Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:08:48
Deutsch
29.Mar 2024

Polynom = x^2+56x-37

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) = 37 = 37
f(1) = 5 = 5
f(2) = 79 = 79
f(3) = 35 = 5*7
f(4) = 203 = 7*29
f(5) = 67 = 67
f(6) = 335 = 5*67
f(7) = 101 = 101
f(8) = 475 = 5*5*19
f(9) = 137 = 137
f(10) = 623 = 7*89
f(11) = 175 = 5*5*7
f(12) = 779 = 19*41
f(13) = 215 = 5*43
f(14) = 943 = 23*41
f(15) = 257 = 257
f(16) = 1115 = 5*223
f(17) = 301 = 7*43
f(18) = 1295 = 5*7*37
f(19) = 347 = 347
f(20) = 1483 = 1483
f(21) = 395 = 5*79
f(22) = 1679 = 23*73
f(23) = 445 = 5*89
f(24) = 1883 = 7*269
f(25) = 497 = 7*71
f(26) = 2095 = 5*419
f(27) = 551 = 19*29
f(28) = 2315 = 5*463
f(29) = 607 = 607
f(30) = 2543 = 2543
f(31) = 665 = 5*7*19
f(32) = 2779 = 7*397
f(33) = 725 = 5*5*29
f(34) = 3023 = 3023
f(35) = 787 = 787
f(36) = 3275 = 5*5*131
f(37) = 851 = 23*37
f(38) = 3535 = 5*7*101
f(39) = 917 = 7*131
f(40) = 3803 = 3803
f(41) = 985 = 5*197
f(42) = 4079 = 4079
f(43) = 1055 = 5*211
f(44) = 4363 = 4363
f(45) = 1127 = 7*7*23
f(46) = 4655 = 5*7*7*19
f(47) = 1201 = 1201
f(48) = 4955 = 5*991
f(49) = 1277 = 1277
f(50) = 5263 = 19*277
f(51) = 1355 = 5*271
f(52) = 5579 = 7*797
f(53) = 1435 = 5*7*41
f(54) = 5903 = 5903
f(55) = 1517 = 37*41
f(56) = 6235 = 5*29*43
f(57) = 1601 = 1601
f(58) = 6575 = 5*5*263
f(59) = 1687 = 7*241
f(60) = 6923 = 7*23*43
f(61) = 1775 = 5*5*71
f(62) = 7279 = 29*251
f(63) = 1865 = 5*373
f(64) = 7643 = 7643
f(65) = 1957 = 19*103
f(66) = 8015 = 5*7*229
f(67) = 2051 = 7*293
f(68) = 8395 = 5*23*73
f(69) = 2147 = 19*113
f(70) = 8783 = 8783
f(71) = 2245 = 5*449
f(72) = 9179 = 67*137
f(73) = 2345 = 5*7*67
f(74) = 9583 = 7*37*37
f(75) = 2447 = 2447
f(76) = 9995 = 5*1999
f(77) = 2551 = 2551
f(78) = 10415 = 5*2083
f(79) = 2657 = 2657
f(80) = 10843 = 7*1549
f(81) = 2765 = 5*7*79
f(82) = 11279 = 11279
f(83) = 2875 = 5*5*5*23
f(84) = 11723 = 19*617
f(85) = 2987 = 29*103
f(86) = 12175 = 5*5*487
f(87) = 3101 = 7*443
f(88) = 12635 = 5*7*19*19
f(89) = 3217 = 3217
f(90) = 13103 = 13103
f(91) = 3335 = 5*23*29
f(92) = 13579 = 37*367
f(93) = 3455 = 5*691
f(94) = 14063 = 7*7*7*41
f(95) = 3577 = 7*7*73
f(96) = 14555 = 5*41*71
f(97) = 3701 = 3701
f(98) = 15055 = 5*3011
f(99) = 3827 = 43*89
f(100) = 15563 = 79*197

