Inhaltsverzeichnis

Development of
Algorithmic Constructions

14:48:33
Deutsch
28.Mar 2024

Polynom = x^2+6x-2

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) = 1 = 1
f(1) = 5 = 5
f(2) = 7 = 7
f(3) = 25 = 5*5
f(4) = 19 = 19
f(5) = 53 = 53
f(6) = 35 = 5*7
f(7) = 89 = 89
f(8) = 55 = 5*11
f(9) = 133 = 7*19
f(10) = 79 = 79
f(11) = 185 = 5*37
f(12) = 107 = 107
f(13) = 245 = 5*7*7
f(14) = 139 = 139
f(15) = 313 = 313
f(16) = 175 = 5*5*7
f(17) = 389 = 389
f(18) = 215 = 5*43
f(19) = 473 = 11*43
f(20) = 259 = 7*37
f(21) = 565 = 5*113
f(22) = 307 = 307
f(23) = 665 = 5*7*19
f(24) = 359 = 359
f(25) = 773 = 773
f(26) = 415 = 5*83
f(27) = 889 = 7*127
f(28) = 475 = 5*5*19
f(29) = 1013 = 1013
f(30) = 539 = 7*7*11
f(31) = 1145 = 5*229
f(32) = 607 = 607
f(33) = 1285 = 5*257
f(34) = 679 = 7*97
f(35) = 1433 = 1433
f(36) = 755 = 5*151
f(37) = 1589 = 7*227
f(38) = 835 = 5*167
f(39) = 1753 = 1753
f(40) = 919 = 919
f(41) = 1925 = 5*5*7*11
f(42) = 1007 = 19*53
f(43) = 2105 = 5*421
f(44) = 1099 = 7*157
f(45) = 2293 = 2293
f(46) = 1195 = 5*239
f(47) = 2489 = 19*131
f(48) = 1295 = 5*7*37
f(49) = 2693 = 2693
f(50) = 1399 = 1399
f(51) = 2905 = 5*7*83
f(52) = 1507 = 11*137
f(53) = 3125 = 5*5*5*5*5
f(54) = 1619 = 1619
f(55) = 3353 = 7*479
f(56) = 1735 = 5*347
f(57) = 3589 = 37*97
f(58) = 1855 = 5*7*53
f(59) = 3833 = 3833
f(60) = 1979 = 1979
f(61) = 4085 = 5*19*43
f(62) = 2107 = 7*7*43
f(63) = 4345 = 5*11*79
f(64) = 2239 = 2239
f(65) = 4613 = 7*659
f(66) = 2375 = 5*5*5*19
f(67) = 4889 = 4889
f(68) = 2515 = 5*503
f(69) = 5173 = 7*739
f(70) = 2659 = 2659
f(71) = 5465 = 5*1093
f(72) = 2807 = 7*401
f(73) = 5765 = 5*1153
f(74) = 2959 = 11*269
f(75) = 6073 = 6073
f(76) = 3115 = 5*7*89
f(77) = 6389 = 6389
f(78) = 3275 = 5*5*131
f(79) = 6713 = 7*7*137
f(80) = 3439 = 19*181
f(81) = 7045 = 5*1409
f(82) = 3607 = 3607
f(83) = 7385 = 5*7*211
f(84) = 3779 = 3779
f(85) = 7733 = 11*19*37
f(86) = 3955 = 5*7*113
f(87) = 8089 = 8089
f(88) = 4135 = 5*827
f(89) = 8453 = 79*107
f(90) = 4319 = 7*617
f(91) = 8825 = 5*5*353
f(92) = 4507 = 4507
f(93) = 9205 = 5*7*263
f(94) = 4699 = 37*127
f(95) = 9593 = 53*181
f(96) = 4895 = 5*11*89
f(97) = 9989 = 7*1427
f(98) = 5095 = 5*1019
f(99) = 10393 = 19*547
f(100) = 5299 = 7*757

