Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:33:52
Deutsch
29.Mar 2024

Polynom = x^2-5

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) = 1 = 1
f(2) = 1 = 1
f(3) = 1 = 1
f(4) = 11 = 11
f(5) = 5 = 5
f(6) = 31 = 31
f(7) = 11 = 11
f(8) = 59 = 59
f(9) = 19 = 19
f(10) = 95 = 5*19
f(11) = 29 = 29
f(12) = 139 = 139
f(13) = 41 = 41
f(14) = 191 = 191
f(15) = 55 = 5*11
f(16) = 251 = 251
f(17) = 71 = 71
f(18) = 319 = 11*29
f(19) = 89 = 89
f(20) = 395 = 5*79
f(21) = 109 = 109
f(22) = 479 = 479
f(23) = 131 = 131
f(24) = 571 = 571
f(25) = 155 = 5*31
f(26) = 671 = 11*61
f(27) = 181 = 181
f(28) = 779 = 19*41
f(29) = 209 = 11*19
f(30) = 895 = 5*179
f(31) = 239 = 239
f(32) = 1019 = 1019
f(33) = 271 = 271
f(34) = 1151 = 1151
f(35) = 305 = 5*61
f(36) = 1291 = 1291
f(37) = 341 = 11*31
f(38) = 1439 = 1439
f(39) = 379 = 379
f(40) = 1595 = 5*11*29
f(41) = 419 = 419
f(42) = 1759 = 1759
f(43) = 461 = 461
f(44) = 1931 = 1931
f(45) = 505 = 5*101
f(46) = 2111 = 2111
f(47) = 551 = 19*29
f(48) = 2299 = 11*11*19
f(49) = 599 = 599
f(50) = 2495 = 5*499
f(51) = 649 = 11*59
f(52) = 2699 = 2699
f(53) = 701 = 701
f(54) = 2911 = 41*71
f(55) = 755 = 5*151
f(56) = 3131 = 31*101
f(57) = 811 = 811
f(58) = 3359 = 3359
f(59) = 869 = 11*79
f(60) = 3595 = 5*719
f(61) = 929 = 929
f(62) = 3839 = 11*349
f(63) = 991 = 991
f(64) = 4091 = 4091
f(65) = 1055 = 5*211
f(66) = 4351 = 19*229
f(67) = 1121 = 19*59
f(68) = 4619 = 31*149
f(69) = 1189 = 29*41
f(70) = 4895 = 5*11*89
f(71) = 1259 = 1259
f(72) = 5179 = 5179
f(73) = 1331 = 11*11*11
f(74) = 5471 = 5471
f(75) = 1405 = 5*281
f(76) = 5771 = 29*199
f(77) = 1481 = 1481
f(78) = 6079 = 6079
f(79) = 1559 = 1559
f(80) = 6395 = 5*1279
f(81) = 1639 = 11*149
f(82) = 6719 = 6719
f(83) = 1721 = 1721
f(84) = 7051 = 11*641
f(85) = 1805 = 5*19*19
f(86) = 7391 = 19*389
f(87) = 1891 = 31*61
f(88) = 7739 = 71*109
f(89) = 1979 = 1979
f(90) = 8095 = 5*1619
f(91) = 2069 = 2069
f(92) = 8459 = 11*769
f(93) = 2161 = 2161
f(94) = 8831 = 8831
f(95) = 2255 = 5*11*41
f(96) = 9211 = 61*151
f(97) = 2351 = 2351
f(98) = 9599 = 29*331
f(99) = 2449 = 31*79
f(100) = 9995 = 5*1999

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^2-5

f(0)=5
f(1)=1
f(2)=1
f(3)=1
f(4)=11
f(5)=1
f(6)=31
f(7)=1
f(8)=59
f(9)=19
f(10)=1
f(11)=29
f(12)=139
f(13)=41
f(14)=191
f(15)=1
f(16)=251
f(17)=71
f(18)=1
f(19)=89
f(20)=79
f(21)=109
f(22)=479
f(23)=131
f(24)=571
f(25)=1
f(26)=61
f(27)=181
f(28)=1
f(29)=1
f(30)=179
f(31)=239
f(32)=1019
f(33)=271
f(34)=1151
f(35)=1
f(36)=1291
f(37)=1
f(38)=1439
f(39)=379
f(40)=1
f(41)=419
f(42)=1759
f(43)=461
f(44)=1931
f(45)=101
f(46)=2111
f(47)=1
f(48)=1
f(49)=599
f(50)=499
f(51)=1
f(52)=2699
f(53)=701
f(54)=1
f(55)=151
f(56)=1
f(57)=811
f(58)=3359
f(59)=1
f(60)=719
f(61)=929
f(62)=349
f(63)=991
