Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:40:06
Deutsch
28.Mar 2024

Polynom = x^2+96x-5

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) = 5 = 5
f(1) = 23 = 23
f(2) = 191 = 191
f(3) = 73 = 73
f(4) = 395 = 5*79
f(5) = 125 = 5*5*5
f(6) = 607 = 607
f(7) = 179 = 179
f(8) = 827 = 827
f(9) = 235 = 5*47
f(10) = 1055 = 5*211
f(11) = 293 = 293
f(12) = 1291 = 1291
f(13) = 353 = 353
f(14) = 1535 = 5*307
f(15) = 415 = 5*83
f(16) = 1787 = 1787
f(17) = 479 = 479
f(18) = 2047 = 23*89
f(19) = 545 = 5*109
f(20) = 2315 = 5*463
f(21) = 613 = 613
f(22) = 2591 = 2591
f(23) = 683 = 683
f(24) = 2875 = 5*5*5*23
f(25) = 755 = 5*151
f(26) = 3167 = 3167
f(27) = 829 = 829
f(28) = 3467 = 3467
f(29) = 905 = 5*181
f(30) = 3775 = 5*5*151
f(31) = 983 = 983
f(32) = 4091 = 4091
f(33) = 1063 = 1063
f(34) = 4415 = 5*883
f(35) = 1145 = 5*229
f(36) = 4747 = 47*101
f(37) = 1229 = 1229
f(38) = 5087 = 5087
f(39) = 1315 = 5*263
f(40) = 5435 = 5*1087
f(41) = 1403 = 23*61
f(42) = 5791 = 5791
f(43) = 1493 = 1493
f(44) = 6155 = 5*1231
f(45) = 1585 = 5*317
f(46) = 6527 = 61*107
f(47) = 1679 = 23*73
f(48) = 6907 = 6907
f(49) = 1775 = 5*5*71
f(50) = 7295 = 5*1459
f(51) = 1873 = 1873
f(52) = 7691 = 7691
f(53) = 1973 = 1973
f(54) = 8095 = 5*1619
f(55) = 2075 = 5*5*83
f(56) = 8507 = 47*181
f(57) = 2179 = 2179
f(58) = 8927 = 79*113
f(59) = 2285 = 5*457
f(60) = 9355 = 5*1871
f(61) = 2393 = 2393
f(62) = 9791 = 9791
f(63) = 2503 = 2503
f(64) = 10235 = 5*23*89
f(65) = 2615 = 5*523
f(66) = 10687 = 10687
f(67) = 2729 = 2729
f(68) = 11147 = 71*157
f(69) = 2845 = 5*569
f(70) = 11615 = 5*23*101
f(71) = 2963 = 2963
f(72) = 12091 = 107*113
f(73) = 3083 = 3083
f(74) = 12575 = 5*5*503
f(75) = 3205 = 5*641
f(76) = 13067 = 73*179
f(77) = 3329 = 3329
f(78) = 13567 = 13567
f(79) = 3455 = 5*691
f(80) = 14075 = 5*5*563
f(81) = 3583 = 3583
f(82) = 14591 = 14591
f(83) = 3713 = 47*79
f(84) = 15115 = 5*3023
f(85) = 3845 = 5*769
f(86) = 15647 = 15647
f(87) = 3979 = 23*173
f(88) = 16187 = 16187
f(89) = 4115 = 5*823
f(90) = 16735 = 5*3347
f(91) = 4253 = 4253
f(92) = 17291 = 17291
f(93) = 4393 = 23*191
f(94) = 17855 = 5*3571
f(95) = 4535 = 5*907
f(96) = 18427 = 18427
f(97) = 4679 = 4679
f(98) = 19007 = 83*229
f(99) = 4825 = 5*5*193
f(100) = 19595 = 5*3919