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+6x-2

f(0)=1
f(1)=5
f(2)=7
f(3)=1
f(4)=19
f(5)=53
f(6)=1
f(7)=89
f(8)=11
f(9)=1
f(10)=79
f(11)=37
f(12)=107
f(13)=1
f(14)=139
f(15)=313
f(16)=1
f(17)=389
f(18)=43
f(19)=1
f(20)=1
f(21)=113
f(22)=307
f(23)=1
f(24)=359
f(25)=773
f(26)=83
f(27)=127
f(28)=1
f(29)=1013
f(30)=1
f(31)=229
f(32)=607
f(33)=257
f(34)=97
f(35)=1433
f(36)=151
f(37)=227
f(38)=167
f(39)=1753
f(40)=919
f(41)=1
f(42)=1
f(43)=421
f(44)=157
f(45)=2293
f(46)=239
f(47)=131
f(48)=1
f(49)=2693
f(50)=1399
f(51)=1
f(52)=137
f(53)=1
f(54)=1619
f(55)=479
f(56)=347
f(57)=1
f(58)=1
f(59)=3833
f(60)=1979
f(61)=1
f(62)=1
f(63)=1
f(64)=2239
f(65)=659
f(66)=1
f(67)=4889
f(68)=503
f(69)=739
f(70)=2659
f(71)=1093
f(72)=401
f(73)=1153
f(74)=269
f(75)=6073
f(76)=1
f(77)=6389
f(78)=1
f(79)=1
f(80)=181
f(81)=1409
f(82)=3607
f(83)=211
f(84)=3779
f(85)=1
f(86)=1
f(87)=8089
f(88)=827
f(89)=1
f(90)=617
f(91)=353
f(92)=4507
f(93)=263
f(94)=1
f(95)=1
f(96)=1
f(97)=1427
f(98)=1019
f(99)=547

b) Substitution of the polynom
The polynom f(x)=x^2+6x-2 could be written as f(y)= y^2-11 with x=y-3

