Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:45:58
Deutsch
28.Mar 2024

Polynom = x^2-47x+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) = 41 = 41
f(2) = 85 = 5*17
f(3) = 127 = 127
f(4) = 167 = 167
f(5) = 205 = 5*41
f(6) = 241 = 241
f(7) = 275 = 5*5*11
f(8) = 307 = 307
f(9) = 337 = 337
f(10) = 365 = 5*73
f(11) = 391 = 17*23
f(12) = 415 = 5*83
f(13) = 437 = 19*23
f(14) = 457 = 457
f(15) = 475 = 5*5*19
f(16) = 491 = 491
f(17) = 505 = 5*101
f(18) = 517 = 11*47
f(19) = 527 = 17*31
f(20) = 535 = 5*107
f(21) = 541 = 541
f(22) = 545 = 5*109
f(23) = 547 = 547
f(24) = 547 = 547
f(25) = 545 = 5*109
f(26) = 541 = 541
f(27) = 535 = 5*107
f(28) = 527 = 17*31
f(29) = 517 = 11*47
f(30) = 505 = 5*101
f(31) = 491 = 491
f(32) = 475 = 5*5*19
f(33) = 457 = 457
f(34) = 437 = 19*23
f(35) = 415 = 5*83
f(36) = 391 = 17*23
f(37) = 365 = 5*73
f(38) = 337 = 337
f(39) = 307 = 307
f(40) = 275 = 5*5*11
f(41) = 241 = 241
f(42) = 205 = 5*41
f(43) = 167 = 167
f(44) = 127 = 127
f(45) = 85 = 5*17
f(46) = 41 = 41
f(47) = 5 = 5
f(48) = 53 = 53
f(49) = 103 = 103
f(50) = 155 = 5*31
f(51) = 209 = 11*19
f(52) = 265 = 5*53
f(53) = 323 = 17*19
f(54) = 383 = 383
f(55) = 445 = 5*89
f(56) = 509 = 509
f(57) = 575 = 5*5*23
f(58) = 643 = 643
f(59) = 713 = 23*31
f(60) = 785 = 5*157
f(61) = 859 = 859
f(62) = 935 = 5*11*17
f(63) = 1013 = 1013
f(64) = 1093 = 1093
f(65) = 1175 = 5*5*47
f(66) = 1259 = 1259
f(67) = 1345 = 5*269
f(68) = 1433 = 1433
f(69) = 1523 = 1523
f(70) = 1615 = 5*17*19
f(71) = 1709 = 1709
f(72) = 1805 = 5*19*19
f(73) = 1903 = 11*173
f(74) = 2003 = 2003
f(75) = 2105 = 5*421
f(76) = 2209 = 47*47
f(77) = 2315 = 5*463
f(78) = 2423 = 2423
f(79) = 2533 = 17*149
f(80) = 2645 = 5*23*23
f(81) = 2759 = 31*89
f(82) = 2875 = 5*5*5*23
f(83) = 2993 = 41*73
f(84) = 3113 = 11*283
f(85) = 3235 = 5*647
f(86) = 3359 = 3359
f(87) = 3485 = 5*17*41
f(88) = 3613 = 3613
f(89) = 3743 = 19*197
f(90) = 3875 = 5*5*5*31
f(91) = 4009 = 19*211
f(92) = 4145 = 5*829
f(93) = 4283 = 4283
f(94) = 4423 = 4423
f(95) = 4565 = 5*11*83
f(96) = 4709 = 17*277
f(97) = 4855 = 5*971
f(98) = 5003 = 5003
f(99) = 5153 = 5153
f(100) = 5305 = 5*1061