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+96x-5

f(0)=5
f(1)=23
f(2)=191
f(3)=73
f(4)=79
f(5)=1
f(6)=607
f(7)=179
f(8)=827
f(9)=47
f(10)=211
f(11)=293
f(12)=1291
f(13)=353
f(14)=307
f(15)=83
f(16)=1787
f(17)=479
f(18)=89
f(19)=109
f(20)=463
f(21)=613
f(22)=2591
f(23)=683
f(24)=1
f(25)=151
f(26)=3167
f(27)=829
f(28)=3467
f(29)=181
f(30)=1
f(31)=983
f(32)=4091
f(33)=1063
f(34)=883
f(35)=229
f(36)=101
f(37)=1229
f(38)=5087
f(39)=263
f(40)=1087
f(41)=61
f(42)=5791
f(43)=1493
f(44)=1231
f(45)=317
f(46)=107
f(47)=1
f(48)=6907
f(49)=71
f(50)=1459
f(51)=1873
f(52)=7691
f(53)=1973
f(54)=1619
f(55)=1
f(56)=1
f(57)=2179
f(58)=113
f(59)=457
f(60)=1871
f(61)=2393
f(62)=9791
f(63)=2503
f(64)=1
f(65)=523
f(66)=10687
f(67)=2729
f(68)=157
f(69)=569
f(70)=1
f(71)=2963
f(72)=1
f(73)=3083
f(74)=503
f(75)=641
f(76)=1
f(77)=3329
f(78)=13567
f(79)=691
f(80)=563
f(81)=3583
f(82)=14591
f(83)=1
f(84)=3023
f(85)=769
f(86)=15647
f(87)=173
f(88)=16187
f(89)=823
f(90)=3347
f(91)=4253
f(92)=17291
f(93)=1
f(94)=3571
f(95)=907
f(96)=18427
f(97)=4679
f(98)=1
f(99)=193

b) Substitution of the polynom
The polynom f(x)=x^2+96x-5 could be written as f(y)= y^2-2309 with x=y-48