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+3
f'(x)>2x+5

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

1, 5, 7, 1, 19, 53, 1, 89, 11, 1, 79, 37, 107, 1, 139, 313, 1, 389, 43, 1, 1, 113, 307, 1, 359, 773, 83, 127, 1, 1013, 1, 229, 607, 257, 97, 1433, 151, 227, 167, 1753, 919, 1, 1, 421, 157, 2293, 239, 131, 1, 2693, 1399, 1, 137, 1, 1619, 479, 347, 1, 1, 3833, 1979, 1, 1, 1, 2239, 659, 1, 4889, 503, 739, 2659, 1093, 401, 1153, 269, 6073, 1, 6389, 1, 1, 181, 1409, 3607, 211, 3779, 1, 1, 8089, 827, 1, 617, 353, 4507, 263, 1, 1, 1, 1427, 1019, 547, 757, 2161, 5507, 449, 1, 271, 1187, 1, 1231, 1, 6379, 1, 6607, 2689, 977, 13913, 283, 14389, 1, 1, 7559, 439, 1, 1, 8059, 2339, 1663, 16889, 1, 1583, 8839, 1, 1301, 3697, 1, 2719, 1931, 1031, 1987, 2879, 929, 829, 1, 4261, 10799, 21893, 317, 523, 1, 3299, 11699, 431, 12007, 1, 1, 24953, 1, 25589, 2591, 709, 1, 1, 1237, 787, 1, 1, 571, 4127, 1, 29573, 2137, 6053, 15307, 563, 2237, 1667, 3203, 661, 1, 33113, 881, 967, 17107, 6917, 1, 397, 3571, 1, 521, 1, 433, 1, 1, 7681, 1021, 509, 1, 39989, 577, 1, 20599, 1, 3001, 1697, 21419, 1, 1, 44089, 4451, 1, 22679, 9157, 3301, 491, 23539, 47513, 1, 1, 1, 7039, 24859, 1, 25307, 1459, 25759, 51973, 1, 52889, 1, 53813, 3877, 10949, 1453, 1, 653, 56633, 5711, 1, 5807, 5323, 4217, 2381, 811, 12101, 4357, 61493, 6199, 1, 6299, 63493, 2909, 1, 32507, 2621, 1, 66553, 1, 67589, 1, 68633, 1, 1, 35107, 14149, 1, 10259, 1447, 72889, 1049, 73973, 1, 15013, 1, 15233, 1, 1, 1, 1823, 1579, 1, 40039, 1, 5801, 1487, 41179, 1, 1193, 84089, 8467, 641, 1, 3457, 1, 2503, 1, 88793, 1277, 89989, 9059, 91193, 1, 18481, 46507, 1, 47119, 8623, 9547, 1, 1, 743, 6997, 19717, 1, 1051, 7177, 101113, 1, 14627, 10303, 2411, 1213, 2999, 52807, 1, 1091, 1109, 1, 1, 1, 1, 55439, 3187, 2953, 1, 56779, 16319, 11491, 115589, 1, 116953, 1, 4733, 8501, 1, 1627, 17299, 1, 122489, 1, 1609, 62299, 1319, 9001, 1, 63719, 128153, 1, 129589, 1, 18719, 1, 26497, 1549, 1, 67339, 7127, 1, 136889, 13763, 138373, 1, 2543, 1, 1, 71059, 142873, 1, 20627, 2903, 1847, 10477, 797, 6737, 1, 1, 150533, 15131, 21727, 15287, 8087, 2087, 887, 78007, 2851, 11257, 158393, 15919, 1151, 2297, 1, 81199, 4663, 82007, 1, 7529, 1, 1, 168089, 1, 169733, 1, 1, 12301, 1, 86939, 2269, 3511, 176389, 1, 25439, 89459, 1, 1, 36293, 91159, 1621, 1, 1, 1, 1, 93739, 1, 1063, 5431, 95479, 191833, 2753, 17599, 19447, 195353, 1, 1, 1, 5683, 1, 2069, 1061, 28927, 1, 4751, 14657, 41221, 1163, 8317, 2131, 5669, 21067, 1, 1, 19403, 5641, 6151, 108107, 1, 1, 11527, 1, 220889, 3169, 222773, 10169, 1, 112807, 1, 113759, 1, 22943, 230389, 1, 12227, 116639, 4259, 1, 47237, 1, 34019, 23911, 240089, 24107, 1, 1, 1, 1, 49201, 123499, 247993, 3557, 249989, 1321, 35999, 126499, 1373, 127507, 1, 128519, 1223, 3701, 260089, 26111, 262133, 18797, 52837, 