Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:43:31
Deutsch
29.Mar 2024

Polynom = x^2+16x-7

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) = 7 = 7
f(1) = 5 = 5
f(2) = 29 = 29
f(3) = 25 = 5*5
f(4) = 73 = 73
f(5) = 49 = 7*7
f(6) = 125 = 5*5*5
f(7) = 77 = 7*11
f(8) = 185 = 5*37
f(9) = 109 = 109
f(10) = 253 = 11*23
f(11) = 145 = 5*29
f(12) = 329 = 7*47
f(13) = 185 = 5*37
f(14) = 413 = 7*59
f(15) = 229 = 229
f(16) = 505 = 5*101
f(17) = 277 = 277
f(18) = 605 = 5*11*11
f(19) = 329 = 7*47
f(20) = 713 = 23*31
f(21) = 385 = 5*7*11
f(22) = 829 = 829
f(23) = 445 = 5*89
f(24) = 953 = 953
f(25) = 509 = 509
f(26) = 1085 = 5*7*31
f(27) = 577 = 577
f(28) = 1225 = 5*5*7*7
f(29) = 649 = 11*59
f(30) = 1373 = 1373
f(31) = 725 = 5*5*29
f(32) = 1529 = 11*139
f(33) = 805 = 5*7*23
f(34) = 1693 = 1693
f(35) = 889 = 7*127
f(36) = 1865 = 5*373
f(37) = 977 = 977
f(38) = 2045 = 5*409
f(39) = 1069 = 1069
f(40) = 2233 = 7*11*29
f(41) = 1165 = 5*233
f(42) = 2429 = 7*347
f(43) = 1265 = 5*11*23
f(44) = 2633 = 2633
f(45) = 1369 = 37*37
f(46) = 2845 = 5*569
f(47) = 1477 = 7*211
f(48) = 3065 = 5*613
f(49) = 1589 = 7*227
f(50) = 3293 = 37*89
f(51) = 1705 = 5*11*31
f(52) = 3529 = 3529
f(53) = 1825 = 5*5*73
f(54) = 3773 = 7*7*7*11
f(55) = 1949 = 1949
f(56) = 4025 = 5*5*7*23
f(57) = 2077 = 31*67
f(58) = 4285 = 5*857
f(59) = 2209 = 47*47
f(60) = 4553 = 29*157
f(61) = 2345 = 5*7*67
f(62) = 4829 = 11*439
f(63) = 2485 = 5*7*71
f(64) = 5113 = 5113
f(65) = 2629 = 11*239
f(66) = 5405 = 5*23*47
f(67) = 2777 = 2777
f(68) = 5705 = 5*7*163
f(69) = 2929 = 29*101
f(70) = 6013 = 7*859
f(71) = 3085 = 5*617
f(72) = 6329 = 6329
f(73) = 3245 = 5*11*59
f(74) = 6653 = 6653
f(75) = 3409 = 7*487
f(76) = 6985 = 5*11*127
f(77) = 3577 = 7*7*73
f(78) = 7325 = 5*5*293
f(79) = 3749 = 23*163
f(80) = 7673 = 7673
f(81) = 3925 = 5*5*157
f(82) = 8029 = 7*31*37
f(83) = 4105 = 5*821
f(84) = 8393 = 7*11*109
f(85) = 4289 = 4289
f(86) = 8765 = 5*1753
f(87) = 4477 = 11*11*37
f(88) = 9145 = 5*31*59
f(89) = 4669 = 7*23*29
f(90) = 9533 = 9533
f(91) = 4865 = 5*7*139
f(92) = 9929 = 9929
f(93) = 5065 = 5*1013
f(94) = 10333 = 10333
f(95) = 5269 = 11*479
f(96) = 10745 = 5*7*307
f(97) = 5477 = 5477
f(98) = 11165 = 5*7*11*29
f(99) = 5689 = 5689
f(100) = 11593 = 11593