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^2-47x+5

f(0)=5
f(1)=41
f(2)=17
f(3)=127
f(4)=167
f(5)=1
f(6)=241
f(7)=11
f(8)=307
f(9)=337
f(10)=73
f(11)=23
f(12)=83
f(13)=19
f(14)=457
f(15)=1
f(16)=491
f(17)=101
f(18)=47
f(19)=31
f(20)=107
f(21)=541
f(22)=109
f(23)=547
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=53
f(49)=103
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=383
f(55)=89
f(56)=509
f(57)=1
f(58)=643
f(59)=1
f(60)=157
f(61)=859
f(62)=1
f(63)=1013
f(64)=1093
f(65)=1
f(66)=1259
f(67)=269
f(68)=1433
f(69)=1523
f(70)=1
f(71)=1709
f(72)=1
f(73)=173
f(74)=2003
f(75)=421
f(76)=1
f(77)=463
f(78)=2423
f(79)=149
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=283
f(85)=647
f(86)=3359
f(87)=1
f(88)=3613
f(89)=197
f(90)=1
f(91)=211
f(92)=829
f(93)=4283
f(94)=4423
f(95)=1
f(96)=277
f(97)=971
f(98)=5003
f(99)=5153

b) Substitution of the polynom
The polynom f(x)=x^2-47x+5 could be written as f(y)= y^2-547.25 with x=y+23.5

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-23.5
f'(x)>2x-48 with x > 23

