Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:56:55
Deutsch
28.Mar 2024

Polynom = x^2+17x-127

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) = 127 = 127
f(1) = 109 = 109
f(2) = 89 = 89
f(3) = 67 = 67
f(4) = 43 = 43
f(5) = 17 = 17
f(6) = 11 = 11
f(7) = 41 = 41
f(8) = 73 = 73
f(9) = 107 = 107
f(10) = 143 = 11*13
f(11) = 181 = 181
f(12) = 221 = 13*17
f(13) = 263 = 263
f(14) = 307 = 307
f(15) = 353 = 353
f(16) = 401 = 401
f(17) = 451 = 11*41
f(18) = 503 = 503
f(19) = 557 = 557
f(20) = 613 = 613
f(21) = 671 = 11*61
f(22) = 731 = 17*43
f(23) = 793 = 13*61
f(24) = 857 = 857
f(25) = 923 = 13*71
f(26) = 991 = 991
f(27) = 1061 = 1061
f(28) = 1133 = 11*103
f(29) = 1207 = 17*71
f(30) = 1283 = 1283
f(31) = 1361 = 1361
f(32) = 1441 = 11*131
f(33) = 1523 = 1523
f(34) = 1607 = 1607
f(35) = 1693 = 1693
f(36) = 1781 = 13*137
f(37) = 1871 = 1871
f(38) = 1963 = 13*151
f(39) = 2057 = 11*11*17
f(40) = 2153 = 2153
f(41) = 2251 = 2251
f(42) = 2351 = 2351
f(43) = 2453 = 11*223
f(44) = 2557 = 2557
f(45) = 2663 = 2663
f(46) = 2771 = 17*163
f(47) = 2881 = 43*67
f(48) = 2993 = 41*73
f(49) = 3107 = 13*239
f(50) = 3223 = 11*293
f(51) = 3341 = 13*257
f(52) = 3461 = 3461
f(53) = 3583 = 3583
f(54) = 3707 = 11*337
f(55) = 3833 = 3833
f(56) = 3961 = 17*233
f(57) = 4091 = 4091
f(58) = 4223 = 41*103
f(59) = 4357 = 4357
f(60) = 4493 = 4493
f(61) = 4631 = 11*421
f(62) = 4771 = 13*367
f(63) = 4913 = 17*17*17
f(64) = 5057 = 13*389
f(65) = 5203 = 11*11*43
f(66) = 5351 = 5351
f(67) = 5501 = 5501
f(68) = 5653 = 5653
f(69) = 5807 = 5807
f(70) = 5963 = 67*89
f(71) = 6121 = 6121
f(72) = 6281 = 11*571
f(73) = 6443 = 17*379
f(74) = 6607 = 6607
f(75) = 6773 = 13*521
f(76) = 6941 = 11*631
f(77) = 7111 = 13*547
f(78) = 7283 = 7283
f(79) = 7457 = 7457
f(80) = 7633 = 17*449
f(81) = 7811 = 73*107
f(82) = 7991 = 61*131
f(83) = 8173 = 11*743
f(84) = 8357 = 61*137
f(85) = 8543 = 8543
f(86) = 8731 = 8731
f(87) = 8921 = 11*811
f(88) = 9113 = 13*701
f(89) = 9307 = 41*227
f(90) = 9503 = 13*17*43
f(91) = 9701 = 89*109
f(92) = 9901 = 9901
f(93) = 10103 = 10103
f(94) = 10307 = 11*937
f(95) = 10513 = 10513
f(96) = 10721 = 71*151
f(97) = 10931 = 17*643
f(98) = 11143 = 11*1013
f(99) = 11357 = 41*277
f(100) = 11573 = 71*163

