Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:27:03
Deutsch
29.Mar 2024

Polynom = x^2+28x-67

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) = 67 = 67
f(1) = 19 = 19
f(2) = 7 = 7
f(3) = 13 = 13
f(4) = 61 = 61
f(5) = 49 = 7*7
f(6) = 137 = 137
f(7) = 89 = 89
f(8) = 221 = 13*17
f(9) = 133 = 7*19
f(10) = 313 = 313
f(11) = 181 = 181
f(12) = 413 = 7*59
f(13) = 233 = 233
f(14) = 521 = 521
f(15) = 289 = 17*17
f(16) = 637 = 7*7*13
f(17) = 349 = 349
f(18) = 761 = 761
f(19) = 413 = 7*59
f(20) = 893 = 19*47
f(21) = 481 = 13*37
f(22) = 1033 = 1033
f(23) = 553 = 7*79
f(24) = 1181 = 1181
f(25) = 629 = 17*37
f(26) = 1337 = 7*191
f(27) = 709 = 709
f(28) = 1501 = 19*79
f(29) = 793 = 13*61
f(30) = 1673 = 7*239
f(31) = 881 = 881
f(32) = 1853 = 17*109
f(33) = 973 = 7*139
f(34) = 2041 = 13*157
f(35) = 1069 = 1069
f(36) = 2237 = 2237
f(37) = 1169 = 7*167
f(38) = 2441 = 2441
f(39) = 1273 = 19*67
f(40) = 2653 = 7*379
f(41) = 1381 = 1381
f(42) = 2873 = 13*13*17
f(43) = 1493 = 1493
f(44) = 3101 = 7*443
f(45) = 1609 = 1609
f(46) = 3337 = 47*71
f(47) = 1729 = 7*13*19
f(48) = 3581 = 3581
f(49) = 1853 = 17*109
f(50) = 3833 = 3833
f(51) = 1981 = 7*283
f(52) = 4093 = 4093
f(53) = 2113 = 2113
f(54) = 4361 = 7*7*89
f(55) = 2249 = 13*173
f(56) = 4637 = 4637
f(57) = 2389 = 2389
f(58) = 4921 = 7*19*37
f(59) = 2533 = 17*149
f(60) = 5213 = 13*401
f(61) = 2681 = 7*383
f(62) = 5513 = 37*149
f(63) = 2833 = 2833
f(64) = 5821 = 5821
f(65) = 2989 = 7*7*61
f(66) = 6137 = 17*19*19
f(67) = 3149 = 47*67
f(68) = 6461 = 7*13*71
f(69) = 3313 = 3313
f(70) = 6793 = 6793
f(71) = 3481 = 59*59
f(72) = 7133 = 7*1019
f(73) = 3653 = 13*281
f(74) = 7481 = 7481
f(75) = 3829 = 7*547
f(76) = 7837 = 17*461
f(77) = 4009 = 19*211
f(78) = 8201 = 59*139
f(79) = 4193 = 7*599
f(80) = 8573 = 8573
f(81) = 4381 = 13*337
f(82) = 8953 = 7*1279
f(83) = 4573 = 17*269
f(84) = 9341 = 9341
f(85) = 4769 = 19*251
f(86) = 9737 = 7*13*107
f(87) = 4969 = 4969
f(88) = 10141 = 10141
f(89) = 5173 = 7*739
f(90) = 10553 = 61*173
f(91) = 5381 = 5381
f(92) = 10973 = 10973
f(93) = 5593 = 7*17*47
f(94) = 11401 = 13*877
f(95) = 5809 = 37*157
f(96) = 11837 = 7*19*89
f(97) = 6029 = 6029
f(98) = 12281 = 12281
f(99) = 6253 = 13*13*37
f(100) = 12733 = 7*17*107