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, 41, 17, 127, 167, 1, 241, 11, 307, 337, 73, 23, 83, 19, 457, 1, 491, 101, 47, 31, 107, 541, 109, 547, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 53, 103, 1, 1, 1, 1, 383, 89, 509, 1, 643, 1, 157, 859, 1, 1013, 1093, 1, 1259, 269, 1433, 1523, 1, 1709, 1, 173, 2003, 421, 1, 463, 2423, 149, 1, 1, 1, 1, 283, 647, 3359, 1, 3613, 197, 1, 211, 829, 4283, 4423, 1, 277, 971, 5003, 5153, 1061, 1, 1123, 251, 349, 1, 569, 257, 347, 6763, 1, 7109, 1, 439, 7643, 313, 8009, 1, 1, 8573, 1753, 1, 1831, 199, 233, 1951, 433, 1, 1, 557, 1, 1, 449, 11443, 1, 2377, 12109, 2467, 739, 1163, 521, 13259, 2699, 443, 1, 2843, 761, 1, 787, 661, 281, 683, 1, 16223, 311, 1, 1, 691, 331, 379, 3617, 1669, 3727, 18913, 1129, 1, 19759, 1, 20333, 503, 1, 1, 1, 21803, 1, 4481, 22709, 4603, 1, 23633, 4789, 1427, 983, 1, 1327, 5107, 1361, 5237, 26513, 1579, 1087, 27509, 5569, 28183, 2593, 1, 29209, 1, 1759, 30253, 6121, 373, 6263, 1667, 1, 1, 1, 1, 33493, 33863, 1, 653, 6997, 35363, 1153, 1, 3319, 1, 1621, 1, 1, 38459, 409, 2309, 2087, 8011, 40459, 743, 1, 571, 8419, 42509, 1, 487, 1, 8837, 1439, 9007, 4133, 45893, 1, 1, 9439, 1, 48073, 1, 1, 1, 49853, 1, 10151, 1249, 10333, 1109, 52583, 1, 733, 1, 54443, 617, 1, 673, 593, 56813, 1, 2311, 1, 1, 59233, 59723, 12043, 5519, 12241, 61703, 3659, 12541, 2039, 12743, 64223, 3407, 13049, 3461, 1, 3929, 811, 13567, 1, 599, 1693, 3041, 2819, 4177, 1, 6553, 72623, 14633, 73709, 14851, 1, 75353, 1, 1, 1, 77573, 7103, 15739, 79259, 1, 4729, 80963, 709, 1747, 719, 1571, 83843, 1, 85009, 1, 86183, 4567, 1, 87959, 1, 89153, 89753, 1063, 8269, 18313, 92173, 1, 18679, 94009, 757, 1, 5639, 839, 1, 1777, 1, 98993, 797, 937, 1187, 101533, 1231, 20563, 1, 1, 1, 967, 21211, 6277, 1, 108023, 1, 1151, 4783, 1, 1, 1, 727, 113359, 22807, 114713, 1, 4643, 2203, 1, 6949, 3833, 1, 120209, 24181, 1, 1, 1, 1, 1, 1, 5471, 25309, 1, 5119, 128693, 1019, 1531, 130859, 26317, 132313, 1, 5351, 4339, 2459, 1, 2909, 1447, 138209, 27791, 139703, 140453, 911, 141959, 1, 13043, 6271, 1, 145759, 5861, 147293, 148063, 1, 1, 1583, 151163, 1, 1, 153509, 30859, 155083, 1, 31333, 1559, 1021, 3001, 159853, 1, 161459, 1, 1, 163883, 32939, 1, 6653, 1, 1663, 33757, 907, 1, 171263, 172093, 6917, 3697, 34919, 175433, 10369, 35423, 1, 3251, 1, 180503, 1, 182209, 1, 1, 1, 1, 4549, 1499, 1, 1, 37997, 1, 1237, 192613, 193493, 1, 195259, 39229, 197033, 947, 2339, 1, 1, 8761, 202403, 1, 3853, 41023, 12119, 206933, 3779, 208759, 8387, 1069, 6823, 42487, 213359, 2521, 1, 1, 1, 19819, 43789, 1, 9601, 2609, 1, 1091, 224603, 4799, 1, 227459, 4153, 229373, 1, 1, 232259, 1, 234193, 12377, 1, 1867, 2801, 1, 240043, 1, 242009, 2113, 1, 10651, 49193, 1, 1, 1033, 1, 1, 1, 50593, 