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+16x-7

f(0)=7
f(1)=5
f(2)=29
f(3)=1
f(4)=73
f(5)=1
f(6)=1
f(7)=11
f(8)=37
f(9)=109
f(10)=23
f(11)=1
f(12)=47
f(13)=1
f(14)=59
f(15)=229
f(16)=101
f(17)=277
f(18)=1
f(19)=1
f(20)=31
f(21)=1
f(22)=829
f(23)=89
f(24)=953
f(25)=509
f(26)=1
f(27)=577
f(28)=1
f(29)=1
f(30)=1373
f(31)=1
f(32)=139
f(33)=1
f(34)=1693
f(35)=127
f(36)=373
f(37)=977
f(38)=409
f(39)=1069
f(40)=1
f(41)=233
f(42)=347
f(43)=1
f(44)=2633
f(45)=1
f(46)=569
f(47)=211
f(48)=613
f(49)=227
f(50)=1
f(51)=1
f(52)=3529
f(53)=1
f(54)=1
f(55)=1949
f(56)=1
f(57)=67
f(58)=857
f(59)=1
f(60)=157
f(61)=1
f(62)=439
f(63)=71
f(64)=5113
f(65)=239
f(66)=1
f(67)=2777
f(68)=163
f(69)=1
f(70)=859
f(71)=617
f(72)=6329
f(73)=1
f(74)=6653
f(75)=487
f(76)=1
f(77)=1
f(78)=293
f(79)=1
f(80)=7673
f(81)=1
f(82)=1
f(83)=821
f(84)=1
f(85)=4289
f(86)=1753
f(87)=1
f(88)=1
f(89)=1
f(90)=9533
f(91)=1
f(92)=9929
f(93)=1013
f(94)=10333
f(95)=479
f(96)=307
f(97)=5477
f(98)=1
f(99)=5689

b) Substitution of the polynom
The polynom f(x)=x^2+16x-7 could be written as f(y)= y^2-71 with x=y-8