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+48
f'(x)>2x+95

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

5, 23, 191, 73, 79, 1, 607, 179, 827, 47, 211, 293, 1291, 353, 307, 83, 1787, 479, 89, 109, 463, 613, 2591, 683, 1, 151, 3167, 829, 3467, 181, 1, 983, 4091, 1063, 883, 229, 101, 1229, 5087, 263, 1087, 61, 5791, 1493, 1231, 317, 107, 1, 6907, 71, 1459, 1873, 7691, 1973, 1619, 1, 1, 2179, 113, 457, 1871, 2393, 9791, 2503, 1, 523, 10687, 2729, 157, 569, 1, 2963, 1, 3083, 503, 641, 1, 3329, 13567, 691, 563, 3583, 14591, 1, 3023, 769, 15647, 173, 16187, 823, 3347, 4253, 17291, 1, 3571, 907, 18427, 4679, 1, 193, 3919, 4973, 331, 1, 4159, 1, 21407, 1, 22027, 1117, 197, 5743, 23291, 5903, 4787, 1213, 1069, 6229, 25247, 1279, 1, 6563, 26591, 6733, 1091, 1381, 27967, 7079, 1, 1451, 1, 7433, 30091, 1, 6163, 1559, 31547, 1, 389, 1, 6607, 8353, 33791, 8543, 6911, 1747, 35327, 8929, 36107, 1, 1, 9323, 37691, 1, 7699, 1, 1709, 9929, 40127, 2027, 8191, 10343, 1, 1, 8527, 2153, 43487, 10979, 727, 2239, 9043, 1, 46091, 11633, 1879, 2371, 1, 257, 48767, 1, 1987, 1, 50591, 12763, 10303, 1, 1, 13229, 271, 2693, 10867, 1, 55291, 1, 11251, 2837, 643, 1, 58207, 587, 11839, 14923, 2617, 15173, 12239, 617, 62207, 15679, 2749, 3187, 1, 16193, 599, 16453, 13267, 3343, 853, 16979, 68447, 3449, 13903, 1, 967, 17783, 1, 1, 72767, 18329, 73867, 1, 2999, 821, 76091, 19163, 15443, 3889, 78347, 1, 787, 4003, 16127, 1, 919, 20593, 1, 4177, 84127, 21179, 3709, 859, 17299, 21773, 87691, 22073, 773, 1, 90107, 22679, 337, 4597, 1, 23293, 1321, 23603, 1, 4783, 1319, 24229, 97547, 4909, 19763, 1, 991, 25183, 811, 5101, 102667, 1123, 103967, 5231, 4211, 373, 106591, 26813, 1, 1, 1021, 27479, 110587, 5563, 367, 1, 1, 28493, 997, 1, 116027, 29179, 1039, 1181, 1033, 29873, 1, 30223, 1, 1223, 123007, 1, 124427, 6257, 25171, 31643, 127291, 32003, 25747, 6473, 1, 1423, 2801, 6619, 1, 1, 134591, 1471, 5443, 6841, 137567, 1, 139067, 6991, 5623, 397, 941, 1, 28723, 7219, 839, 36479, 3121, 1, 1289, 37253, 149791, 37643, 30271, 7607, 1, 1, 421, 1553, 31219, 1, 2221, 39623, 31859, 1601, 160907, 40429, 162527, 8167, 32831, 41243, 1, 1811, 33487, 1, 1, 1, 2339, 1, 34483, 43313, 174091, 433, 1, 8831, 177467, 44579, 179167, 9001, 1447, 45433, 1163, 45863, 1, 1, 8089, 1, 187787, 9433, 37907, 1, 8317, 449, 38611, 9697, 194827, 1, 467, 1, 39679, 49823, 200191, 50273, 1, 2029, 203807, 1, 205627, 1, 41491, 461, 1, 52553, 42227, 1, 212987, 53479, 1187, 10789, 487, 54413, 218591, 1, 8819, 11071, 222367, 55829, 224267, 11261, 1, 56783, 1, 1, 1, 11549, 231947, 58229, 10169, 11743, 1, 1, 237791, 59693, 47951, 12037, 241727, 60679, 243707, 2447, 49139, 61673, 247691, 1, 49939, 1, 251707, 63179, 521, 1, 51151, 2791, 257791, 1, 1, 13043, 261887, 65729, 4327, 13249, 1, 66763, 268091, 1103, 1, 1, 272267, 68329, 1, 1, 11059, 69383, 278591, 1, 2441, 1, 282847, 70979, 547, 14303, 57427, 72053, 289291, 1, 1, 14627, 1, 73679, 1151, 2969, 557, 3251, 300191, 75323, 1, 1, 6481, 3323, 306827, 