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+17x-127

f(0)=127
f(1)=109
f(2)=89
f(3)=67
f(4)=43
f(5)=17
f(6)=11
f(7)=41
f(8)=73
f(9)=107
f(10)=13
f(11)=181
f(12)=1
f(13)=263
f(14)=307
f(15)=353
f(16)=401
f(17)=1
f(18)=503
f(19)=557
f(20)=613
f(21)=61
f(22)=1
f(23)=1
f(24)=857
f(25)=71
f(26)=991
f(27)=1061
f(28)=103
f(29)=1
f(30)=1283
f(31)=1361
f(32)=131
f(33)=1523
f(34)=1607
f(35)=1693
f(36)=137
f(37)=1871
f(38)=151
f(39)=1
f(40)=2153
f(41)=2251
f(42)=2351
f(43)=223
f(44)=2557
f(45)=2663
f(46)=163
f(47)=1
f(48)=1
f(49)=239
f(50)=293
f(51)=257
f(52)=3461
f(53)=3583
f(54)=337
f(55)=3833
f(56)=233
f(57)=4091
f(58)=1
f(59)=4357
f(60)=4493
f(61)=421
f(62)=367
f(63)=1
f(64)=389
f(65)=1
f(66)=5351
f(67)=5501
f(68)=5653
f(69)=5807
f(70)=1
f(71)=6121
f(72)=571
f(73)=379
f(74)=6607
f(75)=521
f(76)=631
f(77)=547
f(78)=7283
f(79)=7457
f(80)=449
f(81)=1
f(82)=1
f(83)=743
f(84)=1
f(85)=8543
f(86)=8731
f(87)=811
f(88)=701
f(89)=227
f(90)=1
f(91)=1
f(92)=9901
f(93)=10103
f(94)=937
f(95)=10513
f(96)=1
f(97)=643
f(98)=1013
f(99)=277

b) Substitution of the polynom
The polynom f(x)=x^2+17x-127 could be written as f(y)= y^2-199.25 with x=y-8.5