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
f'(x)>2x+15

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

7, 5, 29, 1, 73, 1, 1, 11, 37, 109, 23, 1, 47, 1, 59, 229, 101, 277, 1, 1, 31, 1, 829, 89, 953, 509, 1, 577, 1, 1, 1373, 1, 139, 1, 1693, 127, 373, 977, 409, 1069, 1, 233, 347, 1, 2633, 1, 569, 211, 613, 227, 1, 1, 3529, 1, 1, 1949, 1, 67, 857, 1, 157, 1, 439, 71, 5113, 239, 1, 2777, 163, 1, 859, 617, 6329, 1, 6653, 487, 1, 1, 293, 1, 7673, 1, 1, 821, 1, 4289, 1753, 1, 1, 1, 9533, 1, 9929, 1013, 10333, 479, 307, 5477, 1, 5689, 11593, 1181, 523, 1, 12473, 907, 1, 6577, 2677, 619, 1979, 1409, 1, 1, 14813, 7529, 3061, 1, 1, 1, 1483, 1657, 16829, 1709, 1, 383, 1, 313, 1, 9349, 18973, 1, 331, 283, 1, 443, 4133, 10477, 607, 1, 3119, 2213, 2039, 2273, 743, 1667, 4729, 1, 1, 12289, 1, 2521, 521, 1, 3739, 13249, 1, 13577, 1, 1987, 599, 1, 1, 2917, 2683, 14929, 863, 15277, 883, 15629, 1, 1, 2939, 467, 33053, 1, 1, 17077, 1381, 17449, 5039, 1, 5147, 1, 36793, 641, 683, 2711, 7669, 2767, 39133, 1, 39929, 1, 1, 1, 1187, 1907, 1, 1, 919, 1, 44029, 1, 1951, 1, 1, 491, 1, 23509, 6779, 4789, 1559, 4877, 1697, 3547, 911, 1, 1, 2339, 51913, 5237, 7547, 1, 1097, 27109, 10937, 1, 1, 4007, 1, 1, 57529, 5801, 2017, 797, 1699, 967, 1, 30469, 2671, 563, 1, 1, 1, 4567, 12889, 691, 13093, 2999, 1, 6701, 877, 1361, 1459, 34549, 557, 5011, 1, 5087, 593, 7229, 1087, 1, 10559, 631, 2143, 1021, 1, 38329, 77213, 1, 1, 1, 1, 40009, 1, 40577, 1, 41149, 11839, 1669, 7639, 8461, 85193, 1, 751, 6211, 17509, 1, 997, 8933, 1, 823, 1, 45869, 1, 46477, 18713, 1, 94793, 1, 881, 1933, 1, 1579, 1, 4507, 2851, 1, 727, 10169, 102329, 1471, 103613, 677, 20981, 1, 1931, 1, 15359, 1, 2221, 10949, 1867, 827, 2027, 8011, 4513, 1, 1, 2297, 5023, 11621, 16699, 58789, 1, 5407, 23929, 60169, 11003, 1, 5323, 1759, 123833, 853, 1, 887, 1, 63689, 1, 1171, 129529, 1, 130973, 1, 5297, 9511, 26777, 1, 135353, 1, 1777, 13757, 19759, 3023, 27961, 2267, 1, 1, 12983, 1, 1429, 1319, 929, 73309, 4211, 74077, 1, 1, 150473, 1, 152029, 1, 13963, 11027, 31033, 77977, 1, 1, 22619, 15913, 1, 16073, 991, 1, 1, 1, 1063, 11827, 166393, 1, 168029, 1, 24239, 1, 1, 86077, 1193, 1, 174653, 1, 176329, 2531, 16183, 2417, 1, 1, 1, 91129, 1, 18397, 184829, 1, 8111, 1217, 37657, 1, 1, 3079, 191773, 3853, 27647, 19441, 1213, 2087, 3583, 3413, 39769, 1297, 1, 2879, 1, 20333, 204233, 941, 1, 1, 5939, 1, 1733, 21061, 5717, 1, 6883, 15307, 8609, 1, 3947, 1627, 1009, 1999, 31547, 1, 222713, 111829, 1549, 16111, 1, 1, 9931, 22937, 20939, 1, 33179, 4021, 6691, 117577, 1889, 118549, 1, 1, 5107, 1, 241993, 3919, 1, 122477, 7027, 1, 35419, 1, 1, 1091, 1, 1, 50789, 18211, 51193, 128489, 257993, 1, 1, 1, 1291, 11959, 2113, 1, 1, 19087, 268253, 3847, 270329, 2467, 1, 1873, 1, 