13367, 1, 51199, 1637, 10321, 3121, 6343, 1, 262109, 52627, 1, 265193, 1, 267259, 2333, 269333, 1619, 2857, 1, 2879, 3761, 1399, 55331, 1, 1, 279823, 280883, 1, 283009, 1, 1, 1, 57457, 288359, 1, 1, 1, 1, 15461, 1, 26903, 297023, 1, 1, 3533, 301403, 1, 1481, 304709, 1973, 5791, 1, 3637, 1, 1, 16447, 313613, 3313, 1, 63397, 13831, 1, 1, 321509, 1, 1, 4451, 65213, 327209, 3863, 329503, 330653, 66361, 30269, 3517, 335273, 17707, 1, 19927, 13597, 11003, 1, 68687, 14983, 6287, 346963, 20479, 1, 1, 2269, 352883, 354073, 1733, 1, 1, 1, 360053, 72251, 362459, 72733, 364873, 7789, 73459, 1, 1, 1, 1471, 1409, 3709, 75167, 2969, 12203, 1, 1, 4021, 1, 384473, 7013, 1, 77641, 1, 390703, 1, 393209, 78893, 1289, 396983, 3463, 36319, 1, 402043, 21227, 80917, 1, 1, 408413, 409693, 1, 2383, 1, 414833, 3889, 2693, 418709, 1, 421303, 24859, 1, 1, 85303, 1, 1, 1, 431759, 1, 1, 8221, 87407, 1, 1871, 1, 40213, 17747, 26177, 89269, 447683, 6151, 90073, 451709, 1, 11083, 1, 8311, 1, 2243, 20051, 5197, 92779, 465259, 3733, 27529, 469363, 3037, 1, 1, 3187, 476243, 3821, 1483, 96079, 25357, 4691, 1, 485959, 8861, 10399, 1, 5783, 21433, 98873, 495773, 2063, 99719, 500009, 1, 2689, 10729, 5323, 507109, 1, 509963, 12473, 1, 1, 6067, 12613, 47143, 104003, 2647, 4547, 524353, 22861, 6203, 528709, 106033, 531623, 28057, 9719, 28211, 21499, 538943, 1, 108377, 543359, 108967, 17623, 2273, 1, 50069, 1, 553733, 5189, 1, 558209, 1, 29537, 562703, 5939, 1, 10313, 568723, 570233, 2789, 12197, 1, 576293, 1, 1, 580859, 1, 1, 585443, 1, 588509, 6211, 1, 1, 5171, 596209, 119551, 599303, 54623, 1, 35527, 1, 19583, 608633, 122039, 4817, 24533, 614893, 616463, 1, 32611, 1, 1, 1, 25037, 27283, 2677, 1613, 632273, 1, 1, 1, 2741, 4297, 1, 2069, 7589, 13759, 648283, 6841, 651509, 1, 654743, 1, 7741, 659609, 132247, 6563, 1, 1, 29033, 1, 1, 39569, 1, 675959, 2557, 1, 35837, 136511, 36011, 8069, 687523, 62653, 1, 1, 27767, 695843, 697513, 139837, 41227, 1, 704213, 1, 1, 17299, 142189, 37507, 42019, 7537, 2293, 143881, 7001, 2687, 3083, 1, 1, 1, 731333, 146609, 734759, 1, 6899, 1, 6449, 1, 1, 746813, 39397, 30011, 752009, 1, 755483, 6947, 8929, 1, 1, 69473, 765953, 2897, 769459, 154243, 1, 774733, 155299, 40961, 1, 1789, 71233, 6829, 16747, 9281, 10831, 792443, 31769, 796009, 1, 25793, 4027, 1, 1, 3433, 808553, 810353, 1, 8059, 1, 1, 1, 164239, 3253, 32993, 1, 1, 1, 832109, 9811, 4831, 8293, 33577, 841259, 15329, 4289, 1, 1907, 2633, 1, 1, 1, 171571, 859709, 172313, 1, 2213, 173429, 37783, 6967, 1, 11981, 1, 878359, 176047, 2731, 80363, 1, 887759, 177929, 3469, 19009, 1, 1, 179821, 6047, 902903, 16451, 8803, 7901, 910523, 39671, 1, 916259, 1933, 8599, 48527, 4507, 1, 185557, 3217, 1, 1, 2131, 1, 3343, 941323, 188653, 945209, 1, 1, 951053,