f(64)=4091
f(65)=211
f(66)=229
f(67)=1
f(68)=149
f(69)=1
f(70)=1
f(71)=1259
f(72)=5179
f(73)=1
f(74)=5471
f(75)=281
f(76)=199
f(77)=1481
f(78)=6079
f(79)=1559
f(80)=1279
f(81)=1
f(82)=6719
f(83)=1721
f(84)=641
f(85)=1
f(86)=389
f(87)=1
f(88)=1
f(89)=1979
f(90)=1619
f(91)=2069
f(92)=769
f(93)=2161
f(94)=8831
f(95)=1
f(96)=1
f(97)=2351
f(98)=331
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-5 could be written as f(y)= y^2-5 with x=y+0

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+0
f'(x)>2x-1 with x > 2

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, 1, 1, 1, 11, 1, 31, 1, 59, 19, 1, 29, 139, 41, 191, 1, 251, 71, 1, 89, 79, 109, 479, 131, 571, 1, 61, 181, 1, 1, 179, 239, 1019, 271, 1151, 1, 1291, 1, 1439, 379, 1, 419, 1759, 461, 1931, 101, 2111, 1, 1, 599, 499, 1, 2699, 701, 1, 151, 1, 811, 3359, 1, 719, 929, 349, 991, 4091, 211, 229, 1, 149, 1, 1, 1259, 5179, 1, 5471, 281, 199, 1481, 6079, 1559, 1279, 1, 6719, 1721, 641, 1, 389, 1, 1, 1979, 1619, 2069, 769, 2161, 8831, 1, 1, 2351, 331, 1, 1999, 2549, 10399, 241, 569, 1, 1021, 2861, 1, 2969, 1, 3079, 12539, 3191, 1181, 661, 13451, 311, 449, 3539, 2879, 3659, 14879, 1, 809, 1, 269, 1, 1489, 4159, 1, 4289, 17419, 4421, 619, 911, 1, 4691, 1, 439, 3919, 4969, 1061, 1, 20731, 1051, 1, 491, 359, 1, 409, 1, 23099, 5851, 1, 1201, 839, 1, 2269, 1, 5119, 1, 1381, 1, 26891, 1361, 27551, 6971, 28219, 1, 5779, 7309, 2689, 7481, 30271, 1531, 30971, 1, 401, 8009, 1, 431, 33119, 761, 33851, 1, 34591, 8741, 35339, 8929, 7219, 829, 1, 9311, 1, 1901, 541, 1, 39199, 521, 421, 10099, 3709, 10301, 41611, 1, 1, 10711, 1, 1, 8819, 1, 44939, 1031, 1579, 2311, 4241, 1, 1, 631, 9679, 1, 49279, 1, 4561, 2531, 51071, 1171, 881, 13109, 1, 13339, 53819, 1, 54751, 1, 55691, 739, 1, 1, 11519, 14519, 1889, 509, 1009, 3001, 5501, 1, 691, 1409, 1, 15749, 63499, 16001, 2081, 3251, 3449, 1, 659, 1, 1229, 17029, 68639, 17291, 69691, 3511, 1, 1, 6529, 18089, 1, 1669, 2551, 601, 1831, 1, 1, 19181, 77279, 1, 15679, 19739, 7229, 20021, 80651, 1, 919, 1, 82939, 20879, 1, 21169, 85259, 1951, 4549, 1, 1109, 22051, 88799, 22349, 1, 1, 91199, 1, 1, 4651, 859, 23561, 3271, 23869, 19219, 24179, 8849, 1289, 5189, 1, 3221, 25121, 101119, 25439, 20479, 25759, 1, 2371, 104971, 5281, 9661, 26731, 1, 27059, 751, 1, 5801, 1459, 10141, 1, 1429, 1, 1609, 28729, 1, 709, 116959, 29411, 1, 1, 2029, 971, 1, 30449, 24499, 1621, 6521, 31151, 125311, 6301, 1, 1, 128159, 1, 25919, 32579, 2221, 32941, 132491, 6661, 1, 3061, 1, 34039, 1, 1811, 1949, 34781, 139871, 1, 141371, 35531, 1, 1, 28879, 3299, 3559, 36671, 147451, 7411, 148991, 37441, 1, 1, 1601, 38219, 1, 38611, 155231, 1, 821, 1, 1049, 39799, 2909, 1, 161599, 3691, 163211, 1, 164831, 41411, 8761, 1, 33619, 1, 1, 42641, 15581, 1, 