137777, 1129, 138829, 2557, 1, 9059, 4027, 1, 20287, 1, 13007, 11489, 3067, 1, 1, 41647, 1, 1, 13399, 1259, 21211, 5419, 1, 300233, 30133, 302429, 1, 6217, 2591, 1, 4967, 61813, 1, 311293, 4463, 1997, 1, 2897, 158449, 12721, 1, 9151, 160709, 1, 32369, 1, 1, 4481, 23447, 65881, 3373, 1, 166429, 1583, 1, 2089, 33749, 1, 169909, 1451, 171077, 1, 2237, 345673, 1, 1, 1, 1, 7643, 10079, 176977, 1, 178169, 32503, 1237, 359929, 1, 362333, 1, 72949, 2731, 73433, 184189, 1427, 3371, 53147, 1493, 1, 5077, 15077, 27011, 3299, 1, 381853, 1321, 34939, 38557, 55259, 1, 1, 195277, 3407, 3331, 13597, 5651, 3929, 1, 399353, 1, 7307, 201577, 2311, 1861, 1, 1, 409529, 41081, 37463, 29527, 1, 1, 83449, 209269, 1, 1831, 1, 42373, 60719, 19379, 1, 1543, 7823, 1, 432893, 6203, 435529, 8737, 19051, 219749, 1, 221077, 12671, 20219, 6659, 1, 448829, 1, 1, 4621, 90841, 20707, 91381, 7901, 1, 1487, 66047, 1, 12569, 2309, 93557, 1, 1, 1, 2243, 1, 476029, 47741, 68399, 240089, 13759, 10499, 1, 22079, 487133, 1, 4049, 7019, 1, 247069, 99109, 248477, 1, 1, 1, 50261, 1, 1, 506873, 36307, 20389, 1, 3307, 1637, 1, 1, 74047, 51977, 1, 261329, 2833, 5591, 105401, 37747, 3251, 7591, 1, 1, 11399, 24419, 15391, 1, 1, 271549, 2399, 1, 547529, 1, 7541, 1, 1, 4703, 1, 8999, 1, 56093, 80347, 56393, 4673, 1, 4943, 3701, 3089, 40927, 574493, 57601, 577529, 1, 82939, 26459, 1, 292577, 10667, 1, 589753, 8447, 6661, 1, 1, 10301, 10891, 300277, 17203, 1, 2789, 1, 5581, 1, 611453, 43787, 1, 4001, 24709, 13463, 56443, 1, 1, 1, 89599, 1, 1, 315977, 11519, 6481, 17209, 1, 27823, 64153, 1, 10399, 1, 2551, 1, 29599, 1, 2111, 2593, 1879, 1, 47207, 26501, 4549, 133157, 1, 8689, 1, 13721, 1, 23297, 14723, 135781, 48611, 2903, 48847, 685513, 6247, 18617, 2381, 1, 15083, 1, 9421, 27953, 350249, 702173, 2011, 2069, 10103, 708893, 32299, 142453, 356977, 1, 358669, 1741, 72073, 1, 1, 10223, 1, 13259, 1, 1, 367189, 736093, 1, 3643, 1, 9649, 6311, 1, 1, 3191, 1, 753353, 1, 756829, 2447, 20549, 1, 1, 382777, 1993, 1, 770813, 3359, 26701, 11087, 1, 1, 14207, 3877, 31397, 35759, 1, 1, 113147, 1, 34591, 1, 159833, 1, 5179, 57467, 73303, 1, 809929, 81173, 1, 407669, 1, 409477, 14923, 6971, 824393, 1, 828029, 2371, 1, 8867, 33413, 18199, 23971, 38219, 1, 84449, 1, 1, 14407, 1, 170741, 2657, 171481, 13859, 78283, 86297, 123547, 7879, 1, 1, 174457, 437077, 1, 1, 1, 1, 13187, 2393, 7333, 444589, 3637, 446477, 1, 15461, 1, 90053, 82039, 12919, 906233, 5897, 182009, 455977, 182773, 457889, 4229, 91961, 2801, 1, 925373, 463649, 1, 1, 3163, 1, 936953, 93889, 940829, 4099, 12269, 3727, 27103, 43207, 6569, 477229, 25849, 13691, 960329, 1, 964253, 1, 8419, 1, 1, 487049, 139439, 1, 3463, 1, 983993, 10061, 1, 2281, 6841, 45179, 9137, 99793, 1, 100193, 143419, 7507, 201589, 1, 202393, 1,