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+56x-37

f(0)=37
f(1)=5
f(2)=79
f(3)=7
f(4)=29
f(5)=67
f(6)=1
f(7)=101
f(8)=19
f(9)=137
f(10)=89
f(11)=1
f(12)=41
f(13)=43
f(14)=23
f(15)=257
f(16)=223
f(17)=1
f(18)=1
f(19)=347
f(20)=1483
f(21)=1
f(22)=73
f(23)=1
f(24)=269
f(25)=71
f(26)=419
f(27)=1
f(28)=463
f(29)=607
f(30)=2543
f(31)=1
f(32)=397
f(33)=1
f(34)=3023
f(35)=787
f(36)=131
f(37)=1
f(38)=1
f(39)=1
f(40)=3803
f(41)=197
f(42)=4079
f(43)=211
f(44)=4363
f(45)=1
f(46)=1
f(47)=1201
f(48)=991
f(49)=1277
f(50)=277
f(51)=271
f(52)=797
f(53)=1
f(54)=5903
f(55)=1
f(56)=1
f(57)=1601
f(58)=263
f(59)=241
f(60)=1
f(61)=1
f(62)=251
f(63)=373
f(64)=7643
f(65)=103
f(66)=229
f(67)=293
f(68)=1
f(69)=113
f(70)=8783
f(71)=449
f(72)=1
f(73)=1
f(74)=1
f(75)=2447
f(76)=1999
f(77)=2551
f(78)=2083
f(79)=2657
f(80)=1549
f(81)=1
f(82)=11279
f(83)=1
f(84)=617
f(85)=1
f(86)=487
f(87)=443
f(88)=1
f(89)=3217
f(90)=13103
f(91)=1
f(92)=367
f(93)=691
f(94)=1
f(95)=1
f(96)=1
f(97)=3701
f(98)=3011
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+56x-37 could be written as f(y)= y^2-821 with x=y-28