1321, 1499, 1, 43889, 35279, 1, 16189, 1091, 6199, 1, 2039, 2399, 1, 1, 36979, 46439, 186619, 4261, 188351, 9461, 1571, 47741, 4679, 48179, 1249, 48619, 195359, 1, 17921, 9901, 1, 1, 200699, 1, 40499, 50849, 204299, 1, 2609, 941, 207931, 1, 19069, 1699, 1, 53129, 3499, 53591, 1, 1, 1039, 54521, 219019, 4999, 44179, 1789, 222779, 55931, 1129, 1, 226571, 5171, 228479, 1399, 1, 57839, 8011, 58321, 12329, 1, 1, 1, 21649, 59779, 1549, 5479, 242059, 60761, 3089, 12251, 246011, 61751, 247999, 5659, 49999, 1, 1, 3329, 1, 1, 256031, 1, 1709, 1, 4729, 2251, 262139, 5981, 1, 1, 1471, 66821, 1069, 67339, 1319, 1, 14341, 1, 1, 13781, 276671, 69431, 1871, 69959, 56179, 70489, 2339, 1, 285151, 1301, 287291, 72091, 289439, 1231, 2011, 3851, 15461, 6701, 2129, 14851, 1, 1, 300299, 1, 1, 1, 9829, 1, 27901, 15401, 309131, 1, 311359, 1, 3301, 1, 10891, 79241, 1, 1451, 2939, 1, 1, 2791, 1, 81509, 327179, 1, 329471, 1, 30161, 83231, 1, 1, 3541, 84389, 5741, 2741, 5591, 1, 343391, 1, 345739, 86729, 6329, 3011, 350459, 87911, 352831, 1, 355211, 89101, 1, 4721, 71999, 8209, 8839, 90901, 4099, 18301, 367231, 92111, 369659, 8429, 74419, 93329, 1, 93941, 12161, 18911, 379451, 5009, 20101, 95789, 1, 1, 5449, 8821, 1, 19531, 12641, 3389, 9619, 98909, 79379, 9049, 3049, 1, 36541, 20161, 1, 1, 1, 102079, 81919, 1741, 1, 2521, 414731, 1, 417311, 104651, 1, 3631, 84499, 1, 5381, 1, 427711, 1, 1, 1, 432959, 1, 87119, 109229, 4339, 109891, 1, 22111, 443551, 10111, 1, 2729, 89779, 112559, 451579, 1, 23909, 1, 14741, 1, 1, 115259, 92479, 1, 465119, 1, 11411, 1, 1, 117991, 15269, 10789, 95219, 1511, 1, 1, 25349, 24151, 484411, 1, 2309, 122149, 1, 122849, 492799, 123551, 495611, 24851, 8171, 1, 45569, 125669, 2459, 11489, 26681, 6689, 17579, 25561, 8689, 128521, 515519, 1, 1, 129959, 47389, 130681, 524171, 1, 527071, 1861, 529979, 132859, 9689, 1, 28201, 12211, 538751, 27011, 18679, 1, 17569, 1, 109519, 12479, 5051, 4759, 50321, 27751, 3109, 139501, 2341, 140249, 1, 1, 1, 1, 1, 2591, 571531, 143261, 9419, 2441, 1, 144779, 580639, 1, 8221, 1, 1, 2411, 2099, 1, 1, 148609, 20551, 149381, 4951, 1, 1, 13721, 3169, 1, 1, 4919, 6871, 153271, 5639, 2801, 617791, 154841, 2971, 8191, 124819, 156419, 15299, 157211, 21739, 31601, 57601, 1, 8969, 1, 4129, 5531, 643199, 161201, 646411, 32401, 649631, 1, 34361, 1, 1, 164429, 21269, 1, 662591, 33211, 2141, 166871, 1, 2749, 1, 15319, 4861, 5839, 678971, 34031, 1, 8999, 11239, 15619, 4751, 5569, 62929, 4231, 11789, 1, 698891, 175141, 702239, 175979, 12829, 176819, 708959, 1, 712331, 1879, 1, 179351, 17539, 2281, 1, 1, 25031, 1801, 66301, 36551, 732731, 183611, 736159, 6361, 147919, 1, 2179, 186191, 1, 1, 2089, 187921, 1, 2659, 151379, 2131, 3821, 17321, 1, 38281, 69761, 3259, 4259, 1, 2539, 6691, 777919, 