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+28x-67

f(0)=67
f(1)=19
f(2)=7
f(3)=13
f(4)=61
f(5)=1
f(6)=137
f(7)=89
f(8)=17
f(9)=1
f(10)=313
f(11)=181
f(12)=59
f(13)=233
f(14)=521
f(15)=1
f(16)=1
f(17)=349
f(18)=761
f(19)=1
f(20)=47
f(21)=37
f(22)=1033
f(23)=79
f(24)=1181
f(25)=1
f(26)=191
f(27)=709
f(28)=1
f(29)=1
f(30)=239
f(31)=881
f(32)=109
f(33)=139
f(34)=157
f(35)=1069
f(36)=2237
f(37)=167
f(38)=2441
f(39)=1
f(40)=379
f(41)=1381
f(42)=1
f(43)=1493
f(44)=443
f(45)=1609
f(46)=71
f(47)=1
f(48)=3581
f(49)=1
f(50)=3833
f(51)=283
f(52)=4093
f(53)=2113
f(54)=1
f(55)=173
f(56)=4637
f(57)=2389
f(58)=1
f(59)=149
f(60)=401
f(61)=383
f(62)=1
f(63)=2833
f(64)=5821
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=3313
f(70)=6793
f(71)=1
f(72)=1019
f(73)=281
f(74)=7481
f(75)=547
f(76)=461
f(77)=211
f(78)=1
f(79)=599
f(80)=8573
f(81)=337
f(82)=1279
f(83)=269
f(84)=9341
f(85)=251
f(86)=107
f(87)=4969
f(88)=10141
f(89)=739
f(90)=1
f(91)=5381
f(92)=10973
f(93)=1
f(94)=877
f(95)=1
f(96)=1
f(97)=6029
f(98)=12281
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+28x-67 could be written as f(y)= y^2-263 with x=y-14