6. Sequence of the polynom (only primes)

5, 41, 17, 127, 167, 241, 11, 307, 337, 73, 23, 83, 19, 457, 491, 101, 47, 31, 107, 541, 109, 547, 53, 103, 383, 89, 509, 643, 157, 859, 1013, 1093, 1259, 269, 1433, 1523, 1709, 173, 2003, 421, 463, 2423, 149, 283, 647, 3359, 3613, 197, 211, 829, 4283, 4423, 277, 971, 5003, 5153, 1061, 1123, 251, 349, 569, 257, 347, 6763, 7109, 439, 7643, 313, 8009, 8573, 1753, 1831, 199, 233, 1951, 433, 557, 449, 11443, 2377, 12109, 2467, 739, 1163, 521, 13259, 2699, 443, 2843, 761, 787, 661, 281, 683, 16223, 311, 691, 331, 379, 3617, 1669, 3727, 18913, 1129, 19759, 20333, 503, 21803, 4481, 22709, 4603, 23633, 4789, 1427, 983, 1327, 5107, 1361, 5237, 26513, 1579, 1087, 27509, 5569, 28183, 2593, 29209, 1759, 30253, 6121, 373, 6263, 1667, 33493, 33863, 653, 6997, 35363, 1153, 3319, 1621, 38459, 409, 2309, 2087, 8011, 40459, 743, 571, 8419, 42509, 487, 8837, 1439, 9007, 4133, 45893, 9439, 48073, 49853, 10151, 1249, 10333, 1109, 52583, 733, 54443, 617, 673, 593, 56813, 2311, 59233, 59723, 12043, 5519, 12241, 61703, 3659, 12541, 2039, 12743, 64223, 3407, 13049, 3461, 3929, 811, 13567, 599, 1693, 3041, 2819, 4177, 6553, 72623, 14633, 73709, 14851, 75353, 77573, 7103, 15739, 79259, 4729, 80963, 709, 1747, 719, 1571, 83843, 85009, 86183, 4567, 87959, 89153, 89753, 1063, 8269, 18313, 92173, 18679, 94009, 757, 5639, 839, 1777, 98993, 797, 937, 1187, 101533, 1231, 20563, 967, 21211, 6277, 108023, 1151, 4783, 727, 113359, 22807, 114713, 4643, 2203, 6949, 3833, 120209, 24181, 5471, 25309, 5119, 128693, 1019, 1531, 130859, 26317, 132313, 5351, 4339, 2459, 2909, 1447, 138209, 27791, 139703, 140453, 911, 141959, 13043, 6271, 145759, 5861, 147293, 148063, 1583, 151163, 153509, 30859, 155083, 31333, 1559, 1021, 3001, 159853, 161459, 163883, 32939, 6653, 1663, 33757, 907, 171263, 172093, 6917, 3697, 34919, 175433, 10369, 35423, 3251, 180503, 182209, 4549, 1499, 37997, 1237, 192613, 193493, 195259, 39229, 197033, 947, 2339, 8761, 202403, 3853, 41023, 12119, 206933, 3779, 208759, 8387, 1069, 6823, 42487, 213359, 2521, 19819, 43789, 9601, 2609, 1091, 224603, 4799, 227459, 4153, 229373, 232259, 234193, 12377, 1867, 2801, 240043, 242009, 2113, 10651, 49193, 1033, 50593, 13367, 51199, 1637, 10321, 3121, 6343, 262109, 52627, 265193, 267259, 2333, 269333, 1619, 2857, 2879, 3761, 1399, 55331, 279823, 280883, 283009, 57457, 288359, 15461, 26903, 297023, 3533, 301403, 1481, 304709, 1973, 5791, 3637, 16447, 313613, 3313, 63397, 13831, 321509, 4451, 65213, 327209, 3863, 329503, 330653, 66361, 30269, 3517, 335273, 17707, 19927, 13597, 11003, 68687, 14983, 6287, 346963, 20479, 2269, 352883, 354073, 1733, 360053, 72251, 362459, 72733, 364873, 7789, 73459, 1471, 1409, 3709, 75167, 2969, 12203, 4021, 384473, 7013, 77641, 390703, 393209, 78893, 1289, 396983, 3463, 36319, 402043, 21227, 80917, 408413, 409693, 2383, 414833, 3889, 2693, 418709, 421303, 24859, 85303, 431759, 8221, 87407, 1871, 40213, 17747, 26177, 89269, 447683, 6151, 90073, 451709, 11083, 8311, 2243, 20051, 5197, 92779, 465259, 3733, 27529, 469363, 3037, 3187, 476243, 3821, 1483, 96079, 25357, 4691, 485959, 8861, 10399, 5783, 21433, 98873, 495773, 2063, 99719, 500009, 2689, 10729, 5323, 507109, 509963, 12473, 6067, 12613, 47143, 104003, 2647, 4547, 524353, 22861, 6203, 528709, 106033, 531623, 28057, 9719, 28211, 21499, 538943, 108377, 543359, 108967, 17623, 2273, 50069, 553733, 5189, 558209, 29537, 562703, 5939, 10313, 568723, 570233, 2789, 12197, 576293, 580859, 585443, 588509, 6211, 5171, 596209, 119551, 599303, 54623, 35527, 19583, 608633, 122039, 4817, 24533, 614893, 616463, 32611, 25037, 27283, 2677, 1613, 632273, 2741, 4297, 2069, 7589, 13759, 648283, 6841, 651509, 654743, 7741, 659609, 132247, 6563, 29033, 39569, 675959, 2557, 35837, 136511, 36011, 8069, 687523, 62653, 27767, 695843, 697513, 139837, 41227, 704213, 17299, 142189, 37507, 42019, 7537, 2293, 143881, 7001, 2687, 3083, 731333, 146609, 734759, 6899, 6449, 746813, 39397, 30011, 752009, 755483, 6947, 8929, 69473, 765953, 2897, 769459, 154243, 774733, 155299, 40961, 1789, 71233, 6829, 16747, 9281, 10831, 792443, 31769, 796009, 25793, 4027, 3433, 808553, 810353, 8059, 164239, 3253, 32993, 832109, 9811, 4831, 8293, 33577, 841259, 15329, 4289, 1907, 2633, 171571, 859709, 172313, 2213, 173429, 37783, 6967, 11981, 878359, 176047, 2731, 80363, 887759, 177929, 3469, 19009, 179821, 6047, 902903, 16451, 8803, 7901, 910523, 39671, 916259, 1933, 8599, 48527, 4507, 185557, 3217, 2131, 3343, 941323, 188653, 945209, 951053,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-47x+5 and
the reducible primes which appear as divisor for the first time
p | x^2-47x+5 and p < x^2-47x+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)
11010821.0000000.8000001.0000000.0000000.0000000.000000
21005231210.5200000.3100000.5200005.2000003.87500010.500000
31.0006652204450.6650000.2200000.66500012.7884627.09677421.190475
410.0006.9531.5125.4410.6953000.1512000.69530010.4556396.87272712.226966
5100.00069.96511.70958.2560.6996500.1170900.69965010.0625637.74404810.706856
61.000.000699.00294.596604.4060.6990020.0945960.6990029.9907388.07891410.375000
710.000.0006.979.222799.3286.179.8940.6979220.0799330.6979229.9845528.44991310.224740
8100.000.00069.729.4216.925.57162.803.8500.6972940.0692560.6972949.9910028.66424210.162609
91.000.000.000696.730.11361.130.951635.599.1620.6967300.0611310.6967309.9919108.82684610.120386
1010.000.000.0006.962.954.360547.064.8006.415.889.5600.6962950.0547060.6962959.9937618.94906410.094238