10259, 3739, 39161, 1, 17881, 27191, 1, 158419, 198469, 795659, 1, 6101, 1, 802811, 201151, 73309, 202049, 161999, 202949, 42821, 10729, 817211, 1, 1, 205661, 824459, 1, 5711, 207479, 831739, 208391, 835391, 1, 839051, 1, 2999, 1, 1, 11161, 44741, 212981, 27541, 1, 857471, 214831, 1, 1, 4219, 19699, 4799, 1, 872351, 43711, 1, 219491, 879839, 1, 1, 1, 80669, 222311, 1, 44651, 30859, 224201, 1, 225149, 1, 226099, 15361, 20641, 9011, 1, 22291, 228961, 917759, 12101, 1, 1, 6211, 231841, 84481, 1, 933151, 7541, 1, 1, 188179, 5749, 85889, 236681, 948671, 1, 1, 3361, 50341, 12611, 1, 240589, 4211, 21961, 968251, 1, 2851, 1, 3889, 244529, 1, 245519, 984059, 246511, 89821, 1, 992011, 1, 1, 249499,

6. Sequence of the polynom (only primes)

5, 11, 31, 59, 19, 29, 139, 41, 191, 251, 71, 89, 79, 109, 479, 131, 571, 61, 181, 179, 239, 1019, 271, 1151, 1291, 1439, 379, 419, 1759, 461, 1931, 101, 2111, 599, 499, 2699, 701, 151, 811, 3359, 719, 929, 349, 991, 4091, 211, 229, 149, 1259, 5179, 5471, 281, 199, 1481, 6079, 1559, 1279, 6719, 1721, 641, 389, 1979, 1619, 2069, 769, 2161, 8831, 2351, 331, 1999, 2549, 10399, 241, 569, 1021, 2861, 2969, 3079, 12539, 3191, 1181, 661, 13451, 311, 449, 3539, 2879, 3659, 14879, 809, 269, 1489, 4159, 4289, 17419, 4421, 619, 911, 4691, 439, 3919, 4969, 1061, 20731, 1051, 491, 359, 409, 23099, 5851, 1201, 839, 2269, 5119, 1381, 26891, 1361, 27551, 6971, 28219, 5779, 7309, 2689, 7481, 30271, 1531, 30971, 401, 8009, 431, 33119, 761, 33851, 34591, 8741, 35339, 8929, 7219, 829, 9311, 1901, 541, 39199, 521, 421, 10099, 3709, 10301, 41611, 10711, 8819, 44939, 1031, 1579, 2311, 4241, 631, 9679, 49279, 4561, 2531, 51071, 1171, 881, 13109, 13339, 53819, 54751, 55691, 739, 11519, 14519, 1889, 509, 1009, 3001, 5501, 691, 1409, 15749, 63499, 16001, 2081, 3251, 3449, 659, 1229, 17029, 68639, 17291, 69691, 3511, 6529, 18089, 1669, 2551, 601, 1831, 19181, 77279, 15679, 19739, 7229, 20021, 80651, 919, 82939, 20879, 21169, 85259, 1951, 4549, 1109, 22051, 88799, 22349, 91199, 4651, 859, 23561, 3271, 23869, 19219, 24179, 8849, 1289, 5189, 3221, 25121, 101119, 25439, 20479, 25759, 2371, 104971, 5281, 9661, 26731, 27059, 751, 5801, 1459, 10141, 1429, 1609, 28729, 709, 116959, 29411, 2029, 971, 30449, 24499, 1621, 6521, 31151, 125311, 6301, 128159, 25919, 32579, 2221, 32941, 132491, 6661, 3061, 34039, 1811, 1949, 34781, 139871, 141371, 35531, 28879, 3299, 3559, 36671, 147451, 7411, 148991, 37441, 1601, 38219, 38611, 155231, 821, 1049, 39799, 2909, 161599, 3691, 163211, 164831, 41411, 8761, 33619, 42641, 15581, 1321, 1499, 43889, 35279, 16189, 1091, 6199, 2039, 2399, 36979, 46439, 186619, 4261, 188351, 9461, 1571, 47741, 4679, 48179, 1249, 48619, 195359, 17921, 9901, 200699, 40499, 50849, 204299, 