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+8.5
f'(x)>2x+16

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

127, 109, 89, 67, 43, 17, 11, 41, 73, 107, 13, 181, 1, 263, 307, 353, 401, 1, 503, 557, 613, 61, 1, 1, 857, 71, 991, 1061, 103, 1, 1283, 1361, 131, 1523, 1607, 1693, 137, 1871, 151, 1, 2153, 2251, 2351, 223, 2557, 2663, 163, 1, 1, 239, 293, 257, 3461, 3583, 337, 3833, 233, 4091, 1, 4357, 4493, 421, 367, 1, 389, 1, 5351, 5501, 5653, 5807, 1, 6121, 571, 379, 6607, 521, 631, 547, 7283, 7457, 449, 1, 1, 743, 1, 8543, 8731, 811, 701, 227, 1, 1, 9901, 10103, 937, 10513, 1, 643, 1013, 277, 1, 907, 12011, 941, 12457, 1153, 12911, 773, 311, 1237, 1, 14081, 14321, 14563, 1, 15053, 1, 15551, 15803, 16057, 1483, 1, 16831, 17093, 1021, 17623, 17891, 1, 18433, 1439, 463, 1, 19541, 461, 20107, 20393, 20681, 313, 1933, 21557, 1, 1303, 157, 373, 23057, 383, 23671, 23981, 1429, 2237, 24923, 587, 25561, 1, 359, 1, 26861, 27191, 1619, 1, 1, 1, 28871, 1, 2687, 1759, 179, 431, 2381, 31307, 31663, 1, 32381, 1, 33107, 1, 787, 34211, 34583, 2689, 397, 1, 193, 36473, 36857, 37243, 1, 197, 1, 1, 199, 1, 1, 3673, 1, 41213, 41621, 3821, 42443, 2521, 1, 43691, 44111, 44533, 1, 3491, 751, 3557, 4243, 1, 47543, 47981, 1181, 1, 49307, 4523, 2953, 50651, 3931, 1, 4001, 1, 1291, 499, 53857, 54323, 1, 757, 55733, 56207, 5153, 4397, 57641, 1, 569, 59093, 59581, 1, 853, 61057, 61553, 5641, 881, 3709, 4889, 64063, 4967, 1, 1, 66107, 3919, 67141, 6151, 1663, 541, 69233, 69761, 5407, 70823, 1, 4229, 72431, 1697, 1, 719, 1223, 1, 1, 76253, 863, 1, 1163, 6037, 79043, 7237, 80173, 1, 4783, 81883, 82457, 1931, 691, 84191, 6521, 5021, 601, 86531, 87121, 1, 1, 88903, 89501, 8191, 90703, 1, 859, 647, 93131, 7211, 1, 1301, 5623, 96211, 8803, 2377, 2281, 98711, 821, 1, 1, 739, 1, 653, 1453, 9437, 1559, 1, 105751, 1, 107057, 107713, 1, 8387, 2551, 1, 10093, 1831, 112361, 1, 10337, 1069, 115061, 115741, 116423, 1, 1, 10771, 1, 1789, 120557, 1, 121951, 122651, 1, 947, 1, 125471, 11471, 1, 127607, 9871, 11731, 1, 130483, 733, 131933, 132661, 133391, 1, 134857, 135593, 10487, 1, 10601, 1091, 139303, 3257, 1367, 1, 761, 143063, 143821, 887, 1, 11239, 1, 1, 148411, 149183, 8821, 1, 1193, 2273, 153073, 1, 154643, 1, 1, 157013, 1, 797, 1, 160201, 1, 161807, 14783, 9613, 164231, 1093, 165857, 12821, 167491, 1, 9949, 169957, 2549, 15601, 172441, 173273, 4049, 174943, 175781, 1, 1, 178307, 13781, 180001, 1, 1667, 1, 10789, 184271, 185131, 185993, 16987, 1433, 1, 4621, 1, 1, 11299, 192961, 193841, 194723, 4549, 17863, 197381, 1, 1, 1399, 1951, 15527, 202751, 203653, 204557, 205463, 1, 1, 3413, 3121, 1, 2971, 1, 1, 967, 3023, 1373, 19681, 5303, 218357, 219293, 20021, 221171, 222113, 13121, 17231, 224951, 17377, 1, 1451, 2221, 13513, 1, 231643, 232607, 1783, 234541, 5477, 18191, 21587, 18341, 14083, 1, 21943, 242357, 243343, 3347, 245321, 14489, 1109, 22573, 1, 250301, 1487, 22937, 1, 1, 1, 1, 6277, 258373, 23581, 260411, 261433, 1553, 1409, 