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
245411.2500001.0000000.2500001.6666672.0000001.000000
388711.0000000.8750000.1250001.6000001.7500001.000000
416141040.8750000.6250000.2500001.7500001.4285714.000000
532201280.6250000.3750000.2500001.4285711.2000002.000000
664281990.4375000.2968750.1406251.4000001.5833331.125000
71287037330.5468750.2890620.2578122.5000001.9473683.666667
825615762950.6132810.2421880.3710942.2428571.6756762.878788
95123351222130.6542970.2382810.4160162.1337581.9677422.242105
101.0246772234540.6611330.2177730.4433592.0208951.8278692.131455
112.0481.3963861.0100.6816410.1884770.4931642.0620381.7309422.224670
124.0962.8147032.1110.6870120.1716310.5153812.0157591.8212442.090099
138.1925.6861.2714.4150.6940920.1551510.5389402.0206111.8079662.091426
1416.38411.4292.3169.1130.6975710.1413570.5562132.0100251.8221872.064100
1532.76822.9294.29918.6300.6997380.1311950.5685422.0062121.8562182.044332
1665.53645.8337.98737.8460.6993560.1218720.5774841.9989101.8578742.031455
17131.07291.71914.88776.8320.6997600.1135790.5861822.0011561.8639042.030122
18262.144183.39127.835155.5560.6995810.1061820.5933991.9994881.8697522.024625
19524.288366.73052.098314.6320.6994820.0993690.6001131.9997161.8716722.022629
201.048.576732.91198.807634.1040.6989580.0942300.6047291.9985031.8965602.015383
212.097.1521.465.311187.0111.278.3000.6987150.0891740.6095411.9993031.8926902.015915
224.194.3042.929.018355.6072.573.4110.6983320.0847830.6135491.9989051.9015302.013151
238.388.6085.855.713678.2265.177.4870.6980550.0808510.6172051.9992071.9072352.011916
2416.777.21611.707.8091.295.45410.412.3550.6978400.0772150.6206251.9993821.9100622.011083
2533.554.43223.406.6162.481.17920.925.4370.6975720.0739450.6236271.9992311.9152972.009674
2667.108.86446.802.2194.758.05642.044.1630.6974070.0709010.6265071.9995291.9176592.009237
27134.217.72893.577.5929.139.53884.438.0540.6972070.0680950.6291131.9994261.9208562.008318
28268.435.456187.106.13617.585.363169.520.7730.6970250.0655110.6315141.9994761.9240982.007635
29536.870.912374.129.12433.891.032340.238.0920.6968700.0631270.6337431.9995551.9272302.007058
301.073.741.824748.091.50565.399.541682.691.9640.6967150.0609080.6358061.9995541.9297002.006512
312.147.483.6481.495.877.407126.355.6041.369.521.8030.6965720.0588390.6377331.9995911.9320562.006061
324.294.967.2962.991.204.114244.389.5202.746.814.5940.6964440.0569010.6395431.9996321.9341412.005674
338.589.934.5925.981.360.777473.202.2185.508.158.5590.6963220.0550880.6412341.9996501.9362622.005289
3417.179.869.18411.960.825.508917.214.33911.043.611.1690.6962120.0533890.6428231.9996831.9383142.004955