132607, 7607, 12149, 1709, 5387, 1, 27143, 1, 2791, 1, 7253, 55333, 1, 25343, 27983, 40127, 5639, 7649, 142039, 8147, 143107, 57457, 1, 1, 1, 3691, 1, 1759, 147419, 1, 1531, 59621, 1, 42899, 30139, 1, 4337, 3671, 152899, 61381, 1, 2473, 155119, 2341, 31247, 313589, 2861, 45119, 1, 63617, 1, 1, 160739, 322613, 1, 324889, 32603, 1, 1181, 1, 3119, 9479, 8761, 9029, 4789, 7823, 1, 17827, 2207, 68209, 1, 9811, 172279, 345733, 1, 49727, 1, 350453, 25117, 1283, 177007, 3739, 25457, 357593, 35879, 51427, 1901, 362393, 1327, 1489, 1, 1, 26317, 1, 37087, 4483, 5333, 1, 2111, 10771, 1, 6899, 190339, 54559, 1, 20231, 1, 1471, 4513, 1811, 27901, 1, 1, 1, 39563, 2897, 7963, 57059, 2251, 4231, 1, 80897, 202879, 37003, 1, 409589, 1, 1, 1, 1, 2633, 11923, 209299, 419893, 1, 1559, 42379, 11489, 1, 1, 214507, 2459, 2017, 22787, 43427, 5657, 43691, 438233, 31397, 88177, 1741, 2063, 1, 1867, 8951, 9161, 4093, 23767, 1, 12979, 1451, 91393, 1723, 3307, 46103, 12497, 1, 42283, 233239, 13367, 1, 94117, 235979, 67619, 1, 476089, 1, 478853, 1, 3853, 34501, 5099, 4583, 1, 48859, 489989, 49139, 1, 247099, 9011, 1, 19937, 1, 1, 1, 1, 50551, 1, 254179, 101957, 1, 1, 6947, 6211, 1, 518389, 1, 5857, 37337, 1, 262807, 1, 1, 529973, 2797, 1, 10687, 4219, 38377, 1, 1, 1, 3527, 3469, 1, 1, 54907, 550553, 276019, 3163, 1, 111301, 39857, 2677, 56099, 10613, 1, 13151, 1, 1, 285007, 22861, 286519, 1, 5237, 577589, 8273, 580633, 7867, 1, 41801, 1, 1, 12037, 11827, 53899, 59443, 4481, 298759, 119813, 42901, 1, 3821, 1, 8669, 608389, 1, 87359, 1, 122929, 308107, 1, 5843, 620933, 8893, 624089, 1, 1, 44917, 1, 7349, 1, 317599, 1637, 63839, 1, 1, 2029, 1, 129281, 17053, 5197, 1, 652853, 65447, 4933, 1, 659333, 3407, 1721, 2939, 133169, 1, 669113, 2683, 1, 9629, 3733, 338659, 1, 30937, 136453, 341959, 2647, 3617, 688889, 1973, 4409, 346939, 139109, 49801, 1, 9467, 2333, 1, 13313, 1, 101279, 18701, 28493, 51001, 143141, 32609, 37847, 1, 722489, 1, 103699, 363799, 145861, 365507, 1, 367219, 66923, 1, 1, 74131, 39107, 53197, 149297, 10111, 3061, 1, 3319, 1, 108127, 1, 760373, 54437, 4129, 382807, 1, 1, 770873, 77263, 1, 1, 1, 389839, 1, 1, 8263, 56197, 1873, 79031, 1, 1, 795653, 7523, 4567, 2551, 1, 402299, 2351, 80819, 1, 11597, 1999, 407699, 1, 3079, 32833, 1, 117779, 1, 8537, 2243, 118819, 37889, 167077, 8543, 167809, 420439, 1, 1, 846389, 1, 121439, 425959, 1, 9949, 24499, 429659, 7621, 12329, 4099, 3467, 868613, 1, 174469, 1, 25031, 438979, 46307, 2383, 126227, 88547, 10691, 3343, 7129, 2957, 1, 9151, 1, 90059, 1, 90439, 906293, 1, 26003, 456007, 1, 1, 917753, 91967, 921589, 1, 1, 463679, 3793, 1, 186629, 1, 1, 1, 940889, 13469, 944773, 1, 189733, 67901, 1, 477259, 1, 8713, 960389, 1, 1, 483139, 193649, 1, 1, 487079, 5393, 1, 1, 98207, 1, 4363, 39521, 26053, 4049, 496999, 7603, 1, 52631, 9109, 2287,