1, 1, 77543, 311291, 1, 1, 15733, 1, 79229, 571, 15959, 64063, 80363, 322591, 80933, 1, 16301, 6961, 1, 577, 1, 1, 83233, 4229, 83813, 1, 16879, 338747, 84979, 341087, 1, 68687, 1, 345791, 86743, 1, 17467, 4937, 3823, 352907, 3541, 71059, 89123, 4019, 1, 72019, 3613, 1583, 1, 4999, 18307, 73471, 92143, 369791, 92753, 1, 1, 374687, 93979, 377147, 18919, 3301, 95213, 2111, 2039, 15383, 1, 16829, 1, 389567, 19541, 15683, 1, 2003, 98963, 79423, 19919, 399647, 1373, 1, 20173, 1327, 101503, 407291, 4441, 81971, 1, 4969, 1, 4663, 1, 1777, 104723, 420191, 105373, 84559, 4241, 2459, 1, 428027, 21467, 86131, 1367, 433291, 1, 87187, 21863, 19069, 1549, 441247, 22129, 1889, 709, 19417, 1, 17971, 22531, 451967, 113329, 1481, 1, 3659, 1571, 460091, 115363, 1, 23209, 1, 1, 1, 1, 94207, 1, 4691, 1, 95311, 1, 479327, 2557, 4423, 1, 96979, 1993, 487691, 122273, 1, 4919, 8087, 1, 496127, 24877, 1, 125093, 21817, 125803, 100927, 25303, 2657, 2707, 22189, 25589, 102643, 128663, 516091, 1, 20759, 26021, 5167, 130829, 2719, 1, 21107, 132283, 3067, 133013, 106703, 1, 536447, 1511, 1, 27043, 1, 1, 545291, 136693, 2333, 27487, 3511, 138179, 554207, 5557, 111439, 1, 560191, 140423, 1543, 5647, 1, 1999, 24749, 28537, 114451, 143443, 9431, 144203, 1, 1, 1, 2389, 584447, 1, 117503, 147263, 2579, 1, 23747, 29761, 596767, 149579, 1, 30071, 1, 6571, 606091, 2081, 121843, 30539, 5419, 6673, 2917, 30853, 123727, 3299, 1667, 1, 124991, 31327, 628127, 157429, 631307, 6329, 1, 159023, 637691, 1, 5573, 1, 644107, 1, 647327, 1, 5657, 3469, 7877, 163853, 1, 32933, 3457, 165479, 663547, 1, 1, 167113, 2287, 2753, 5387, 33751, 7603, 1, 1, 1, 1, 171233, 6299, 7481, 137983, 1, 1, 173729, 14821, 34913, 139987, 175403, 3929, 176243, 1789, 1, 710027, 177929, 713407, 7151, 1, 179623, 8677, 180473, 144719, 7253, 1, 182179, 15541, 36607, 1, 1, 737291, 184753, 148147, 37123, 744187, 186479, 747647, 1, 150223, 1759, 1, 8221, 30323, 37991, 761567, 190829, 2909, 1, 1, 192583, 772091, 193463, 2543, 1, 1, 195229, 782687, 1, 157247, 197003, 1, 197893, 1, 1, 34649, 199679, 10133, 1, 2203, 201473, 35117, 1, 1, 1, 1, 1, 1823, 41017, 164431, 205993, 825791, 1831, 165887, 1, 7643, 208729, 836747, 1, 168083, 1, 1, 2677, 33911, 1847, 1, 213329, 8467, 1, 6871, 215183, 18353, 216113, 173263, 1, 870047, 217979, 873787, 43783, 1, 2017, 38317, 220793, 177011, 1, 888827, 222679, 1, 1, 2939, 3163, 1, 225523, 180799, 9059, 907807, 1, 10243, 45677, 183091, 229343, 1, 230303, 184627, 2011, 2383, 1, 930847, 46639, 2633, 10181, 938591, 3221, 37699, 47221, 946367, 3001, 1, 1013, 38167, 1, 2843, 1, 1, 1, 6397, 241979, 42169, 48593, 194767, 243953, 977791, 244943, 8537, 1, 2129, 246929, 3851, 1, 1, 1, 1, 249923, 2251, 10037, 16487, 1, 1009727, 50587, 202751, 1, 9007, 1, 204367, 51193, 1025887, 11173, 12409, 51599, 206803, 1, 14621, 1, 41687, 1, 1, 262079, 1050367, 1, 42179, 264133, 1058591, 265163, 9241, 53239, 1066847, 267229, 1070987, 53653, 9349, 3793, 1079291, 270343, 1, 54277, 1, 3061, 1091807, 10939,