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
122021010
244131012
387342212
41610554312
53212754422
664191184654
712837142381397
825662243813191812
9512122418128343327
101.0242237215154616048
112.04838613025698999693
124.096703245458169182179173
138.1921.271437834315317336303
1416.3842.3168011.515570579610557
1532.7684.2991.4552.8441.0891.0681.0971.045
1665.5367.9872.6745.3131.9791.9962.0451.967
17131.07214.8875.0069.8813.7313.6923.7693.695
18262.14427.8359.34618.4896.9846.8716.9906.990
19524.28852.09817.32834.77012.98012.93413.07213.112
201.048.57698.80732.88565.92224.61324.64624.81024.738
212.097.152187.01162.467124.54446.74646.71946.86746.679
224.194.304355.607118.567237.04088.96588.89088.96988.783
238.388.608678.226226.081452.145169.635169.324169.968169.299
2416.777.2161.295.454431.734863.720324.045323.831323.879323.699
2533.554.4322.481.179826.2901.654.889620.635619.999620.275620.270
2667.108.8644.758.0561.584.7193.173.3371.189.9061.189.0151.189.8431.189.292
27134.217.7289.139.5383.045.4466.094.0922.285.6332.284.5792.285.0912.284.235
28268.435.45617.585.3635.861.70611.723.6574.396.7284.396.9954.397.5274.394.113
29536.870.91233.891.03211.298.42222.592.6108.474.7368.474.5658.472.7998.468.932
301.073.741.82465.399.54121.799.66543.599.87616.352.86616.351.99716.349.22116.345.457
312.147.483.648126.355.60442.120.17084.235.43431.592.76531.586.52931.588.85031.587.460
324.294.967.296244.389.52081.457.514162.932.00661.105.50061.094.02361.091.31161.098.686
338.589.934.592473.202.218157.734.514315.467.704118.297.959118.303.695118.296.468118.304.096
3417.179.869.184917.214.339305.735.523611.478.816229.304.189229.320.633229.292.477229.297.040


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
121011000
241011000
381011000
4164132101
5328262222
6649362232
712833171678117
825695514420272523
95122131169745575160
101.024454248206102117116119
112.0481.010540470248259253250
124.0962.1111.119992519532537523
138.1924.4152.3392.0761.0951.1291.1171.074
1416.3849.1134.8104.3032.2602.3192.2942.240
1532.76818.6309.7658.8654.6264.7084.6764.620
1665.53637.84619.75818.0889.3999.4559.4769.516
17131.07276.83239.91336.91919.24219.07219.28019.238
18262.144155.55680.50975.04738.79638.71139.12138.928
19524.288314.632162.578152.05478.80178.33278.99278.507
201.048.576634.104327.360306.744158.465158.601158.682158.356
212.097.1521.278.300658.033620.267319.052320.189319.706319.353
224.194.3042.573.4111.322.3411.251.070643.030643.868643.346643.167
238.388.6085.177.4872.657.3612.520.1261.294.4451.295.7101.293.7251.293.607
2416.777.21610.412.3555.337.5185.074.8372.602.1382.603.8922.602.8822.603.443
2533.554.43220.925.43710.713.92810.211.5095.228.9215.233.8825.231.0685.231.566
2667.108.86442.044.16321.503.54520.540.61810.503.88710.512.44610.511.93910.515.891
27134.217.72884.438.05443.137.72341.300.33121.102.26721.113.17421.107.82021.114.793
28268.435.456169.520.77386.529.86382.990.91042.376.61242.382.09542.373.77642.388.290
29536.870.912340.238.092173.513.658166.724.43485.057.38985.066.97085.049.78485.063.949
301.073.741.824682.691.964347.888.485334.803.479170.662.582170.674.100170.668.574170.686.708
312.147.483.6481.369.521.803697.392.819672.128.984342.369.674342.382.467342.383.949342.385.713
324.294.967.2962.746.814.5941.397.822.5451.348.992.049686.706.843686.688.919686.689.266686.729.566
338.589.934.5925.508.158.5592.801.318.7232.706.839.8361.377.066.6571.377.007.2401.377.010.6781.377.073.984
3417.179.869.18411.043.611.1695.613.317.4045.430.293.7652.760.953.7012.760.827.8162.760.867.4432.760.962.209


8. Check for existing Integer Sequences by OEIS

Found in Database : 5, 41, 17, 127, 167, 1, 241, 11, 307, 337, 73, 23, 83, 19, 457, 1, 491, 101, 47, 31,
Found in Database : 5, 41, 17, 127, 167, 241, 11, 307, 337, 73, 23, 83, 19, 457, 491, 101, 47, 31, 107, 541, 109, 547,
Found in Database : 5, 11, 17, 19, 23, 31, 41, 47, 53, 73, 83, 89, 101, 103, 107, 109, 127, 149,