20347, 265541, 2099, 1, 268643, 4421, 24611, 1, 272807, 1, 1, 21227, 277003, 1, 6491, 1, 1, 2333, 1583, 284423, 4021, 1, 1, 1, 22291, 290861, 1321, 1637, 26737, 1811, 3329, 297371, 1, 1, 7333, 301751, 302851, 1, 2851, 2141, 2819, 1, 309493, 1, 7603, 312841, 313961, 315083, 1747, 317333, 1, 319591, 24671, 4409, 29363, 324131, 325271, 7591, 327557, 1279, 19403, 30091, 1721, 25639, 5483, 2347, 5521, 1, 339107, 1499, 1, 1, 31253, 344957, 346133, 1, 2437, 1, 1, 352043, 1, 354421, 1787, 1, 21059, 8761, 4937, 1, 1, 28001, 1, 1, 1, 368857, 1979, 371291, 372511, 1, 1, 376183, 1, 22273, 29221, 1709, 29411, 34871, 384821, 2459, 387307, 35323, 3643, 23003, 3089, 393557, 394813, 30467, 1, 30661, 1, 4507, 1, 403661, 1319, 406207, 407483, 6701, 1, 1, 1867, 3259, 1, 37861, 417763, 419057, 420353, 1459, 2801, 1, 3517, 426863, 4157, 33037, 39163, 1, 6469, 1, 436061, 1579, 438707, 1, 441361, 442691, 26119, 40487, 34361, 448031, 2659, 10993, 6367, 6211, 41341, 10607, 1, 458807, 3803, 461521, 462881, 1, 1, 2113, 4547, 42701, 1, 472457, 473833, 43201, 476591, 477973, 479357, 28279, 37087, 3691, 3391, 486307, 487703, 489101, 1, 5527, 8087, 494713, 2741, 1, 3061, 3499, 3323, 38707, 29683, 1, 12377, 6971, 510311, 1, 513173, 30271, 1, 517481, 1, 520363, 1, 1889, 524701, 7853, 527603, 31121, 530513, 1, 1, 4999, 536357, 3761, 31723, 41597, 1609, 543707, 1753, 546661, 49831, 549623, 551107, 1, 2963, 4241, 1, 7867, 1, 561551, 563051, 3019, 566057, 1, 569071, 51871, 572093, 5569, 575123, 44357, 3547, 1, 52837, 1, 584261, 585791, 1, 2239, 34729, 3067, 4673, 1901, 1, 4943, 1, 601201, 602753, 1, 1, 607421, 1, 610543, 612107, 613673, 55931, 2791, 1, 1, 56503, 3163, 7019, 3989, 9371, 2011, 15391, 1, 3187, 1, 49031, 5281, 49277, 4253, 37871, 1, 1, 648631, 59113, 651857, 653473, 1, 1, 1, 38821, 50891, 9341, 9923, 2143, 60737, 9433, 1, 2297, 61333, 11087, 677953, 1, 681251, 1, 684557, 62383, 1, 2683, 691193, 62987, 694523, 1, 17021, 41149, 53939, 6569, 1, 3167, 707923, 709607, 64663, 712981, 17431, 42139, 16699, 719753, 721451, 1, 724853, 55889, 42839, 66361, 731681, 733393, 735107, 736823, 1, 7187, 67453, 6823, 3373, 747161, 5237, 5479, 3229, 754093, 4637, 44563, 759313, 1, 7129, 764551, 766301, 1, 769807, 59351, 8689, 1, 7127, 11621, 1, 782141, 1, 19163, 4211, 1, 1, 1, 61121, 10909, 798143, 1, 1, 803513, 1, 1, 4519, 9109, 812503, 62639, 816113, 3701, 74521, 1, 823357, 1, 75181, 828811, 1973, 832457, 834283, 1, 1, 76343, 1, 843443, 2207, 7001, 49939, 850807, 5231, 4721, 856351, 12809, 1907, 66301, 863771, 66587, 4639, 11909, 20261, 873091, 5573, 876833, 2999, 1, 882461, 884341, 68171, 80737, 1, 14621, 8353, 14683, 897557, 1, 1, 903251, 1, 6007, 82633, 70067, 53693, 1, 6997, 1, 920441, 1, 924283, 4799, 9011, 84551, 1, 933923, 1, 1, 72287, 941671, 1, 55621, 1, 949451, 86491, 22171, 955307, 957263, 1, 1, 963143, 1, 4909, 969041, 1, 1, 974957, 976933, 1, 8999, 1, 1, 1, 4969, 1, 4261, 90437, 996803, 1, 3613, 23321, 2531, 7349, 91711, 1, 4583, 1,