6. Sequence of the polynom (only primes)

5, 23, 191, 73, 79, 607, 179, 827, 47, 211, 293, 1291, 353, 307, 83, 1787, 479, 89, 109, 463, 613, 2591, 683, 151, 3167, 829, 3467, 181, 983, 4091, 1063, 883, 229, 101, 1229, 5087, 263, 1087, 61, 5791, 1493, 1231, 317, 107, 6907, 71, 1459, 1873, 7691, 1973, 1619, 2179, 113, 457, 1871, 2393, 9791, 2503, 523, 10687, 2729, 157, 569, 2963, 3083, 503, 641, 3329, 13567, 691, 563, 3583, 14591, 3023, 769, 15647, 173, 16187, 823, 3347, 4253, 17291, 3571, 907, 18427, 4679, 193, 3919, 4973, 331, 4159, 21407, 22027, 1117, 197, 5743, 23291, 5903, 4787, 1213, 1069, 6229, 25247, 1279, 6563, 26591, 6733, 1091, 1381, 27967, 7079, 1451, 7433, 30091, 6163, 1559, 31547, 389, 6607, 8353, 33791, 8543, 6911, 1747, 35327, 8929, 36107, 9323, 37691, 7699, 1709, 9929, 40127, 2027, 8191, 10343, 8527, 2153, 43487, 10979, 727, 2239, 9043, 46091, 11633, 1879, 2371, 257, 48767, 1987, 50591, 12763, 10303, 13229, 271, 2693, 10867, 55291, 11251, 2837, 643, 58207, 587, 11839, 14923, 2617, 15173, 12239, 617, 62207, 15679, 2749, 3187, 16193, 599, 16453, 13267, 3343, 853, 16979, 68447, 3449, 13903, 967, 17783, 72767, 18329, 73867, 2999, 821, 76091, 19163, 15443, 3889, 78347, 787, 4003, 16127, 919, 20593, 4177, 84127, 21179, 3709, 859, 17299, 21773, 87691, 22073, 773, 90107, 22679, 337, 4597, 23293, 1321, 23603, 4783, 1319, 24229, 97547, 4909, 19763, 991, 25183, 811, 5101, 102667, 1123, 103967, 5231, 4211, 373, 106591, 26813, 1021, 27479, 110587, 5563, 367, 28493, 997, 116027, 29179, 1039, 1181, 1033, 29873, 30223, 1223, 123007, 124427, 6257, 25171, 31643, 127291, 32003, 25747, 6473, 1423, 2801, 6619, 134591, 1471, 5443, 6841, 137567, 139067, 6991, 5623, 397, 941, 28723, 7219, 839, 36479, 3121, 1289, 37253, 149791, 37643, 30271, 7607, 421, 1553, 31219, 2221, 39623, 31859, 1601, 160907, 40429, 162527, 8167, 32831, 41243, 1811, 33487, 2339, 34483, 43313, 174091, 433, 8831, 177467, 44579, 179167, 9001, 1447, 45433, 1163, 45863, 8089, 187787, 9433, 37907, 8317, 449, 38611, 9697, 194827, 467, 39679, 49823, 200191, 50273, 2029, 203807, 205627, 41491, 461, 52553, 42227, 212987, 53479, 1187, 10789, 487, 54413, 218591, 8819, 11071, 222367, 55829, 224267, 11261, 56783, 11549, 231947, 58229, 10169, 11743, 237791, 59693, 47951, 12037, 241727, 60679, 243707, 2447, 49139, 61673, 247691, 49939, 251707, 63179, 521, 51151, 2791, 257791, 13043, 261887, 65729, 4327, 13249, 66763, 268091, 1103, 272267, 68329, 11059, 69383, 278591, 2441, 282847, 70979, 547, 14303, 57427, 72053, 289291, 14627, 73679, 1151, 2969, 557, 3251, 300191, 75323, 6481, 3323, 306827, 77543, 311291, 15733, 79229, 571, 15959, 64063, 80363, 322591, 80933, 16301, 6961, 577, 