6. Sequence of the polynom (only primes)

5, 7, 19, 53, 89, 11, 79, 37, 107, 139, 313, 389, 43, 113, 307, 359, 773, 83, 127, 1013, 229, 607, 257, 97, 1433, 151, 227, 167, 1753, 919, 421, 157, 2293, 239, 131, 2693, 1399, 137, 1619, 479, 347, 3833, 1979, 2239, 659, 4889, 503, 739, 2659, 1093, 401, 1153, 269, 6073, 6389, 181, 1409, 3607, 211, 3779, 8089, 827, 617, 353, 4507, 263, 1427, 1019, 547, 757, 2161, 5507, 449, 271, 1187, 1231, 6379, 6607, 2689, 977, 13913, 283, 14389, 7559, 439, 8059, 2339, 1663, 16889, 1583, 8839, 1301, 3697, 2719, 1931, 1031, 1987, 2879, 929, 829, 4261, 10799, 21893, 317, 523, 3299, 11699, 431, 12007, 24953, 25589, 2591, 709, 1237, 787, 571, 4127, 29573, 2137, 6053, 15307, 563, 2237, 1667, 3203, 661, 33113, 881, 967, 17107, 6917, 397, 3571, 521, 433, 7681, 1021, 509, 39989, 577, 20599, 3001, 1697, 21419, 44089, 4451, 22679, 9157, 3301, 491, 23539, 47513, 7039, 24859, 25307, 1459, 25759, 51973, 52889, 53813, 3877, 10949, 1453, 653, 56633, 5711, 5807, 5323, 4217, 2381, 811, 12101, 4357, 61493, 6199, 6299, 63493, 2909, 32507, 2621, 66553, 67589, 68633, 35107, 14149, 10259, 1447, 72889, 1049, 73973, 15013, 15233, 1823, 1579, 40039, 5801, 1487, 41179, 1193, 84089, 8467, 641, 3457, 2503, 88793, 1277, 89989, 9059, 91193, 18481, 46507, 47119, 8623, 9547, 743, 6997, 19717, 1051, 7177, 101113, 14627, 10303, 2411, 1213, 2999, 52807, 1091, 1109, 55439, 3187, 2953, 56779, 16319, 11491, 115589, 116953, 4733, 8501, 1627, 17299, 122489, 1609, 62299, 1319, 9001, 63719, 128153, 129589, 18719, 26497, 1549, 67339, 7127, 136889, 13763, 138373, 2543, 71059, 142873, 20627, 2903, 1847, 10477, 797, 6737, 150533, 15131, 21727, 15287, 8087, 2087, 887, 78007, 2851, 11257, 158393, 15919, 1151, 2297, 81199, 4663, 82007, 7529, 168089, 169733, 12301, 86939, 2269, 3511, 176389, 25439, 89459, 36293, 91159, 1621, 93739, 1063, 5431, 95479, 191833, 2753, 17599, 19447, 195353, 5683, 2069, 1061, 28927, 4751, 14657, 41221, 1163, 8317, 2131, 5669, 21067, 19403, 5641, 6151, 108107, 11527, 220889, 3169, 222773, 10169, 112807, 113759, 22943, 230389, 12227, 116639, 4259, 47237, 34019, 23911, 240089, 24107, 49201, 123499, 247993, 3557, 249989, 1321, 35999, 126499, 1373, 127507, 128519, 1223, 3701, 260089, 26111, 262133, 18797, 52837, 132607, 7607, 12149, 1709, 5387, 27143, 2791, 7253, 55333, 25343, 27983, 40127, 5639, 7649, 142039, 8147, 143107, 57457, 3691, 1759, 147419, 1531, 59621, 42899, 30139, 4337, 3671, 152899, 61381, 2473, 155119, 2341, 31247, 313589, 2861, 45119, 63617, 160739, 322613, 324889, 32603, 1181, 3119, 9479, 8761, 9029, 4789, 7823, 17827, 2207, 68209, 9811, 172279, 345733, 49727, 350453, 25117, 1283, 177007, 3739, 25457, 357593, 35879, 51427, 1901, 362393, 1327, 1489, 26317, 37087, 4483, 5333, 2111, 10771, 6899, 190339, 54559, 20231, 1471, 4513, 1811, 27901, 39563, 2897, 7963, 57059, 2251, 4231, 80897, 202879, 37003, 409589, 2633, 11923, 209299, 419893, 1559, 42379, 11489, 214507, 2459, 2017, 22787, 43427, 5657, 43691, 438233, 31397, 88177, 1741, 2063, 1867, 8951, 9161, 4093, 23767, 12979, 1451, 91393, 1723, 3307, 46103, 12497, 42283, 233239, 13367, 94117, 235979, 67619, 476089, 478853, 3853, 34501, 5099, 4583, 48859, 489989, 49139, 247099, 9011, 19937, 50551, 254179, 101957, 6947, 6211, 518389, 5857, 37337, 262807, 529973, 2797, 10687, 4219, 38377, 3527, 3469, 54907, 550553, 276019, 3163, 111301, 39857, 2677, 56099, 10613, 13151, 285007, 22861, 286519, 5237, 577589, 8273, 580633, 7867, 41801, 12037, 11827, 53899, 59443, 4481, 298759, 119813, 42901, 3821, 8669, 608389, 87359, 122929, 308107, 5843, 620933, 8893, 624089, 44917, 7349, 317599, 1637, 63839, 2029, 129281, 17053, 5197, 652853, 65447, 4933, 659333, 3407, 1721, 2939, 133169, 669113, 2683, 9629, 3733, 338659, 30937, 136453, 341959, 2647, 3617, 688889, 1973, 4409, 346939, 139109, 49801, 9467, 2333, 13313, 101279, 18701, 28493, 51001, 143141, 32609, 37847, 722489, 103699, 363799, 145861, 365507, 367219, 66923, 74131, 39107, 53197, 149297, 10111, 3061, 3319, 108127, 760373, 54437, 4129, 382807, 770873, 77263, 389839, 8263, 56197, 1873, 79031, 795653, 7523, 4567, 2551, 402299, 2351, 80819, 11597, 1999, 407699, 3079, 32833, 117779, 8537, 2243, 118819, 37889, 167077, 8543, 167809, 420439, 846389, 121439, 425959, 9949, 24499, 429659, 7621, 12329, 4099, 3467, 868613, 174469, 25031, 438979, 46307, 2383, 126227, 88547, 10691, 3343, 7129, 2957, 9151, 90059, 90439, 906293, 26003, 456007, 917753, 91967, 921589, 463679, 3793, 186629, 940889, 13469, 944773, 189733, 67901, 477259, 8713, 960389, 483139, 193649, 487079, 5393, 98207, 4363, 39521, 26053, 4049, 496999, 7603, 52631, 9109, 2287,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+6x-2 and
the reducible primes which appear as divisor for the first time
p | x^2+6x-2 and p < x^2+6x-2