6. Sequence of the polynom (only primes)

127, 109, 89, 67, 43, 17, 11, 41, 73, 107, 13, 181, 263, 307, 353, 401, 503, 557, 613, 61, 857, 71, 991, 1061, 103, 1283, 1361, 131, 1523, 1607, 1693, 137, 1871, 151, 2153, 2251, 2351, 223, 2557, 2663, 163, 239, 293, 257, 3461, 3583, 337, 3833, 233, 4091, 4357, 4493, 421, 367, 389, 5351, 5501, 5653, 5807, 6121, 571, 379, 6607, 521, 631, 547, 7283, 7457, 449, 743, 8543, 8731, 811, 701, 227, 9901, 10103, 937, 10513, 643, 1013, 277, 907, 12011, 941, 12457, 1153, 12911, 773, 311, 1237, 14081, 14321, 14563, 15053, 15551, 15803, 16057, 1483, 16831, 17093, 1021, 17623, 17891, 18433, 1439, 463, 19541, 461, 20107, 20393, 20681, 313, 1933, 21557, 1303, 157, 373, 23057, 383, 23671, 23981, 1429, 2237, 24923, 587, 25561, 359, 26861, 27191, 1619, 28871, 2687, 1759, 179, 431, 2381, 31307, 31663, 32381, 33107, 787, 34211, 34583, 2689, 397, 193, 36473, 36857, 37243, 197, 199, 3673, 41213, 41621, 3821, 42443, 2521, 43691, 44111, 44533, 3491, 751, 3557, 4243, 47543, 47981, 1181, 49307, 4523, 2953, 50651, 3931, 4001, 1291, 499, 53857, 54323, 757, 55733, 56207, 5153, 4397, 57641, 569, 59093, 59581, 853, 61057, 61553, 5641, 881, 3709, 4889, 64063, 4967, 66107, 3919, 67141, 6151, 1663, 541, 69233, 69761, 5407, 70823, 4229, 72431, 1697, 719, 1223, 76253, 863, 1163, 6037, 79043, 7237, 80173, 4783, 81883, 82457, 1931, 691, 84191, 6521, 5021, 601, 86531, 87121, 88903, 89501, 8191, 90703, 859, 647, 93131, 7211, 1301, 5623, 96211, 8803, 2377, 2281, 98711, 821, 739, 653, 1453, 9437, 1559, 105751, 107057, 107713, 8387, 2551, 10093, 1831, 112361, 10337, 1069, 115061, 115741, 116423, 10771, 1789, 120557, 121951, 122651, 947, 125471, 11471, 127607, 9871, 11731, 130483, 733, 131933, 132661, 133391, 134857, 135593, 10487, 10601, 1091, 139303, 3257, 1367, 761, 143063, 143821, 887, 11239, 148411, 149183, 8821, 1193, 2273, 153073, 154643, 157013, 797, 160201, 161807, 14783, 9613, 164231, 1093, 165857, 12821, 167491, 9949, 169957, 2549, 15601, 172441, 173273, 4049, 174943, 175781, 178307, 13781, 180001, 1667, 10789, 184271, 185131, 185993, 16987, 1433, 4621, 11299, 192961, 193841, 194723, 4549, 17863, 197381, 1399, 1951, 15527, 202751, 203653, 204557, 205463, 3413, 3121, 2971, 967, 3023, 1373, 19681, 5303, 218357, 219293, 20021, 221171, 222113, 13121, 17231, 224951, 17377, 1451, 2221, 13513, 231643, 232607, 1783, 234541, 5477, 18191, 21587, 18341, 14083, 21943, 242357, 243343, 3347, 245321, 14489, 1109, 22573, 250301, 1487, 22937, 6277, 258373, 23581, 260411, 261433, 1553, 1409, 20347, 265541, 2099, 268643, 4421, 24611, 272807, 21227, 277003, 6491, 2333, 1583, 284423, 4021, 22291, 