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+28
f'(x)>2x+55

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

37, 5, 79, 7, 29, 67, 1, 101, 19, 137, 89, 1, 41, 43, 23, 257, 223, 1, 1, 347, 1483, 1, 73, 1, 269, 71, 419, 1, 463, 607, 2543, 1, 397, 1, 3023, 787, 131, 1, 1, 1, 3803, 197, 4079, 211, 4363, 1, 1, 1201, 991, 1277, 277, 271, 797, 1, 5903, 1, 1, 1601, 263, 241, 1, 1, 251, 373, 7643, 103, 229, 293, 1, 113, 8783, 449, 1, 1, 1, 2447, 1999, 2551, 2083, 2657, 1549, 1, 11279, 1, 617, 1, 487, 443, 1, 3217, 13103, 1, 367, 691, 1, 1, 1, 3701, 3011, 1, 1, 1, 2297, 1, 16603, 4217, 149, 1, 1, 641, 18223, 1, 1, 953, 1, 701, 569, 5051, 4099, 5197, 727, 1069, 163, 157, 22283, 5647, 1, 5801, 4703, 1, 3449, 1223, 349, 1, 25423, 1, 1, 1, 5347, 1, 409, 1, 653, 1, 587, 383, 1, 7451, 1, 1, 4409, 1, 1373, 1597, 32303, 8167, 6607, 1193, 193, 8537, 1, 1, 35279, 1783, 1, 1301, 199, 1, 1, 9497, 1, 1, 1, 1979, 39983, 439, 1, 10301, 1, 1, 42443, 2143, 1, 1, 44123, 1, 1, 11351, 1, 1, 46703, 2357, 971, 1, 48463, 12227, 9871, 12451, 1, 1811, 7309, 1, 2741, 1, 53003, 13367, 1, 1, 1, 1, 55823, 563, 56779, 1, 1, 14557, 11743, 1, 11939, 1, 8669, 1, 1667, 3109, 62683, 15797, 12739, 2293, 1, 709, 2267, 3313, 1553, 673, 9689, 2441, 1, 17351, 1, 1, 1, 1, 1471, 3631, 3181, 18427, 14851, 18701, 2153, 2711, 76463, 3851, 3373, 3907, 1, 1, 2281, 20101, 1, 1, 2003, 827, 11897, 599, 84443, 733, 17123, 937, 17359, 3121, 12569, 1, 1, 1, 1, 1, 2617, 1, 977, 23357, 1, 4733, 95279, 1, 13789, 1, 3911, 337, 683, 24917, 1, 1, 647, 1, 3547, 1, 1, 1, 1, 1, 1, 1, 108079, 5437, 15629, 3931, 22147, 27851, 4483, 1, 1123, 1, 863, 1, 1, 29207, 1237, 1019, 1, 4271, 120283, 1, 1, 1, 123083, 4421, 3557, 1, 25183, 31657, 127343, 1, 18397, 1, 130223, 1723, 1, 1, 26627, 1, 1, 1, 3319, 6841, 5981, 1, 1, 4993, 28111, 35327, 7477, 1, 4951, 1031, 1091, 36467, 29327, 857, 5927, 1619, 1, 1, 151279, 7603, 152843, 1, 1, 1, 4457, 2063, 4259, 7919, 159179, 421, 1, 1, 32479, 40801, 32803, 1, 2333, 1, 1039, 1, 1, 42437, 6823, 1, 1, 883, 7561, 8737, 9241, 8821, 1721, 6361, 5113, 1, 36131, 45377, 4241, 9161, 26297, 1321, 1, 2029, 37507, 1, 1, 6791, 941, 1, 1, 1, 194543, 48857, 1, 7043, 39619, 49747, 1459, 10039, 4919, 1447, 1, 1381, 2161, 51551, 1801, 1, 1571, 1499, 1609, 1, 1427, 1, 1, 1, 1, 1873, 1931, 10957, 2179, 1, 1, 1, 44771, 1307, 1, 1, 227663, 1, 32797, 11527, 231503, 1, 46687, 58601, 1, 1, 3251, 2383, 1, 1, 12697, 1, 6949, 61051, 1, 61547, 247183, 12409, 35597, 1787, 1, 1, 1, 1, 51043, 9151, 36749, 1, 11273, 1, 261323, 65587, 1, 1, 53087, 66617, 6221, 1, 269579, 1933, 1, 1, 54751, 1, 55171, 1871, 39709, 1993, 14741, 14057, 6883, 3079, 1, 10193, 1637, 71887, 1, 2897, 2879, 14593, 41849, 10501, 59023, 74051, 1607, 74597, 1, 1, 1, 15139, 1327, 4013, 2111, 76801, 1, 1, 1, 15583, 312779, 1, 1493, 1613, 1, 79601, 63907, 80167, 16937, 1, 1, 1, 1, 1997, 65731, 2011, 66191, 1, 47609, 1, 335579, 1, 337903, 1, 9721, 1, 1, 4523, 2633, 3461, 1291, 1, 1, 1, 3061, 88301, 70879, 88897, 50969, 2557, 8353, 1, 1, 90697, 1693, 13043, 1, 1259, 3581, 18503, 19541, 1, 1, 1913, 1, 94351, 1847, 4129, 381103, 2731, 1481, 1, 2459, 96827, 77711, 1, 11173, 14011, 3821, 1, 396079, 19867, 1439, 14281, 1, 3469, 1, 1511, 10979, 1, 2539, 1, 1, 2399, 4357, 103801, 83299, 1, 1, 21019, 1, 21149, 1, 106397, 12197, 1, 85903, 1, 432143, 21673, 2207, 1, 1, 1, 1, 677, 88547, 5843, 63629, 3191, 15451, 1, 4463, 113027, 1, 1, 13033, 114377, 458863, 23011, 6323, 1, 3491, 16631, 93407, 117101, 1, 117787, 472523, 1, 1579, 23833, 6733, 4133, 1, 120551, 1, 17321, 11863, 1, 489179, 1, 491983, 1, 1, 6529, 99523, 1709, 500443, 1, 10271, 1, 13679, 1, 20359, 127601, 102367, 1, 73529, 1, 27241, 25951, 1, 130477, 1, 18743, 105251, 131927, 1, 1, 3571, 1, 3323, 3119, 1, 134851, 4327, 3307, 77689, 1, 23773, 1, 5443, 7253, 110543, 19793, 15877, 139297, 558683, 757, 6311, 1, 80669, 1, 113539, 1, 114143, 4933, 30197, 1, 1, 5783, 30517, 1, 4663, 1, 16741, 20981, 15919, 29527, 8837, 1, 13841, 1, 17093, 1, 2797, 7933, 1, 1, 1, 1, 610703, 153067, 1, 1, 1, 22091, 88589, 6217, 623279, 1, 626443, 157007, 17989, 22543, 6661, 158597, 635983, 1, 33641, 1, 1, 160997, 129119, 4373, 129763, 1, 1901, 1, 2719, 6569, 1, 165037, 1, 1, 19001, 166667, 668303, 1, 8501, 821, 1, 1, 5897, 169951, 1867, 170777, 4201, 4903, 98297, 34487, 1, 173267, 1, 174101, 3989, 1, 1, 1, 704779, 35323, 19139, 1, 1, 178301, 142979, 7789, 3221, 35999, 1, 5167, 8147, 1, 1, 7937, 146383, 1, 1, 1, 1, 1, 1, 185987, 4261, 26693, 149827, 6473, 752603, 37717, 1, 5413, 5711, 6563, 1, 1, 8069, 1, 4783, 1, 773579, 38767, 777103, 194717, 156127, 27943, 4481, 1, 6971, 1, 1, 2087, 1, 1, 159683, 10529, 5531, 5431, 805583, 1, 115597, 1, 28027, 4967, 2437, 1, 3347, 1, 823643, 1, 43541, 8291, 830923, 29741, 1, 1, 1, 2039, 10657, 42187, 1, 6053, 6199, 212777, 7417, 7369, 2347, 30661, 1, 2269, 864079, 1, 867803, 11443, 1, 31193, 1, 2129, 879023, 1, 882779, 1, 3089, 222107, 1, 223051, 178819, 9739, 1, 6427, 901679, 45179, 47657, 6131, 6271, 4649, 3727, 228757, 4751, 45943, 31751, 9227, 5743, 33091, 37139, 1, 1, 1, 936203, 6701, 5839, 1, 1, 236477, 1, 2351, 1, 34061, 12097, 47881, 959579, 1, 2081, 1, 1, 1, 1, 5659, 975323, 1, 1, 1, 983243, 10709, 1, 247301, 1, 1, 142169, 1, 5021, 1, 1, 251297, 4111, 1, 202243, 253307, 1, 2677, 4451, 1, 1, 256337, 8219, 257351, 8969, 2903, 5101, 7411, 10093, 52081, 1043663, 261427, 1, 37493, 1, 7121,