83233, 4229, 83813, 16879, 338747, 84979, 341087, 68687, 345791, 86743, 17467, 4937, 3823, 352907, 3541, 71059, 89123, 4019, 72019, 3613, 1583, 4999, 18307, 73471, 92143, 369791, 92753, 374687, 93979, 377147, 18919, 3301, 95213, 2111, 2039, 15383, 16829, 389567, 19541, 15683, 2003, 98963, 79423, 19919, 399647, 1373, 20173, 1327, 101503, 407291, 4441, 81971, 4969, 4663, 1777, 104723, 420191, 105373, 84559, 4241, 2459, 428027, 21467, 86131, 1367, 433291, 87187, 21863, 19069, 1549, 441247, 22129, 1889, 709, 19417, 17971, 22531, 451967, 113329, 1481, 3659, 1571, 460091, 115363, 23209, 94207, 4691, 95311, 479327, 2557, 4423, 96979, 1993, 487691, 122273, 4919, 8087, 496127, 24877, 125093, 21817, 125803, 100927, 25303, 2657, 2707, 22189, 25589, 102643, 128663, 516091, 20759, 26021, 5167, 130829, 2719, 21107, 132283, 3067, 133013, 106703, 536447, 1511, 27043, 545291, 136693, 2333, 27487, 3511, 138179, 554207, 5557, 111439, 560191, 140423, 1543, 5647, 1999, 24749, 28537, 114451, 143443, 9431, 144203, 2389, 584447, 117503, 147263, 2579, 23747, 29761, 596767, 149579, 30071, 6571, 606091, 2081, 121843, 30539, 5419, 6673, 2917, 30853, 123727, 3299, 1667, 124991, 31327, 628127, 157429, 631307, 6329, 159023, 637691, 5573, 644107, 647327, 5657, 3469, 7877, 163853, 32933, 3457, 165479, 663547, 167113, 2287, 2753, 5387, 33751, 7603, 171233, 6299, 7481, 137983, 173729, 14821, 34913, 139987, 175403, 3929, 176243, 1789, 710027, 177929, 713407, 7151, 179623, 8677, 180473, 144719, 7253, 182179, 15541, 36607, 737291, 184753, 148147, 37123, 744187, 186479, 747647, 150223, 1759, 8221, 30323, 37991, 761567, 190829, 2909, 192583, 772091, 193463, 2543, 195229, 782687, 157247, 197003, 197893, 34649, 199679, 10133, 2203, 201473, 35117, 1823, 41017, 164431, 205993, 825791, 1831, 165887, 7643, 208729, 836747, 168083, 2677, 33911, 1847, 213329, 8467, 6871, 215183, 18353, 216113, 173263, 870047, 217979, 873787, 43783, 2017, 38317, 220793, 177011, 888827, 222679, 2939, 3163, 225523, 180799, 9059, 907807, 10243, 45677, 183091, 229343, 230303, 184627, 2011, 2383, 930847, 46639, 2633, 10181, 938591, 3221, 37699, 47221, 946367, 3001, 1013, 38167, 2843, 6397, 241979, 42169, 48593, 194767, 243953, 977791, 244943, 8537, 2129, 246929, 3851, 249923, 2251, 10037, 16487, 1009727, 50587, 202751, 9007, 204367, 51193, 1025887, 11173, 12409, 51599, 206803, 14621, 41687, 262079, 1050367, 42179, 264133, 1058591, 265163, 9241, 53239, 1066847, 267229, 1070987, 53653, 9349, 3793, 1079291, 270343, 54277, 3061, 1091807, 10939,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+96x-5 and
the reducible primes which appear as divisor for the first time
p | x^2+96x-5 and p < x^2+96x-5