290861, 1321, 1637, 26737, 1811, 3329, 297371, 7333, 301751, 302851, 2851, 2141, 2819, 309493, 7603, 312841, 313961, 315083, 1747, 317333, 319591, 24671, 4409, 29363, 324131, 325271, 7591, 327557, 1279, 19403, 30091, 1721, 25639, 5483, 2347, 5521, 339107, 1499, 31253, 344957, 346133, 2437, 352043, 354421, 1787, 21059, 8761, 4937, 28001, 368857, 1979, 371291, 372511, 376183, 22273, 29221, 1709, 29411, 34871, 384821, 2459, 387307, 35323, 3643, 23003, 3089, 393557, 394813, 30467, 30661, 4507, 403661, 1319, 406207, 407483, 6701, 1867, 3259, 37861, 417763, 419057, 420353, 1459, 2801, 3517, 426863, 4157, 33037, 39163, 6469, 436061, 1579, 438707, 441361, 442691, 26119, 40487, 34361, 448031, 2659, 10993, 6367, 6211, 41341, 10607, 458807, 3803, 461521, 462881, 2113, 4547, 42701, 472457, 473833, 43201, 476591, 477973, 479357, 28279, 37087, 3691, 3391, 486307, 487703, 489101, 5527, 8087, 494713, 2741, 3061, 3499, 3323, 38707, 29683, 12377, 6971, 510311, 513173, 30271, 517481, 520363, 1889, 524701, 7853, 527603, 31121, 530513, 4999, 536357, 3761, 31723, 41597, 1609, 543707, 1753, 546661, 49831, 549623, 551107, 2963, 4241, 7867, 561551, 563051, 3019, 566057, 569071, 51871, 572093, 5569, 575123, 44357, 3547, 52837, 584261, 585791, 2239, 34729, 3067, 4673, 1901, 4943, 601201, 602753, 607421, 610543, 612107, 613673, 55931, 2791, 56503, 3163, 7019, 3989, 9371, 2011, 15391, 3187, 49031, 5281, 49277, 4253, 37871, 648631, 59113, 651857, 653473, 38821, 50891, 9341, 9923, 2143, 60737, 9433, 2297, 61333, 11087, 677953, 681251, 684557, 62383, 2683, 691193, 62987, 694523, 17021, 41149, 53939, 6569, 3167, 707923, 709607, 64663, 712981, 17431, 42139, 16699, 719753, 721451, 724853, 55889, 42839, 66361, 731681, 733393, 735107, 736823, 7187, 67453, 6823, 3373, 747161, 5237, 5479, 3229, 754093, 4637, 44563, 759313, 7129, 764551, 766301, 769807, 59351, 8689, 7127, 11621, 782141, 19163, 4211, 61121, 10909, 798143, 803513, 4519, 9109, 812503, 62639, 816113, 3701, 74521, 823357, 75181, 828811, 1973, 832457, 834283, 76343, 843443, 2207, 7001, 49939, 850807, 5231, 4721, 856351, 12809, 1907, 66301, 863771, 66587, 4639, 11909, 20261, 873091, 5573, 876833, 2999, 882461, 884341, 68171, 80737, 14621, 8353, 14683, 897557, 903251, 6007, 82633, 70067, 53693, 6997, 920441, 924283, 4799, 9011, 84551, 933923, 72287, 941671, 55621, 949451, 86491, 22171, 955307, 957263, 963143, 4909, 969041, 974957, 976933, 8999, 4969, 4261, 90437, 996803, 3613, 23321, 2531, 7349, 91711, 4583,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+17x-127 and
the reducible primes which appear as divisor for the first time
p | x^2+17x-127 and p < x^2+17x-127