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+14
f'(x)>2x+27

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

67, 19, 7, 13, 61, 1, 137, 89, 17, 1, 313, 181, 59, 233, 521, 1, 1, 349, 761, 1, 47, 37, 1033, 79, 1181, 1, 191, 709, 1, 1, 239, 881, 109, 139, 157, 1069, 2237, 167, 2441, 1, 379, 1381, 1, 1493, 443, 1609, 71, 1, 3581, 1, 3833, 283, 4093, 2113, 1, 173, 4637, 2389, 1, 149, 401, 383, 1, 2833, 5821, 1, 1, 1, 1, 3313, 6793, 1, 1019, 281, 7481, 547, 461, 211, 1, 599, 8573, 337, 1279, 269, 9341, 251, 107, 4969, 10141, 739, 1, 5381, 10973, 1, 877, 1, 1, 6029, 12281, 1, 1, 6481, 1, 1, 719, 6949, 1, 1, 14621, 7433, 127, 7681, 1201, 7933, 1, 431, 131, 1, 1, 8713, 1361, 1283, 18233, 487, 2683, 733, 317, 577, 2843, 10093, 347, 1483, 569, 821, 1, 1567, 601, 1, 1, 163, 499, 1, 1, 1, 1453, 1787, 1949, 12829, 25981, 1879, 26633, 1, 557, 727, 27961, 14149, 4091, 14489, 1, 1, 30013, 1, 30713, 1, 2417, 15889, 4591, 16249, 1933, 449, 4799, 16981, 1, 1, 35081, 17729, 35837, 199, 2153, 18493, 1, 1, 38153, 19273, 5563, 1, 503, 1, 571, 1, 3181, 1, 1, 1, 6143, 1277, 743, 22129, 491, 1187, 45533, 1, 2729, 1801, 1, 3407, 1301, 227, 1, 24733, 1, 1, 1, 25633, 51721, 3727, 4049, 1, 2819, 1, 54493, 27481, 7919, 27953, 4337, 28429, 8191, 28909, 1, 1, 59273, 29881, 677, 4339, 1303, 30869, 523, 1, 63241, 31873, 1, 32381, 5021, 1, 66301, 33409, 1, 1, 1, 263, 1, 34981, 1, 2089, 10223, 1, 3823, 5227, 73721, 1, 947, 769, 75913, 1, 11003, 1, 78137, 1, 1, 1, 4729, 5783, 81533, 673, 929, 1, 6449, 42209, 12143, 42793, 1, 1, 12479, 43973, 2393, 6367, 4723, 2657, 1, 1, 92153, 46381, 13339, 46993, 1, 47609, 13691, 2837, 97081, 997, 919, 49481, 1, 7159, 1, 2671, 14591, 1, 103421, 853, 2137, 1, 106013, 1, 1, 4153, 108637, 1, 109961, 55313, 1223, 1, 691, 1, 857, 57329, 115337, 8287, 1, 58693, 1663, 1, 6287, 4621, 1, 907, 1, 61469, 1039, 787, 125053, 1, 126473, 3347, 1619, 9187, 9949, 65029, 1, 1399, 132233, 3499, 1, 67213, 1, 1, 983, 1, 3733, 1, 937, 70181, 1061, 797, 1123, 4217, 1, 5573, 145661, 10459, 1, 1, 1, 1, 2243, 1607, 1, 1, 1171, 77093, 1, 4099, 9209, 11239, 158141, 6113, 159737, 11467, 161341, 1, 23279, 2213, 827, 6361, 23743, 1, 3571, 1721, 13037, 5009, 9007, 1, 2579, 86813, 24923, 87649, 1, 1, 1, 1, 2017, 991, 181213, 91033, 182921, 13127, 10861, 1, 1, 1, 1, 883, 1, 1, 14737, 1, 193337, 1, 5273, 13999, 196873, 1621, 1, 5869, 1, 100669, 1, 1, 10739, 14639, 12109, 1543, 207673, 1, 1, 8093, 1, 106129, 213181, 107053, 1, 107981, 2437, 15559, 218761, 109849, 3617, 1, 17117, 111733, 32059, 1, 226313, 8741, 32603, 1, 230137, 971, 2129, 1, 233993, 1291, 3323, 118453, 1999, 119429, 1, 1, 34543, 1, 1, 17483, 245753, 123373, 1, 1, 249737, 1, 35963, 9721, 5399, 1, 1, 128393, 3631, 1, 259837, 1, 261881, 1, 263933, 7793, 1, 7027, 1481, 1, 1, 135589, 1, 1, 21101, 1051, 1, 19819, 1, 10753, 1, 2309, 