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

- no title specified

A

B

C

D

E

F

G

H

I

J

K

 

10^n

x

all Primes

P(x)=x^2+6x-2

P(x) | x^2+6x-2

C/B

D/B

E/B

C(n) / C(n-1)

D(n) / D(n-1)

E(n) / E(n-1)

 

1

10

8

4

4

0,800000

0,400000

0,400000

   

 

2

100

71

17

54

0,710000

0,170000

0,540000

8,875000

4,250000

13,500000

 

3

1.000

696

104

592

0,696000

0,104000

0,592000

9,802817

6,117647

10,962963

 

4

10.000

7.001

724

6.277

0,700100

0,072400

0,627700

10,058908

6,961538

10,603041

 

5

100.000

69.944

5.523

64.421

0,699440

0,055230

0,644210

9,990573

7,628453

10,263024

 

6

1.000.000

697.842

45.100

652.742

0,697842

0,045100

0,652742

9,977153

8,165852

10,132441

 

7

10.000.000

6.968.656

381.917

6.586.739

0,696866

0,038192

0,658674

9,986008

8,468226

10,090877

 

8

100.000.000

69.617.899

3.308.104

66.309.795

0,696179

0,033081

0,663098

9,990147

8,661840

10,067166

 

9

1.000.000.000

695.678.933

29.187.747

666.491.186

0,695679

0,029188

0,666491

9,992817

8,823104

10,051172

 

10

10.000.000.000

6.953.068.193

261.206.564

6.691.861.629

0,695307

0,026121

0,669186

9,994651

8,949186

10,040435

 

11

100.000.000.000

69.502.420.813

2.363.706.298

67.138.714.515

0,695024

0,023637

0,671387

9,995935

9,049184

10,032890

 

12

1.000.000.000.000

694.800.894.771

21.586.726.901

673.214.167.870

0,694801

0,021587

0,673214

9,996787

9,132576

10,027213

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

B

C

D

E

F

G

H

I

J

K

 

2^n

x

all Primes

P(x)=x^2+6x-2

P(x) | x^2+6x-2

C/B

D/B

E/B

C(n) / C(n-1)

D(n) / D(n-1)

E(n) / E(n-1)

 

1

2

3

2

1

1,500000

1,000000

0,500000

   

 

2

4

4

2

2

1,000000

0,500000

0,500000

1,333333

1,000000

 

 

3

8

7

4

3

0,875000

0,500000

0,375000

1,750000

2,000000

1,500000

 

4

16

12

5

7

0,750000

0,312500

0,437500

1,714286

1,250000

2,333333

 

5

32

23

8

15

0,718750

0,250000

0,468750

1,916667

1,600000

2,142857

 

6

64

45

13

32

0,703125

0,203125

0,500000

1,956522

1,625000

2,133333

 

7

128

90

20

70

0,703125

0,156250

0,546875

2,000000

1,538462

2,187500

 

8

256

182

35

147

0,710938

0,136719

0,574219

2,022222

1,750000

2,100000

 

9

512

357

67

290

0,697266

0,130859

0,566406

1,961538

1,914286

1,972789

 

10

1.024

715

105

610

0,698242

0,102539

0,595703

2,002801

1,567164

2,103448

 

11

2.048

1.448

181

1.267

0,707031

0,088379

0,618652

2,025175

1,723810

2,077049

 

12

4.096

2.879

337

2.542

0,702881

0,082275

0,620605

1,988260

1,861878

2,006314

 

13

8.192

5.729

607

5.122

0,699341

0,074097

0,625244

1,989927

1,801187

2,014949

 

14

16.384

11.469

1.093

10.376

0,700012

0,066711

0,633301

2,001920

1,800659

2,025771

 

15

32.768

22.918

2.048

20.870

0,699402

0,062500

0,636902

1,998256

1,873742

2,011372

 