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)
1109900.9000000.9000000.0000000.0000000.0000000.000000
21008050300.8000000.5000000.3000008.8888895.555555inf
31.0007513284230.7510000.3280000.4230009.3875006.56000014.100000
410.0007.3612.3634.9980.7361000.2363000.4998009.8015987.20426811.815603
5100.00072.99118.14954.8420.7299100.1814900.5484209.9159087.68049110.972789
61.000.000724.093147.500576.5930.7240930.1475000.5765939.9203058.12717010.513712
710.000.0007.194.1321.249.8205.944.3120.7194130.1249820.5944319.9353708.47335610.309373
8100.000.00071.584.59010.829.03760.755.5530.7158460.1082900.6075569.9504148.66447710.220788
91.000.000.000713.206.57595.546.552617.660.0230.7132070.0955470.6176609.9631308.82318110.166314
1010.000.000.0007.111.030.067855.053.9416.255.976.1260.7111030.0855050.6255989.9705068.94908210.128510


ABCDEFGHIJK
exponent =log2 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
123301.5000001.5000000.0000000.0000000.0000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
388801.0000001.0000000.0000001.6000001.600000-nan
416141400.8750000.8750000.0000001.7500001.750000-nan
532262240.8125000.6875000.1250001.8571431.571429inf
6645337160.8281250.5781250.2500002.0384611.6818184.000000
712810365380.8046880.5078120.2968751.9433961.7567572.375000
8256197111860.7695310.4335940.3359381.9126211.7076922.263158
95123901941960.7617190.3789060.3828121.9796951.7477482.279070
101.0247693364330.7509770.3281250.4228521.9717951.7319592.209184
112.0481.5305979330.7470700.2915040.4555661.9895971.7767862.154734
124.0963.0351.0841.9510.7409670.2646480.4763181.9836601.8157452.091104
138.1926.0271.9844.0430.7357180.2421880.4935301.9858321.8302582.072271
1416.38412.0413.6358.4060.7349240.2218630.5130621.9978431.8321572.079149
1532.76823.9886.67417.3140.7320560.2036740.5283811.9921931.8360382.059719
1665.53647.92912.40335.5260.7313390.1892550.5420841.9980411.8584062.051866
17131.07295.56923.19472.3750.7291340.1769560.5521771.9939701.8700312.037240
18262.144190.69043.296147.3940.7274250.1651610.5622631.9953121.8666902.036532
19524.288380.43781.534298.9030.7256260.1555140.5701121.9950551.8831762.027918
201.048.576759.053154.116604.9370.7238890.1469760.5769131.9952131.8902052.023857
212.097.1521.514.782292.9481.221.8340.7223040.1396880.5826161.9956211.9008282.019771
224.194.3043.024.095556.6982.467.3970.7210000.1327270.5882731.9963901.9003302.019421
238.388.6086.037.2981.060.7504.976.5480.7197020.1264510.5932511.9963981.9054322.016922
2416.777.21612.054.0102.025.88310.028.1270.7184750.1207520.5977231.9965901.9098592.015077
2533.554.43224.072.0303.878.70520.193.3250.7174020.1155940.6018081.9970141.9145752.013669
2667.108.86448.075.5317.437.68040.637.8510.7163810.1108300.6055511.9971531.9175682.012440
27134.217.72896.032.17814.291.13081.741.0480.7154950.1064770.6090181.9975271.9214502.011451
28268.435.456191.835.68127.495.171164.340.5100.7146440.1024270.6122161.9976191.9239332.010502
29536.870.912383.252.62752.981.383330.271.2440.7138640.0986860.6151781.9978171.9269342.009676
301.073.741.824765.722.737102.221.257663.501.4800.7131350.0952010.6179341.9979581.9293812.008960
312.147.483.6481.529.986.353197.487.3411.332.499.0120.7124550.0919620.6204931.9980941.9319602.008283
324.294.967.2963.057.258.241381.976.7092.675.281.5320.7118230.0889360.6228871.9982261.9341832.007717
338.589.934.5926.109.406.893739.610.6415.369.796.2520.7112290.0861020.6251271.9983291.9362722.007189
3417.179.869.18412.209.236.1481.433.572.47210.775.663.6760.7106710.0834450.6272261.9984321.9382802.006717