16633, 1, 40699, 142981, 15107, 1583, 289181, 8537, 291337, 20887, 1, 147293, 42239, 148381, 297853, 7867, 42863, 1, 1, 1, 1823, 4129, 306653, 1, 18169, 1, 1, 156109, 1, 157229, 45083, 1, 5387, 22783, 1, 2633, 1, 3301, 324637, 162889, 46703, 9649, 17327, 165181, 3643, 3539, 333821, 1, 2141, 12973, 338461, 1427, 340793, 8999, 49019, 172153, 1997, 1, 1, 1601, 5741, 1321, 1, 2239, 354953, 25439, 7603, 179269, 3023, 1, 1, 1307, 2741, 1, 366973, 1, 3389, 185309, 371837, 26647, 19699, 1, 53819, 14537, 6427, 1, 7789, 191449, 2273, 27527, 386621, 1, 1, 27883, 391613, 5309, 1, 197689, 396637, 1, 1, 15401, 23629, 1, 1, 202753, 3203, 29147, 409337, 1, 1, 206593, 4657, 4423, 4583, 209173, 3019, 1, 1, 12457, 424841, 1, 1, 214381, 1, 215693, 25453, 16693, 62191, 11491, 2687, 31379, 11909, 1613, 443293, 1, 1, 1, 64091, 4787, 1, 1, 1, 1, 2909, 32719, 459421, 230389, 1871, 33107, 7879, 233113, 1, 1, 470333, 1, 3557, 3889, 475837, 1, 1, 18461, 481373, 34483, 484153, 242773, 5351, 1, 4493, 6637, 4139, 1, 1, 1, 38321, 13147, 501001, 2111, 1873, 19433, 72383, 254053, 8353, 1, 10457, 2357, 1, 1, 518137, 3659, 521021, 1, 1, 262681, 1, 264133, 1, 1, 76091, 267049, 3169, 1, 9127, 269981, 31849, 1, 14713, 5807, 78191, 4651, 1, 16229, 79039, 1, 556253, 39839, 7079, 1549, 1, 1, 565241, 1, 11597, 284881, 1, 286393, 6311, 2267, 33961, 1, 580381, 22381, 30707, 1, 586493, 294013, 84223, 1, 1, 22853, 85103, 1, 1889, 1, 46301, 17749, 605021, 1, 6833, 304849, 87323, 16127, 1, 2351, 1877, 1, 3119, 1, 9311, 312709, 8831, 1, 1951, 315881, 1, 24421, 636541, 319069, 1, 320669, 1, 6577, 1, 1987, 1, 46499, 652601, 3001, 7207, 1, 1, 17387, 1931, 25537, 17989, 47659, 1, 335249, 672137, 1, 7589, 26041, 96959, 2087, 6257, 4327, 1, 343489, 688637, 49307, 1, 1, 695293, 49783, 1, 1, 1, 351829, 37123, 1, 1, 355193, 1, 2999, 715453, 358573, 718841, 1, 722237, 19051, 1, 9829, 56081, 21493, 104639, 5479, 735901, 1, 739337, 370529, 3361, 1, 746233, 373981, 107099, 28901, 753161, 4241, 5689, 379189, 1, 54419, 763613, 29437, 1913, 54919, 40559, 22717, 1, 2791, 777661, 389713, 111599, 5843, 1, 56179, 60637, 1, 21401, 56687, 6263, 30661, 1, 400381, 2777, 1, 115163, 403969, 809737, 1, 3583, 407573, 3037, 1, 1, 411193, 6197, 413009, 827837, 1, 1, 24509, 1, 1, 1, 420313, 1, 4639, 846137, 423989, 121403, 1, 12739, 427681, 1, 1, 860921, 61627, 1, 433249, 66797, 1, 1, 1, 125119, 438853, 879581, 3217, 1, 442609, 887101, 63499, 2011, 34337, 47087, 3767, 1, 450169, 18413, 2707, 13523, 1, 6841, 455881, 8539, 3847, 1, 1973, 1, 9421, 925181, 1, 1, 24499, 71761, 3137, 1, 2713, 1, 5179, 944521, 24907, 1, 67883, 952313, 8087, 136603, 1, 960137, 28297, 137723, 6803, 3919, 1, 971933, 2447, 5987, 69847, 1, 1, 1, 492893, 16193, 2591, 141679, 1, 58573, 10181, 1, 13537, 1003741, 1, 1007753, 1, 144539, 506893, 59753, 508909, 1, 26891, 2221, 1,