16

65.536

45.898

3.728

42.170

0,700348

0,056885

0,643463

2,002705

1,820313

2,020604

 

17

131.072

91.691

7.055

84.636

0,699547

0,053825

0,645721

1,997712

1,892436

2,007019

 

18

262.144

183.165

13.226

169.939

0,698719

0,050453

0,648266

1,997633

1,874699

2,007881

 

19

524.288

365.873

25.000

340.873

0,697847

0,047684

0,650164

1,997505

1,890216

2,005855

 

20

1.048.576

731.845

47.133

684.712

0,697942

0,044950

0,652992

2,000271

1,885320

2,008701

 

21

2.097.152

1.462.745

89.571

1.373.174

0,697491

0,042711

0,654780

1,998709

1,900388

2,005477

 

22

4.194.304

2.924.091

169.872

2.754.219

0,697158

0,040501

0,656657

1,999044

1,896507

2,005732

 

23

8.388.608

5.846.110

324.452

5.521.658

0,696911

0,038678

0,658233

1,999291

1,909979

2,004800

 

24

16.777.216

11.688.531

618.970

11.069.561

0,696691

0,036893

0,659797

1,999369

1,907740

2,004753

 

25

33.554.432

23.370.006

1.185.366

22.184.640

0,696480

0,035327

0,661154

1,999396

1,915062

2,004112

 

26

67.108.864

46.726.723

2.273.082

44.453.641

0,696282

0,033872

0,662411

1,999431

1,917620

2,003803

 

27

134.217.728

93.428.229

4.365.930

89.062.299

0,696095

0,032529

0,663566

1,999460

1,920709

2,003487

 

28

268.435.456

186.816.810

8.398.616

178.418.194

0,695947

0,031287

0,664660

1,999576

1,923672

2,003297

 

29

536.870.912

373.555.468

16.183.498

357.371.970

0,695801

0,030144

0,665657

1,999582

1,926924

2,003002

 

30

1.073.741.824

746.968.831

31.225.725

715.743.106

0,695669

0,029081

0,666588

1,999620

1,929479

2,002796

 

31

2.147.483.648

1.493.673.241

60.332.875

1.433.340.366

0,695546

0,028095

0,667451

1,999646

1,932153

2,002591

 

32

4.294.967.296

2.986.854.625

116.691.003

2.870.163.622

0,695431

0,027169

0,668262

1,999671

1,934120

2,002430

 

33

8.589.934.592

5.972.830.633

225.937.943

5.746.892.690

0,695329

0,026303

0,669026

1,999706

1,936207

2,002287

 

34

17.179.869.184

11.944.036.512

437.939.940

11.506.096.572

0,695234

0,025491

0,669743

1,999728

1,938320

2,002142

 

35

34.359.738.368

23.885.076.589

849.639.960

23.035.436.629

0,695147

0,024728

0,670419

1,999749

1,940083

2,002020

 

36

68.719.476.736

47.764.566.376

1.649.863.576

46.114.702.800

0,695066

0,024009

0,671057

1,999766

1,941838

2,001903

 

37

137.438.953.472

95.518.682.613

3.206.591.923

92.312.090.690

0,694990

0,023331

0,671659

1,999781

1,943550

2,001793

 

38

274.877.906.944

191.018.013.504

6.237.159.361

184.780.854.143

0,694919

0,022691

0,672229

1,999797

1,945105

2,001697

 

39

549.755.813.888

381.999.933.299

12.141.088.699

369.858.844.600

0,694854

0,022085

0,672769

1,999811

1,946573

2,001608

 

40

1.099.511.627.776

763.932.794.221

23.650.285.030

740.282.509.191

0,694793

0,021510

0,673283

1,999824

1,947954

2,001527

 

           

 

           

 

            
            


8. Check for existing Integer Sequences by OEIS

Found in Database : 1, 5, 7, 1, 19, 53, 1, 89, 11, 1, 79, 37, 107, 1, 139, 313, 1, 389, 43, 1,
Found in Database : 5, 7, 19, 53, 89, 11, 79, 37, 107, 139, 313, 389, 43, 113, 307, 359, 773, 83, 127, 1013, 229, 607, 257, 97, 1433, 151, 227, 167, 1753,
Found in Database : 5, 7, 11, 19, 37, 43, 53, 79, 83, 89, 97, 107, 113, 127, 131, 137, 139,