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
123211011
245411211
388534211
41614776422
532229138554
664371423108109
712865263918141518
8256111377430262827
95121946812649445051
101.02433611721981838884
112.048597203394145147155150
124.0961.084372712258280279267
138.1921.9846771.307504501497482
1416.3843.6351.2082.427916913915891
1532.7686.6742.2394.4351.6731.6571.6841.660
1665.53612.4034.1338.2703.1053.0893.1063.103
17131.07223.1947.73115.4635.8015.7865.7875.820
18262.14443.29614.42228.87410.76310.85710.82910.847
19524.28881.53427.17254.36220.27920.46220.40520.388
201.048.576154.11651.268102.84838.43438.52838.62138.533
212.097.152292.94897.477195.47173.01373.35273.30373.280
224.194.304556.698185.390371.308138.853139.286139.364139.195
238.388.6081.060.750353.291707.459264.839265.383265.303265.225
2416.777.2162.025.883674.4551.351.428506.080506.647506.561506.595
2533.554.4323.878.7051.292.5352.586.170969.216969.826969.323970.340
2667.108.8647.437.6802.478.7054.958.9751.858.8741.859.1951.859.1681.860.443
27134.217.72814.291.1304.762.6249.528.5063.571.7193.573.3353.572.2333.573.843
28268.435.45627.495.1719.164.86318.330.3086.871.7726.875.1956.873.1776.875.027
29536.870.91252.981.38317.660.26635.321.11713.242.70813.246.55313.244.45813.247.664
301.073.741.824102.221.25734.074.73668.146.52125.555.48525.558.07525.549.53325.558.164
312.147.483.648197.487.34165.822.630131.664.71149.376.11849.368.23049.368.89349.374.100
324.294.967.296381.976.709127.320.385254.656.32495.492.93095.487.67595.501.45295.494.652
338.589.934.592739.610.641246.535.091493.075.550184.889.961184.896.672184.910.213184.913.795
3417.179.869.1841.433.572.472477.860.384955.712.088358.379.567358.386.732358.402.772358.403.401


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
240000000
380000000
4160000000
5324220112
66416884246
7128382117810119
825686473919192622
9512196989842426052
101.02443321821592117118106
112.048933466467215235238245
124.0961.951964987468489487507
138.1924.0432.0292.0141.0171.0091.020997
1416.3848.4064.2594.1472.1202.1182.0862.082
1532.76817.3148.8048.5104.3704.3914.2634.290
1665.53635.52618.07817.4488.8808.9258.8348.887
17131.07272.37536.72135.65418.06118.08018.06518.169
18262.144147.39474.76172.63336.88336.78136.83836.892
19524.288298.903150.895148.00874.47174.56074.87974.993
201.048.576604.937305.409299.528150.683151.343151.506151.405
212.097.1521.221.834616.577605.257304.378305.588305.777306.091
224.194.3042.467.3971.244.5271.222.870615.405616.892616.909618.191
238.388.6084.976.5482.509.1012.467.4471.242.8051.243.7491.244.4361.245.558
2416.777.21610.028.1275.054.7094.973.4182.504.2882.507.8422.508.0482.507.949
2533.554.43220.193.32510.175.58210.017.7435.046.3965.049.7315.047.8735.049.325
2667.108.86440.637.85120.470.95220.166.89910.155.88210.161.14010.157.44910.163.380
27134.217.72881.741.04841.159.44340.581.60520.432.77320.435.25520.430.73620.442.284
28268.435.456164.340.51082.722.71081.617.80041.080.11541.088.23741.079.16341.092.995
29536.870.912330.271.244166.200.969164.070.27582.565.69382.570.24982.559.65682.575.646
301.073.741.824663.501.480333.800.845329.700.635165.870.360165.875.060165.858.550165.897.510
312.147.483.6481.332.499.012670.213.129662.285.883333.121.667333.113.060333.101.018333.163.267
324.294.967.2962.675.281.5321.345.251.9271.330.029.605668.813.904668.814.817668.793.369668.859.442
338.589.934.5925.369.796.2522.699.574.2862.670.221.9661.342.445.1031.342.446.5191.342.432.1551.342.472.475
3417.179.869.18410.775.663.6765.416.255.5815.359.408.0952.693.952.2982.693.925.2272.693.881.8972.693.904.254


8. Check for existing Integer Sequences by OEIS

Found in Database : 127, 109, 89, 67, 43, 17, 11, 41, 73, 107, 13, 181, 1, 263, 307, 353, 401, 1, 503, 557,
Found in Database : 127, 109, 89, 67, 43, 17, 11, 41, 73, 107, 13, 181, 263, 307, 353, 401, 503, 557, 613, 61, 857, 71, 991, 1061, 103, 1283, 1361, 131, 1523, 1607, 1693, 137, 1871, 151,
Found in Database : 11, 13, 17, 41, 43, 61, 67, 71, 73, 89, 103, 107, 109, 127, 131, 137,