6. Sequence of the polynom (only primes)

67, 19, 7, 13, 61, 137, 89, 17, 313, 181, 59, 233, 521, 349, 761, 47, 37, 1033, 79, 1181, 191, 709, 239, 881, 109, 139, 157, 1069, 2237, 167, 2441, 379, 1381, 1493, 443, 1609, 71, 3581, 3833, 283, 4093, 2113, 173, 4637, 2389, 149, 401, 383, 2833, 5821, 3313, 6793, 1019, 281, 7481, 547, 461, 211, 599, 8573, 337, 1279, 269, 9341, 251, 107, 4969, 10141, 739, 5381, 10973, 877, 6029, 12281, 6481, 719, 6949, 14621, 7433, 127, 7681, 1201, 7933, 431, 131, 8713, 1361, 1283, 18233, 487, 2683, 733, 317, 577, 2843, 10093, 347, 1483, 569, 821, 1567, 601, 163, 499, 1453, 1787, 1949, 12829, 25981, 1879, 26633, 557, 727, 27961, 14149, 4091, 14489, 30013, 30713, 2417, 15889, 4591, 16249, 1933, 449, 4799, 16981, 35081, 17729, 35837, 199, 2153, 18493, 38153, 19273, 5563, 503, 571, 3181, 6143, 1277, 743, 22129, 491, 1187, 45533, 2729, 1801, 3407, 1301, 227, 24733, 25633, 51721, 3727, 4049, 2819, 54493, 27481, 7919, 27953, 4337, 28429, 8191, 28909, 59273, 29881, 677, 4339, 1303, 30869, 523, 63241, 31873, 32381, 5021, 66301, 33409, 263, 34981, 2089, 10223, 3823, 5227, 73721, 947, 769, 75913, 11003, 78137, 4729, 5783, 81533, 673, 929, 6449, 42209, 12143, 42793, 12479, 43973, 2393, 6367, 4723, 2657, 92153, 46381, 13339, 46993, 47609, 13691, 2837, 97081, 997, 919, 49481, 7159, 2671, 14591, 103421, 853, 2137, 106013, 4153, 108637, 109961, 55313, 1223, 691, 857, 57329, 115337, 8287, 58693, 1663, 6287, 4621, 907, 61469, 1039, 787, 125053, 126473, 3347, 1619, 9187, 9949, 65029, 1399, 132233, 3499, 67213, 983, 3733, 937, 70181, 1061, 797, 1123, 4217, 5573, 145661, 10459, 2243, 1607, 1171, 77093, 4099, 9209, 11239, 158141, 6113, 159737, 11467, 161341, 23279, 2213, 827, 6361, 23743, 3571, 1721, 13037, 5009, 9007, 2579, 86813, 24923, 87649, 2017, 991, 181213, 91033, 182921, 13127, 10861, 883, 14737, 193337, 5273, 13999, 196873, 1621, 5869, 100669, 10739, 14639, 12109, 1543, 207673, 8093, 106129, 213181, 107053, 107981, 2437, 15559, 218761, 109849, 3617, 17117, 111733, 32059, 226313, 8741, 32603, 230137, 971, 2129, 233993, 1291, 3323, 118453, 1999, 119429, 34543, 17483, 245753, 123373, 249737, 35963, 9721, 5399, 128393, 3631, 259837, 261881, 263933, 7793, 7027, 1481, 135589, 21101, 1051, 19819, 10753, 2309, 16633, 40699, 142981, 15107, 1583, 289181, 8537, 291337, 20887, 147293, 42239, 148381, 297853, 7867, 42863, 1823, 4129, 306653, 18169, 156109, 