6. Sequence of the polynom (only primes)

7, 5, 29, 73, 11, 37, 109, 23, 47, 59, 229, 101, 277, 31, 829, 89, 953, 509, 577, 1373, 139, 1693, 127, 373, 977, 409, 1069, 233, 347, 2633, 569, 211, 613, 227, 3529, 1949, 67, 857, 157, 439, 71, 5113, 239, 2777, 163, 859, 617, 6329, 6653, 487, 293, 7673, 821, 4289, 1753, 9533, 9929, 1013, 10333, 479, 307, 5477, 5689, 11593, 1181, 523, 12473, 907, 6577, 2677, 619, 1979, 1409, 14813, 7529, 3061, 1483, 1657, 16829, 1709, 383, 313, 9349, 18973, 331, 283, 443, 4133, 10477, 607, 3119, 2213, 2039, 2273, 743, 1667, 4729, 12289, 2521, 521, 3739, 13249, 13577, 1987, 599, 2917, 2683, 14929, 863, 15277, 883, 15629, 2939, 467, 33053, 17077, 1381, 17449, 5039, 5147, 36793, 641, 683, 2711, 7669, 2767, 39133, 39929, 1187, 1907, 919, 44029, 1951, 491, 23509, 6779, 4789, 1559, 4877, 1697, 3547, 911, 2339, 51913, 5237, 7547, 1097, 27109, 10937, 4007, 57529, 5801, 2017, 797, 1699, 967, 30469, 2671, 563, 4567, 12889, 691, 13093, 2999, 6701, 877, 1361, 1459, 34549, 557, 5011, 5087, 593, 7229, 1087, 10559, 631, 2143, 1021, 38329, 77213, 40009, 40577, 41149, 11839, 1669, 7639, 8461, 85193, 751, 6211, 17509, 997, 8933, 823, 45869, 46477, 18713, 94793, 881, 1933, 1579, 4507, 2851, 727, 10169, 102329, 1471, 103613, 677, 20981, 1931, 15359, 2221, 10949, 1867, 827, 2027, 8011, 4513, 2297, 5023, 11621, 16699, 58789, 5407, 23929, 60169, 11003, 5323, 1759, 123833, 853, 887, 63689, 1171, 129529, 130973, 5297, 9511, 26777, 135353, 1777, 13757, 19759, 3023, 27961, 2267, 12983, 1429, 1319, 929, 73309, 4211, 74077, 150473, 152029, 13963, 11027, 31033, 77977, 22619, 15913, 16073, 991, 1063, 11827, 166393, 168029, 24239, 86077, 1193, 174653, 176329, 2531, 16183, 2417, 91129, 18397, 184829, 8111, 1217, 37657, 3079, 191773, 3853, 27647, 19441, 1213, 2087, 3583, 3413, 39769, 1297, 2879, 20333, 204233, 941, 5939, 1733, 21061, 5717, 6883, 15307, 8609, 3947, 1627, 1009, 1999, 31547, 222713, 111829, 1549, 16111, 9931, 22937, 20939, 33179, 4021, 6691, 117577, 1889, 118549, 5107, 241993, 3919, 122477, 7027, 35419, 1091, 50789, 18211, 51193, 128489, 257993, 1291, 11959, 2113, 19087, 268253, 3847, 270329, 2467, 1873, 137777, 1129, 138829, 2557, 9059, 4027, 20287, 13007, 11489, 3067, 41647, 13399, 1259, 21211, 5419, 300233, 30133, 302429, 6217, 2591, 4967, 61813, 311293, 