1277, 4157, 131, 499, 59, 179, 1669, 1787, 1847,
Found in Database : 3, 11, 59, 71, 73, 79, 97, 103, 107, 113, 131,