157229, 45083, 5387, 22783, 2633, 3301, 324637, 162889, 46703, 9649, 17327, 165181, 3643, 3539, 333821, 2141, 12973, 338461, 1427, 340793, 8999, 49019, 172153, 1997, 1601, 5741, 1321, 2239, 354953, 25439, 7603, 179269, 3023, 1307, 2741, 366973, 3389, 185309, 371837, 26647, 19699, 53819, 14537, 6427, 7789, 191449, 2273, 27527, 386621, 27883, 391613, 5309, 197689, 396637, 15401, 23629, 202753, 3203, 29147, 409337, 206593, 4657, 4423, 4583, 209173, 3019, 12457, 424841, 214381, 215693, 25453, 16693, 62191, 11491, 2687, 31379, 11909, 1613, 443293, 64091, 4787, 2909, 32719, 459421, 230389, 1871, 33107, 7879, 233113, 470333, 3557, 3889, 475837, 18461, 481373, 34483, 484153, 242773, 5351, 4493, 6637, 4139, 38321, 13147, 501001, 2111, 1873, 19433, 72383, 254053, 8353, 10457, 2357, 518137, 3659, 521021, 262681, 264133, 76091, 267049, 3169, 9127, 269981, 31849, 14713, 5807, 78191, 4651, 16229, 79039, 556253, 39839, 7079, 1549, 565241, 11597, 284881, 286393, 6311, 2267, 33961, 580381, 22381, 30707, 586493, 294013, 84223, 22853, 85103, 1889, 46301, 17749, 605021, 6833, 304849, 87323, 16127, 2351, 1877, 3119, 9311, 312709, 8831, 1951, 315881, 24421, 636541, 319069, 320669, 6577, 1987, 46499, 652601, 3001, 7207, 17387, 1931, 25537, 17989, 47659, 335249, 672137, 7589, 26041, 96959, 2087, 6257, 4327, 343489, 688637, 49307, 695293, 49783, 351829, 37123, 355193, 2999, 715453, 358573, 718841, 722237, 19051, 9829, 56081, 21493, 104639, 5479, 735901, 739337, 370529, 3361, 746233, 373981, 107099, 28901, 753161, 4241, 5689, 379189, 54419, 763613, 29437, 1913, 54919, 40559, 22717, 2791, 777661, 389713, 111599, 5843, 56179, 60637, 21401, 56687, 6263, 30661, 400381, 2777, 115163, 403969, 809737, 3583, 407573, 3037, 411193, 6197, 413009, 827837, 24509, 420313, 4639, 846137, 423989, 121403, 12739, 427681, 860921, 61627, 433249, 66797, 125119, 438853, 879581, 3217, 442609, 887101, 63499, 2011, 34337, 47087, 3767, 450169, 18413, 2707, 13523, 6841, 455881, 8539, 3847, 1973, 9421, 925181, 24499, 71761, 3137, 2713, 5179, 944521, 24907, 67883, 952313, 8087, 136603, 960137, 28297, 137723, 6803, 3919, 971933, 2447, 5987, 69847, 492893, 16193, 2591, 141679, 58573, 10181, 13537, 1003741, 1007753, 144539, 506893, 59753, 508909, 26891, 2221,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+28x-67 and
the reducible primes which appear as divisor for the first time
p | x^2+28x-67 and p < x^2+28x-67