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)
11010731.0000000.7000000.3000000.0000000.0000000.000000
21008347360.8300000.4700000.3600008.3000006.71428612.000000
31.0007423294130.7420000.3290000.4130008.9397597.00000011.472222
410.0007.2722.3354.9370.7272000.2335000.4937009.8005397.09726411.953995
5100.00071.88117.92753.9540.7188100.1792700.5395409.8846267.67751610.928499
61.000.000713.917145.881568.0360.7139170.1458810.5680369.9319308.13750210.528153
710.000.0007.107.0151.228.0895.878.9260.7107020.1228090.5878939.9549608.41843010.349566
8100.000.00070.828.64810.608.18960.220.4590.7082860.1060820.6022059.9660208.63796410.243445
91.000.000.000706.431.19093.394.345613.036.8450.7064310.0933940.6130379.9738058.80398610.179877
1010.000.000.0007.049.754.375834.081.6796.215.672.6960.7049750.0834080.6215679.9793948.93075210.139151


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
245411.2500001.0000000.2500001.6666671.333333inf
388711.0000000.8750000.1250001.6000001.7500001.000000
416161151.0000000.6875000.3125002.0000001.5714295.000000
5323020100.9375000.6250000.3125001.8750001.8181822.000000
6645533220.8593750.5156250.3437501.8333331.6500002.200000
712810560450.8203120.4687500.3515621.9090911.8181822.045455
8256203106970.7929690.4140620.3789061.9333331.7666672.155555
95123831931900.7480470.3769530.3710941.8867001.8207551.958763
101.0247603344260.7421880.3261720.4160161.9843341.7305702.242105
112.0481.5165979190.7402340.2915040.4487301.9947371.7874252.157277
124.0963.0031.0801.9230.7331540.2636720.4694821.9808711.8090452.092492
138.1925.9671.9743.9930.7283940.2409670.4874271.9870131.8277782.076443
1416.38411.8903.5848.3060.7257080.2187500.5069581.9926261.8156032.080140
1532.76823.6856.60717.0780.7228090.2016300.5211791.9920101.8434712.056104
1665.53647.19412.30434.8900.7201230.1877440.5323791.9925691.8622672.042979
17131.07294.08422.92971.1550.7178040.1749340.5428701.9935591.8635402.039410
18262.144187.72843.008144.7200.7161250.1640620.5520631.9953231.8757032.033870
19524.288374.78180.794293.9870.7148380.1541020.5607361.9964041.8785812.031419
201.048.576748.421152.420596.0010.7137500.1453590.5683911.9969561.8865262.027304
212.097.1521.494.814288.6231.206.1910.7127830.1376260.5751571.9972901.8936032.023807
224.194.3042.985.285547.9162.437.3690.7117470.1306330.5811141.9970951.8983802.020716
238.388.6085.963.1241.042.7304.920.3940.7108600.1243030.5865571.9975061.9030842.018732
2416.777.21611.912.9361.990.2069.922.7300.7100660.1186260.5914411.9977681.9086492.016654
2533.554.43223.799.7243.807.59419.992.1300.7092870.1134750.5958121.9978051.9131662.014781
2667.108.86447.556.2357.291.77540.264.4600.7086430.1086560.5999871.9981841.9150612.014015
27134.217.72895.027.61313.996.39381.031.2200.7080110.1042810.6037301.9982161.9194772.012475
28268.435.456189.898.06026.910.968162.987.0920.7074250.1002510.6071741.9983461.9227072.011411
29536.870.912379.504.16551.817.395327.686.7700.7068820.0965170.6103641.9984631.9255122.010508
301.073.741.824758.470.23899.910.577658.559.6610.7063800.0930490.6133311.9985821.9281282.009723
312.147.483.6481.515.925.594192.884.4971.323.041.0970.7059080.0898190.6160891.9986621.9305712.008992
324.294.967.2963.029.969.930372.861.9642.657.107.9660.7054700.0868140.6186561.9987591.9330842.008334
338.589.934.5926.056.433.417721.566.4505.334.866.9670.7050620.0840010.6210601.9988431.9352112.007772
3417.179.869.18412.106.237.8021.397.894.28310.708.343.5190.7046760.0813680.6233081.9989051.9373052.007237