4463, 1997, 2897, 158449, 12721, 9151, 160709, 32369, 4481, 23447, 65881, 3373, 166429, 1583, 2089, 33749, 169909, 1451, 171077, 2237, 345673, 7643, 10079, 176977, 178169, 32503, 1237, 359929, 362333, 72949, 2731, 73433, 184189, 1427, 3371, 53147, 1493, 5077, 15077, 27011, 3299, 381853, 1321, 34939, 38557, 55259, 195277, 3407, 3331, 13597, 5651, 3929, 399353, 7307, 201577, 2311, 1861, 409529, 41081, 37463, 29527, 83449, 209269, 1831, 42373, 60719, 19379, 1543, 7823, 432893, 6203, 435529, 8737, 19051, 219749, 221077, 12671, 20219, 6659, 448829, 4621, 90841, 20707, 91381, 7901, 1487, 66047, 12569, 2309, 93557, 2243, 476029, 47741, 68399, 240089, 13759, 10499, 22079, 487133, 4049, 7019, 247069, 99109, 248477, 50261, 506873, 36307, 20389, 3307, 1637, 74047, 51977, 261329, 2833, 5591, 105401, 37747, 3251, 7591, 11399, 24419, 15391, 271549, 2399, 547529, 7541, 4703, 8999, 56093, 80347, 56393, 4673, 4943, 3701, 3089, 40927, 574493, 57601, 577529, 82939, 26459, 292577, 10667, 589753, 8447, 6661, 10301, 10891, 300277, 17203, 2789, 5581, 611453, 43787, 4001, 24709, 13463, 56443, 89599, 315977, 11519, 6481, 17209, 27823, 64153, 10399, 2551, 29599, 2111, 2593, 1879, 47207, 26501, 4549, 133157, 8689, 13721, 23297, 14723, 135781, 48611, 2903, 48847, 685513, 6247, 18617, 2381, 15083, 9421, 27953, 350249, 702173, 2011, 2069, 10103, 708893, 32299, 142453, 356977, 358669, 1741, 72073, 10223, 13259, 367189, 736093, 3643, 9649, 6311, 3191, 753353, 756829, 2447, 20549, 382777, 1993, 770813, 3359, 26701, 11087, 14207, 3877, 31397, 35759, 113147, 34591, 159833, 5179, 57467, 73303, 809929, 81173, 407669, 409477, 14923, 6971, 824393, 828029, 2371, 8867, 33413, 18199, 23971, 38219, 84449, 14407, 170741, 2657, 171481, 13859, 78283, 86297, 123547, 7879, 174457, 437077, 13187, 2393, 7333, 444589, 3637, 446477, 15461, 90053, 82039, 12919, 906233, 5897, 182009, 455977, 182773, 457889, 4229, 91961, 2801, 925373, 463649, 3163, 936953, 93889, 940829, 4099, 12269, 3727, 27103, 43207, 6569, 477229, 25849, 13691, 960329, 964253, 8419, 487049, 139439, 3463, 983993, 10061, 2281, 6841, 45179, 9137, 99793, 100193, 143419, 7507, 201589, 202393,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+16x-7 and
the reducible primes which appear as divisor for the first time
p | x^2+16x-7 and p < x^2+16x-7