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)
1109810.9000000.8000000.1000000.0000000.0000000.000000
21007142290.7100000.4200000.2900007.8888895.25000029.000000
31.0006862644220.6860000.2640000.4220009.6619726.28571414.551724
410.0006.9601.8745.0860.6960000.1874000.50860010.1457737.09848512.052133
5100.00069.65914.47855.1810.6965900.1447800.55181010.0084777.72572010.849587
61.000.000695.897117.283578.6140.6958970.1172830.5786149.9900518.10077410.485747
710.000.0006.950.591988.9965.961.5950.6950590.0989000.5961609.9879608.43256110.303233
8100.000.00069.471.7318.557.22860.914.5030.6947170.0855720.6091459.9950828.65243910.217820
91.000.000.000694.427.69575.430.135618.997.5600.6944280.0754300.6189989.9958318.81478610.161743
1010.000.000.0006.942.217.112674.438.8526.267.778.2600.6942220.0674440.6267789.9970348.94123810.125690


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
388711.0000000.8750000.1250001.6000001.400000inf
416131120.8125000.6875000.1250001.6250001.5714292.000000
532241770.7500000.5312500.2187501.8461541.5454553.500000
6644831170.7500000.4843750.2656252.0000001.8235292.428571
71289050400.7031250.3906250.3125001.8750001.6129032.352941
825617487870.6796880.3398440.3398441.9333331.7400002.175000
95123451491960.6738280.2910160.3828121.9827591.7126442.252874
101.0247052684370.6884770.2617190.4267582.0434781.7986582.229592
112.0481.4084819270.6875000.2348630.4526371.9971631.7947762.121281
124.0962.8348711.9630.6918950.2126460.4792482.0127841.8108112.117584
138.1925.6881.5904.0980.6943360.1940920.5002442.0070571.8254882.087621
1416.38411.4102.8938.5170.6964110.1765750.5198362.0059771.8194972.078331
1532.76822.8145.32117.4930.6962280.1623840.5338441.9994741.8392672.053892
1665.53645.6449.92035.7240.6964720.1513670.5451052.0007011.8643112.042188
17131.07291.28718.43272.8550.6964650.1406250.5558401.9999781.8580652.039385
18262.144182.49734.564147.9330.6961710.1318510.5643201.9991561.8752172.030513
19524.288364.82465.013299.8110.6958470.1240020.5718441.9990681.8809452.026668
201.048.576729.629122.441607.1880.6958280.1167690.5790601.9999481.8833312.025236
212.097.1521.458.356232.0381.226.3180.6953980.1106440.5847541.9987641.8951002.019668
224.194.3042.915.904440.5402.475.3640.6952060.1050330.5901731.9994461.8985682.018533
238.388.6085.830.601839.2174.991.3840.6950620.1000420.5950191.9995861.9049732.016424
2416.777.21611.659.3221.604.03310.055.2890.6949500.0956080.5993421.9996781.9113452.014529
2533.554.43223.315.4793.068.37120.247.1080.6948550.0914450.6034111.9997291.9129102.013578
2667.108.86446.623.7775.880.26240.743.5150.6947480.0876230.6071261.9996921.9164122.012313
27134.217.72893.237.11411.291.01181.946.1030.6946710.0841250.6105461.9997761.9201542.011267
28268.435.456186.449.14721.716.589164.732.5580.6945770.0809010.6136771.9997311.9233522.010255
29536.870.912372.855.63341.833.998331.021.6350.6944980.0779220.6165761.9997711.9263622.009449
301.073.741.824745.627.57580.701.316664.926.2590.6944200.0751590.6192611.9997761.9290842.008709
312.147.483.6481.491.110.391155.858.8701.335.251.5210.6943520.0725770.6217751.9998061.9313052.008120
324.294.967.2962.981.955.794301.379.2322.680.576.5620.6942910.0701700.6241201.9998221.9336682.007544
338.589.934.5925.963.415.074583.419.9065.379.995.1680.6942330.0679190.6263141.9998331.9358332.007029
3417.179.869.18411.925.925.4151.130.528.57110.795.396.8440.6941800.0658050.6283751.9998481.9377612.006581
3534.359.738.36823.850.283.3972.192.819.86321.657.463.5340.6941350.0638190.6303151.9998691.9396412.006176
3668.719.476.73647.697.568.7664.257.235.15943.440.333.6070.6940910.0619510.6321401.9998741.9414432.005790


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
123300201
245500221
387522221
41611745231
532171078261
664311813132151
7128502723232241
8256875235432411
95121498069762701
101.02426814911912721381
112.04848125522624222361
124.09687145142043724311
138.1921.59079579579627911
1416.3842.8931.4661.4271.46521.4251
1532.7685.3212.6892.6322.66422.6541
1665.5369.9205.0114.9094.93224.9851
17131.07218.4329.2849.1489.20329.2261
18262.14434.56417.37417.19017.274217.2871
19524.28865.01332.66532.34832.507232.5031
201.048.576122.44161.61660.82561.343261.0951
212.097.152232.038116.543115.495116.0842115.9511
224.194.304440.540221.167219.373220.3292220.2081
238.388.608839.217420.666418.551419.6872419.5271
2416.777.2161.604.033804.818799.215801.7242802.3061
2533.554.4323.068.3711.539.2621.529.1091.534.49821.533.8701
2667.108.8645.880.2622.949.9612.930.3012.940.23122.940.0281
27134.217.72811.291.0115.664.1935.626.8185.645.04625.645.9621
28268.435.45621.716.58910.892.26310.824.32610.859.234210.857.3521
29536.870.91241.833.99820.977.66120.856.33720.919.334220.914.6611
301.073.741.82480.701.31640.466.16240.235.15440.350.569240.350.7441
312.147.483.648155.858.87078.153.50677.705.36477.924.083277.934.7841
324.294.967.296301.379.232151.116.084150.263.148150.685.1192150.694.1101
338.589.934.592583.419.906292.511.191290.908.715291.699.8132291.720.0901
3417.179.869.1841.130.528.571566.765.517563.763.054565.250.3412565.278.2271
3534.359.738.3682.192.819.8631.099.213.7711.093.606.0921.096.382.00221.096.437.8581
3668.719.476.7364.257.235.1592.133.863.8882.123.371.2712.128.591.13322.128.644.0231