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
123030012
244131012
387251213
41611382423
532205152747
664331122410712
71286020406181125
8256106337315361540
95121936013323712673
101.0243341062284211943130
112.0485972013967521372237
124.0961.080366714142400125413
138.1921.9746701.304256715249754
1416.3843.5841.2032.3814801.3044521.348
1532.7686.6072.1864.4218682.4428492.448
1665.53612.3044.1138.1911.5894.5901.5364.589
17131.07222.9297.70215.2273.0108.5322.8918.496
18262.14443.00814.25628.7525.62515.9915.52015.872
19524.28880.79426.92253.87210.53730.03610.35129.870
201.048.576152.42050.857101.56319.69756.59019.54856.585
212.097.152288.62396.321192.30237.290107.30937.048106.976
224.194.304547.916182.941364.97570.528203.85970.101203.428
238.388.6081.042.730347.769694.961133.856388.068133.206387.600
2416.777.2161.990.206663.3781.326.828255.199739.911254.125740.971
2533.554.4323.807.5941.270.0222.537.572487.3541.416.585486.1281.417.527
2667.108.8647.291.7752.431.7394.860.036931.5742.713.794930.4782.715.929
27134.217.72813.996.3934.666.7599.329.6341.785.0885.211.5911.784.5825.215.132
28268.435.45626.910.9688.970.92517.940.0433.428.73010.024.9373.429.30410.027.997
29536.870.91251.817.39517.274.31834.543.0776.598.49719.306.6356.596.77919.315.484
301.073.741.82499.910.57733.307.83366.602.74412.714.93637.236.39412.710.95637.248.291
312.147.483.648192.884.49764.300.844128.583.65324.528.36171.909.80924.528.41571.917.912
324.294.967.296372.861.964124.291.315248.570.64947.383.661139.041.34447.388.412139.048.547
338.589.934.592721.566.450240.520.321481.046.12991.654.874269.117.41791.653.876269.140.283
3417.179.869.1841.397.894.283465.963.857931.930.426177.473.331521.464.548177.472.163521.484.241


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
241100001
381100001
4165320302
53210731324
664221392758
71284528175161014
825697613611351932
95121901147630663361
101.0244262481787312984140
112.048919511408182275189273
124.0961.9231.079844411561394557
138.1923.9932.1931.8008481.1318741.140
1416.3848.3064.4863.8201.8162.3301.8312.329
1532.76817.0789.1167.9623.8394.7313.8024.706
1665.53634.89018.53616.3547.8289.5347.9369.592
17131.07271.15537.59233.56316.17019.28216.25619.447
18262.144144.72076.27668.44433.20239.08933.30539.124
19524.288293.987154.340139.64767.91079.04568.14578.887
201.048.576596.001311.672284.329138.395159.422138.891159.293
212.097.1521.206.191628.888577.303281.525320.939283.036320.691
224.194.3042.437.3691.266.8481.170.521572.351645.473574.072645.473
238.388.6084.920.3942.552.0212.368.3731.160.8791.299.5971.161.2501.298.668
2416.777.2169.922.7305.138.8784.783.8522.349.0412.613.3652.348.2692.612.055
2533.554.43219.992.13010.332.1099.660.0214.747.0045.251.6964.743.2625.250.168
2667.108.86440.264.46020.777.22919.487.2319.584.94010.549.2259.581.74610.548.549
27134.217.72881.031.22041.754.61639.276.60419.334.87421.181.10219.332.92821.182.316
28268.435.456162.987.09283.871.27679.115.81638.969.88942.523.00738.966.00842.528.188
29536.870.912327.686.770168.412.537159.274.23378.503.33985.341.20778.501.38085.340.844
301.073.741.824658.559.661338.065.301320.494.360158.052.178171.235.803158.043.543171.228.137
312.147.483.6481.323.041.097678.452.130644.588.967318.046.139343.476.138318.037.589343.481.231
324.294.967.2962.657.107.9661.361.262.3741.295.845.592639.721.991688.832.114639.730.882688.822.979
338.589.934.5925.334.866.9672.730.613.4692.604.253.4981.286.190.4541.381.219.0291.286.223.1051.381.234.379
3417.179.869.18410.708.343.5195.476.470.3925.231.873.1272.585.080.1372.769.091.5242.585.094.4972.769.077.361


8. Check for existing Integer Sequences by OEIS

Found in Database : 5, 23, 191, 73, 79, 1, 607, 179, 827, 47, 211, 293, 1291, 353, 307, 83, 1787, 479, 89, 109,
Found in Database : 5, 23, 191, 73, 79, 607, 179, 827, 47, 211, 293, 1291, 353, 307, 83, 1787, 479, 89, 109, 463, 613, 2591, 683, 151, 3167, 829, 3467, 181, 983, 4091, 1063, 883, 229, 101, 1229, 5087, 263,
Found in Database : 5, 23, 47, 61, 71, 73, 79, 83, 89, 101, 107, 109, 113,