2609, 941, 207931, 19069, 1699, 53129, 3499, 53591, 1039, 54521, 219019, 4999, 44179, 1789, 222779, 55931, 1129, 226571, 5171, 228479, 1399, 57839, 8011, 58321, 12329, 21649, 59779, 1549, 5479, 242059, 60761, 3089, 12251, 246011, 61751, 247999, 5659, 49999, 3329, 256031, 1709, 4729, 2251, 262139, 5981, 1471, 66821, 1069, 67339, 1319, 14341, 13781, 276671, 69431, 1871, 69959, 56179, 70489, 2339, 285151, 1301, 287291, 72091, 289439, 1231, 2011, 3851, 15461, 6701, 2129, 14851, 300299, 9829, 27901, 15401, 309131, 311359, 3301, 10891, 79241, 1451, 2939, 2791, 81509, 327179, 329471, 30161, 83231, 3541, 84389, 5741, 2741, 5591, 343391, 345739, 86729, 6329, 3011, 350459, 87911, 352831, 355211, 89101, 4721, 71999, 8209, 8839, 90901, 4099, 18301, 367231, 92111, 369659, 8429, 74419, 93329, 93941, 12161, 18911, 379451, 5009, 20101, 95789, 5449, 8821, 19531, 12641, 3389, 9619, 98909, 79379, 9049, 3049, 36541, 20161, 102079, 81919, 1741, 2521, 414731, 417311, 104651, 3631, 84499, 5381, 427711, 432959, 87119, 109229, 4339, 109891, 22111, 443551, 10111, 2729, 89779, 112559, 451579, 23909, 14741, 115259, 92479, 465119, 11411, 117991, 15269, 10789, 95219, 1511, 25349, 24151, 484411, 2309, 122149, 122849, 492799, 123551, 495611, 24851, 8171, 45569, 125669, 2459, 11489, 26681, 6689, 17579, 25561, 8689, 128521, 515519, 129959, 47389, 130681, 524171, 527071, 1861, 529979, 132859, 9689, 28201, 12211, 538751, 27011, 18679, 17569, 109519, 12479, 5051, 4759, 50321, 27751, 3109, 139501, 2341, 140249, 2591, 571531, 143261, 9419, 2441, 144779, 580639, 8221, 2411, 2099, 148609, 20551, 149381, 4951, 13721, 3169, 4919, 6871, 153271, 5639, 2801, 617791, 154841, 2971, 8191, 124819, 156419, 15299, 157211, 21739, 31601, 57601, 8969, 4129, 5531, 643199, 161201, 646411, 32401, 649631, 34361, 164429, 21269, 662591, 33211, 2141, 166871, 2749, 15319, 4861, 5839, 678971, 34031, 8999, 11239, 15619, 4751, 5569, 62929, 4231, 11789, 698891, 175141, 702239, 175979, 12829, 176819, 708959, 712331, 1879, 179351, 17539, 2281, 25031, 1801, 66301, 36551, 732731, 183611, 736159, 6361, 147919, 2179, 186191, 2089, 187921, 2659, 151379, 2131, 3821, 17321, 38281, 69761, 3259, 4259, 2539, 6691, 777919, 10259, 3739, 39161, 17881, 27191, 158419, 198469, 795659, 6101, 802811, 201151, 73309, 202049, 161999, 202949, 42821, 10729, 817211, 205661, 824459, 5711, 207479, 831739, 208391, 835391, 839051, 2999, 11161, 44741, 212981, 27541, 857471, 214831, 4219, 19699, 4799, 872351, 43711, 219491, 879839, 80669, 222311, 44651, 30859, 224201, 225149, 226099, 15361, 20641, 9011, 22291, 228961, 917759, 12101, 6211, 231841, 84481, 933151, 7541, 188179, 5749, 85889, 236681, 948671, 3361, 50341, 12611, 240589, 4211, 21961, 968251, 2851, 3889, 244529, 245519, 984059, 246511, 89821, 992011, 249499,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x-5 and