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)
1107520.7000000.5000000.2000000.0000000.0000000.000000
21006030300.6000000.3000000.3000008.5714286.00000015.000000
31.0006721744980.6720000.1740000.49800011.2000005.80000016.600000
410.0006.7451.2405.5050.6745000.1240000.55050010.0372037.12643711.054217
5100.00068.0589.52758.5310.6805800.0952700.58531010.0901417.68306410.632335
61.000.000681.98377.653604.3300.6819830.0776530.60433010.0206158.15083410.324956
710.000.0006.834.589654.7606.179.8290.6834590.0654760.61798310.0216428.43187010.225918
8100.000.00068.452.5695.671.27162.781.2980.6845260.0567130.62781310.0156098.66160310.159067
91.000.000.000685.370.55449.997.456635.373.0980.6853710.0499970.63537310.0123428.81591710.120420
1010.000.000.0006.860.707.384447.074.6486.413.632.7360.6860710.0447070.64136310.0102168.94194810.094277


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
244401.0000001.0000000.0000001.3333331.333333-nan
385410.6250000.5000000.1250001.2500001.000000inf
41611650.6875000.3750000.3125002.2000001.5000005.000000
532191270.5937500.3750000.2187501.7272732.0000001.400000
6643819190.5937500.2968750.2968752.0000001.5833332.714286
71287835430.6093750.2734380.3359382.0526321.8421052.263158
8256166571090.6484380.2226560.4257812.1282051.6285712.534884
9512338992390.6601560.1933590.4667972.0361441.7368422.192661
101.0246871775100.6708980.1728520.4980472.0325441.7878792.133891
112.0481.3673321.0350.6674800.1621090.5053711.9898111.8757062.029412
124.0962.7605892.1710.6738280.1437990.5300292.0190201.7740962.097584
138.1925.5261.0404.4860.6745610.1269530.5476072.0021741.7657052.066329
1416.38411.0911.9229.1690.6769410.1173100.5596312.0070571.8480772.043914
1532.76822.2393.50918.7300.6786800.1070860.5715942.0051391.8257022.042753
1665.53644.5446.52238.0220.6796880.0995180.5801702.0029681.8586492.030005
17131.07289.24012.12877.1120.6808470.0925290.5883182.0034121.8595522.028089
18262.144178.47822.786155.6920.6808400.0869220.5939181.9999781.8787932.019037
19524.288357.39142.926314.4650.6816690.0818750.5997942.0024371.8838762.019789
201.048.576715.24881.094634.1540.6821140.0773370.6047762.0013041.8891582.016612
212.097.1521.431.444153.6441.277.8000.6825660.0732630.6093032.0013251.8946412.014968
224.194.3042.864.643291.7032.572.9400.6829840.0695470.6134372.0012261.8985642.013570
238.388.6085.732.389556.0435.176.3460.6833540.0662850.6170692.0010831.9061962.011841
2416.777.21611.470.5231.062.95210.407.5710.6836960.0633570.6203402.0010021.9116362.010602
2533.554.43222.952.7652.033.37620.919.3890.6840460.0605990.6234462.0010221.9129522.010016
2667.108.86445.925.4453.897.08742.028.3580.6843420.0580710.6262712.0008681.9165602.009063
27134.217.72891.892.2947.482.63584.409.6590.6846510.0557500.6289012.0009021.9200582.008398
28268.435.456183.855.43114.393.790169.461.6410.6849150.0536210.6312942.0007711.9236262.007610
29536.870.912367.841.07727.728.255340.112.8220.6851570.0516480.6335102.0007081.9264042.007020
301.073.741.824735.940.22953.488.637682.451.5920.6853980.0498150.6355832.0007011.9290302.006545
312.147.483.6481.472.361.267103.312.7201.369.048.5470.6856220.0481090.6375132.0006531.9314892.006074
324.294.967.2962.945.621.656199.775.8162.745.845.8400.6858310.0465140.6393172.0006111.9337002.005660
338.589.934.5925.892.936.907386.742.3205.506.194.5870.6860280.0450230.6410052.0005751.9358822.005282
3417.179.869.18411.789.133.989749.423.89911.039.710.0900.6862180.0436220.6425962.0005531.9377862.004962
3534.359.738.36823.584.484.8461.453.615.88922.130.868.9570.6863990.0423060.6440932.0005271.9396442.004660
3668.719.476.73647.180.785.9142.822.109.57844.358.676.3360.6865710.0410670.6455042.0005011.9414412.004380


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
123120021
244221021
384221021
4166421041
53212753081
6641911870111
7128351520170171
8256573324260301
9512995346490491
101.0241778790850911
112.04833216716516301681
124.09658929029929602921
138.1921.04052551552505141
1416.3841.92297594795309681
1532.7683.5091.7671.7421.72701.7811
1665.5366.5223.2773.2453.24903.2721
17131.07212.1286.0876.0416.11606.0111
18262.14422.78611.52511.26111.402011.3831
19524.28842.92621.62221.30421.470021.4551
201.048.57681.09440.77740.31740.576040.5171
212.097.152153.64477.21976.42576.843076.8001
224.194.304291.703146.626145.077145.8210145.8811
238.388.608556.043279.168276.875277.8730278.1691
2416.777.2161.062.952533.377529.575531.6600531.2911
2533.554.4322.033.3761.020.1681.013.2081.016.94201.016.4331
2667.108.8643.897.0871.956.1261.940.9611.949.18401.947.9021
27134.217.7287.482.6353.754.9233.727.7123.742.31503.740.3191
28268.435.45614.393.7907.221.3007.172.4907.197.18407.196.6051
29536.870.91227.728.25513.908.55613.819.69913.863.643013.864.6111
301.073.741.82453.488.63726.824.73226.663.90526.743.507026.745.1291
312.147.483.648103.312.72051.804.97551.507.74551.658.219051.654.5001
324.294.967.296199.775.816100.160.06399.615.75399.888.816099.886.9991
338.589.934.592386.742.320193.881.451192.860.869193.371.8190193.370.5001
3417.179.869.184749.423.899375.670.217373.753.682374.712.0760374.711.8221
3534.359.738.3681.453.615.889728.619.864724.996.025726.811.3420726.804.5461
3668.719.476.7362.822.109.5781.414.479.1791.407.630.3991.411.040.01601.411.069.5611