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
381011000
4162021100
5327251114
664176112546
7128401723713911
825687375017251926
95121969410242593956
101.0244371992389312591128
112.048927452475201263193270
124.0961.963972991421542442558
138.1924.0982.0422.0569111.1339091.145
1416.3848.5174.2534.2641.8902.3201.9622.345
1532.76817.4938.8008.6933.9444.7394.0414.769
1665.53635.72417.91817.8068.2049.6098.2869.625
17131.07272.85536.45336.40216.93219.44516.93519.543
18262.144147.93374.18973.74434.45139.37334.64439.465
19524.288299.811150.099149.71270.42779.64470.45379.287
201.048.576607.188303.969303.219143.275160.186143.606160.121
212.097.1521.226.318613.391612.927290.879322.008290.824322.607
224.194.3042.475.3641.238.1981.237.166589.423647.942590.009647.990
238.388.6084.991.3842.496.8082.494.5761.191.7611.304.1881.193.3151.302.120
2416.777.21610.055.2895.031.1675.024.1222.405.7422.621.6492.409.2122.618.686
2533.554.43220.247.10810.128.75710.118.3514.857.5095.264.7734.860.8205.264.006
2667.108.86440.743.51520.380.00820.363.5079.796.89210.573.9449.803.64310.569.036
27134.217.72881.946.10340.981.42740.964.67619.743.18421.228.01719.752.55821.222.344
28268.435.456164.732.55882.387.08882.345.47039.763.94442.596.01139.777.12342.595.480
29536.870.912331.021.635165.558.167165.463.46880.031.50385.471.10280.044.79085.474.240
301.073.741.824664.926.259332.553.582332.372.677161.013.170171.452.494161.014.790171.445.805
312.147.483.6481.335.251.521667.766.744667.484.777323.759.578343.862.759323.769.899343.859.285
324.294.967.2962.680.576.5621.340.574.7491.340.001.813650.736.318689.549.371650.752.583689.538.290
338.589.934.5925.379.995.1682.690.501.6902.689.493.4781.307.515.8191.382.489.7231.307.532.3961.382.457.230
3417.179.869.18410.795.396.8445.398.627.5105.396.769.3342.626.373.6142.771.338.2272.626.417.1902.771.267.813
3534.359.738.36821.657.463.53410.830.469.42110.826.994.1135.274.053.3515.554.657.3455.274.052.9805.554.699.858
3668.719.476.73643.440.333.60721.723.472.31421.716.861.29310.588.063.58311.132.038.95710.588.041.32611.132.189.741


8. Check for existing Integer Sequences by OEIS

Found in Database : 67, 19, 7, 13, 61, 1, 137, 89, 17, 1, 313, 181, 59, 233, 521, 1, 1, 349, 761, 1,
Found in Database : 67, 19, 7, 13, 61, 137, 89, 17, 313, 181, 59, 233, 521, 349, 761, 47, 37, 1033, 79, 1181, 191, 709, 239, 881, 109, 139, 157, 1069, 2237, 167, 2441,
Found in Database : 7, 13, 17, 19, 37, 47, 59, 61, 67, 71, 79, 89, 107, 109, 127, 131, 137, 139, 149,