the reducible primes which appear as divisor for the first time
p | x^2x-5 and p < x^2x-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)
1105500.5000000.5000000.5000000.0000000.0000000.000000
21007050200.7000000.5000000.70000014.00000010.000000inf
31.0007213283930.7210000.3280000.72100010.3000006.56000019.650000
410.0007.1732.2724.9010.7173000.2272000.7173009.9486836.92682912.470737
5100.00071.12617.71253.4140.7112600.1771200.7112609.9157957.79577410.898592
61.000.000707.827143.540564.2870.7078270.1435400.7078279.9517348.10411010.564403
710.000.0007.053.0021.207.6915.845.3110.7053000.1207690.7053009.9643028.41362010.358755
8100.000.00070.352.85310.431.31959.921.5340.7035290.1043130.7035299.9748818.63740710.251214
91.000.000.000702.187.79991.820.600610.367.1990.7021880.0918210.7021889.9809428.80239610.186107
1010.000.000.0007.011.511.538820.089.0196.191.422.5190.7011510.0820090.7011519.9852378.93142710.143767


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)
121100.5000000.5000000.0000000.0000000.0000000.000000
242200.5000000.5000000.0000002.0000002.000000-nan
384400.5000000.5000000.0000002.0000002.000000-nan
416101000.6250000.6250000.0000002.5000002.500000-nan
532221930.6875000.5937500.0937502.2000001.900000inf
664453780.7031250.5781250.1250002.0454551.9473682.666667
71289261310.7187500.4765620.2421882.0444451.6486493.875000
8256185103820.7226560.4023440.3203122.0108701.6885252.645161
95123701891810.7226560.3691410.3535162.0000001.8349512.207317
101.0247403394010.7226560.3310550.3916022.0000001.7936512.215470
112.0481.4676078600.7163090.2963870.4199221.9824321.7905602.144638
124.0962.9351.0651.8700.7165530.2600100.4565432.0006821.7545302.174419
138.1925.8721.9183.9540.7167970.2341310.4826662.0006811.8009392.114439
1416.38411.7083.5608.1480.7146000.2172850.4973141.9938691.8561002.060698
1532.76823.3756.51016.8650.7133480.1986690.5146791.9964981.8286522.069833
1665.53646.66612.09034.5760.7120670.1844790.5275881.9964061.8571432.050163
17131.07293.17522.56370.6120.7108690.1721420.5387271.9966361.8662532.042226
18262.144186.00142.310143.6910.7095380.1614000.5481381.9962541.8751942.034937
19524.288371.55579.465292.0900.7086850.1515670.5571171.9975971.8781612.032765
201.048.576742.020149.920592.1000.7076450.1429750.5646711.9970661.8866172.027115
212.097.1521.482.615283.6141.199.0010.7069660.1352380.5717281.9980801.8917692.024997
224.194.3042.961.929538.2052.423.7240.7061790.1283180.5778611.9977741.8976672.021453
238.388.6085.918.6891.025.1904.893.4990.7055630.1222120.5833511.9982551.9048322.019000
2416.777.21611.825.9831.956.5389.869.4450.7048840.1166190.5882651.9980751.9084642.016848