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
381100010
4165140122
5327251222
664199105554
71284322211013128
8256109496022362526
951223912411550735363
101.024510251259111148113138
112.0481.035497538240288237270
124.0962.1711.0401.131503585507576
138.1924.4862.2032.2831.0591.1891.0641.174
1416.3849.1694.5494.6202.1882.3952.2132.373
1532.76818.7309.3089.4224.4604.8604.5374.873
1665.53638.02218.94519.0779.1279.9649.1739.758
17131.07277.11238.54338.56918.58519.98118.68019.866
18262.144155.69277.79177.90137.62840.08537.80540.174
19524.288314.465156.910157.55576.51980.75876.38280.806
201.048.576634.154316.973317.181154.742162.587154.393162.432
212.097.1521.277.800638.812638.988311.953326.631311.688327.528
224.194.3042.572.9401.286.4041.286.536629.313657.372627.915658.340
238.388.6085.176.3462.587.9422.588.4041.267.2591.320.5671.266.5001.322.020
2416.777.21610.407.5715.201.0235.206.5482.550.6202.652.2072.550.8902.653.854
2533.554.43220.919.38910.455.57810.463.8115.133.0675.327.1095.133.6015.325.612
2667.108.86442.028.35821.009.03821.019.32010.324.66810.692.31810.323.56210.687.810
27134.217.72884.409.65942.201.96042.207.69920.753.43121.454.03020.754.86221.447.336
28268.435.456169.461.64184.725.26384.736.37841.700.73343.039.81141.695.00743.026.090
29536.870.912340.112.822170.048.492170.064.33083.749.28686.304.86583.755.81786.302.854
301.073.741.824682.451.592341.194.154341.257.438168.160.492173.069.626168.161.315173.060.159
312.147.483.6481.369.048.547684.470.594684.577.953337.551.690347.000.451337.545.877346.950.529
324.294.967.2962.745.845.8401.372.803.5621.373.042.278677.386.343695.556.129677.383.640695.519.728
338.589.934.5925.506.194.5872.752.831.6892.753.362.8981.359.062.4661.394.088.9211.359.043.3631.393.999.837
3417.179.869.18411.039.710.0905.519.406.0225.520.304.0682.726.157.0362.793.681.6392.726.223.4302.793.647.985
3534.359.738.36822.130.868.95711.064.597.88611.066.271.0715.467.481.4665.597.997.1345.467.502.1165.597.888.241
3668.719.476.73644.358.676.33622.177.712.25522.180.964.08110.963.405.69011.216.046.55410.963.376.91911.215.847.173


8. Check for existing Integer Sequences by OEIS

Found in Database : 7, 5, 29, 1, 73, 1, 1, 11, 37, 109, 23, 1, 47, 1, 59, 229, 101, 277, 1, 1,
Found in Database : 7, 5, 29, 73, 11, 37, 109, 23, 47, 59, 229, 101, 277, 31, 829, 89, 953, 509, 577, 1373, 139, 1693, 127, 373, 977, 409, 1069,
Found in Database : 5, 7, 11, 23, 29, 31, 37, 47, 59, 67, 71, 73, 89, 101, 109, 127, 139,