2533.554.43223.632.5963.742.32919.890.2670.7043060.1115300.5927761.9983621.9127302.015338
2667.108.86447.230.0097.168.68740.061.3220.7037820.1068220.5969601.9985111.9155682.014117
27134.217.72894.400.15713.762.39880.637.7590.7033360.1025380.6007981.9987331.9197932.012858
28268.435.456188.686.63726.458.304162.228.3330.7029130.0985650.6043481.9987961.9225072.011816
29536.870.912377.159.36350.943.386326.215.9770.7025140.0948890.6076251.9988661.9254212.010844
301.073.741.824753.930.43498.226.781655.703.6530.7021520.0914810.6106721.9989711.9281562.010029
312.147.483.6481.507.145.912189.637.9491.317.507.9630.7018190.0883070.6135121.9990521.9306142.009304
324.294.967.2963.012.937.517366.592.8492.646.344.6680.7015040.0853540.6161501.9991011.9331202.008599
338.589.934.5926.023.372.344709.449.0155.313.923.3290.7012130.0825910.6186221.9991691.9352512.008024
3417.179.869.18412.042.058.5961.374.424.26110.667.634.3350.7009400.0800020.6209381.9992221.9373122.007487
3534.359.738.36824.075.312.8902.665.326.17121.409.986.7190.7006840.0775710.6231131.9992691.9392312.007004
3668.719.476.73648.134.227.5435.173.440.60642.960.786.9370.7004450.0752830.6251621.9993191.9410162.006577


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
121010010
242020110
384130211
41610371522
532196132845
664371225315613
7128611744722923
8256103356813401535
95121896312627752364
101.0243391122274612247124
112.0486071994088123082214
124.0961.065361704137406141381
138.1921.9186471.271242726262688
1416.3843.5601.2002.3604561.3284701.306
1532.7686.5102.1564.3548302.4338662.381
1665.53612.0903.9848.1061.5534.5131.5984.426
17131.07222.5637.45915.1042.8778.4242.9748.288
18262.14442.31014.02928.2815.42615.7925.48415.608
19524.28879.46526.42853.03710.13629.52710.30429.498
201.048.576149.92050.02299.89819.14155.61119.50055.668
212.097.152283.61494.620188.99436.383105.36836.674105.189
224.194.304538.205179.852358.35368.835200.09069.448199.832
238.388.6081.025.190341.834683.356130.979381.222131.539381.450
2416.777.2161.956.538652.5221.304.016249.797728.217250.439728.085
2533.554.4323.742.3291.248.5312.493.798477.3751.393.606478.0671.393.281
2667.108.8647.168.6872.391.1974.777.490914.3412.669.023915.2242.670.099
27134.217.72813.762.3984.589.9409.172.4581.754.7265.125.2041.755.5785.126.890
28268.435.45626.458.3048.823.56217.634.7423.370.9779.855.6313.373.6499.858.047
29536.870.91250.943.38616.982.13833.961.2486.486.06618.982.0116.489.17218.986.137
301.073.741.82498.226.78132.748.11265.478.66912.496.47336.611.49412.502.98136.615.833
312.147.483.648189.637.94963.219.125126.418.82424.112.27570.699.62924.117.61370.708.432
324.294.967.296366.592.849122.204.946244.387.90346.590.812136.700.23046.592.773136.709.034
338.589.934.592709.449.015236.484.817472.964.19890.117.543264.596.92590.114.904264.619.643
3417.179.869.1841.374.424.261458.152.758916.271.503174.495.473512.693.013174.499.432512.736.343
3534.359.738.3682.665.326.171888.452.0441.776.874.127338.235.895994.402.463338.228.666994.459.147
3668.719.476.7365.173.440.6061.724.471.0453.448.969.561656.210.9241.930.453.881656.210.0161.930.565.785


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
5323210111
6648530233
712831161585108
825682433924182020
95121811047745445042
101.02440122517610210510193
112.048860464396216211233200
124.0961.8701.003867473441499457
138.1923.9542.0891.8651.0099471.038960
1416.3848.1484.3143.8342.0431.9932.0942.018
1532.76816.8658.9027.9634.2514.1194.3424.153
1665.53634.57618.14216.4348.7938.3998.8358.549
17131.07270.61236.93533.67717.93217.17018.02917.481
18262.144143.69174.69968.99236.41335.21036.65635.412
19524.288292.090151.602140.48874.12171.63774.38171.951
201.048.576592.100306.449285.651150.111145.626150.465145.898
212.097.1521.199.001619.372579.629303.838295.230304.473295.460
224.194.3042.423.7241.248.5741.175.150613.380597.242614.484598.618
238.388.6084.893.4992.516.3152.377.1841.238.8941.206.1451.239.3231.209.137
2416.777.2169.869.4455.066.5944.802.8512.495.9042.438.2062.494.7582.440.577
2533.554.43219.890.26710.197.6649.692.6035.024.1254.922.1365.024.9004.919.106
2667.108.86440.061.32220.516.41119.544.91110.115.9069.918.10610.114.3569.912.954
27134.217.72880.637.75941.247.72039.390.03920.350.10919.972.06920.348.62819.966.953
28268.435.456162.228.33382.892.60079.335.73340.924.01940.193.33740.919.21840.191.759
29536.870.912326.215.977166.524.083159.691.89482.254.75780.856.35582.245.72580.859.140
301.073.741.824655.703.653334.442.736321.260.917165.258.649162.596.485165.256.839162.591.680
312.147.483.6481.317.507.963671.441.218646.066.745331.922.809326.824.791331.942.042326.818.321
324.294.967.2962.646.344.6681.347.678.1641.298.666.504666.485.801656.671.166666.524.702656.662.999
338.589.934.5925.313.923.3292.704.307.0012.609.616.3281.337.935.8951.318.990.1931.338.018.2341.318.979.007
3417.179.869.18410.667.634.3355.425.418.9085.242.215.4272.685.128.2902.648.586.6622.685.295.6652.648.623.718
3534.359.738.36821.409.986.71910.882.391.81310.527.594.9065.387.799.3645.317.185.6935.387.898.3545.317.103.308
3668.719.476.73642.960.786.93721.824.258.82421.136.528.11310.808.450.76210.671.903.57810.808.621.18610.671.811.411


8. Check for existing Integer Sequences by OEIS

Found in Database : 5, 1, 1, 1, 11, 1, 31, 1, 59, 19, 1, 29, 139, 41, 191, 1, 251, 71, 1, 89,
Found in Database : 5, 11, 31, 59, 19, 29, 139, 41, 191, 251, 71, 89, 79, 109, 479, 131, 571, 61, 181, 179, 239, 1019, 271, 1151, 1291, 1439, 379,
Found in Database : 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149,