6. Sequence of the polynom (only primes)

37, 5, 79, 7, 29, 67, 101, 19, 137, 89, 41, 43, 23, 257, 223, 347, 1483, 73, 269, 71, 419, 463, 607, 2543, 397, 3023, 787, 131, 3803, 197, 4079, 211, 4363, 1201, 991, 1277, 277, 271, 797, 5903, 1601, 263, 241, 251, 373, 7643, 103, 229, 293, 113, 8783, 449, 2447, 1999, 2551, 2083, 2657, 1549, 11279, 617, 487, 443, 3217, 13103, 367, 691, 3701, 3011, 2297, 16603, 4217, 149, 641, 18223, 953, 701, 569, 5051, 4099, 5197, 727, 1069, 163, 157, 22283, 5647, 5801, 4703, 3449, 1223, 349, 25423, 5347, 409, 653, 587, 383, 7451, 4409, 1373, 1597, 32303, 8167, 6607, 1193, 193, 8537, 35279, 1783, 1301, 199, 9497, 1979, 39983, 439, 10301, 42443, 2143, 44123, 11351, 46703, 2357, 971, 48463, 12227, 9871, 12451, 1811, 7309, 2741, 53003, 13367, 55823, 563, 56779, 14557, 11743, 11939, 8669, 1667, 3109, 62683, 15797, 12739, 2293, 709, 2267, 3313, 1553, 673, 9689, 2441, 17351, 1471, 3631, 3181, 18427, 14851, 18701, 2153, 2711, 76463, 3851, 3373, 3907, 2281, 20101, 2003, 827, 11897, 599, 84443, 733, 17123, 937, 17359, 3121, 12569, 2617, 977, 23357, 4733, 95279, 13789, 3911, 337, 683, 24917, 647, 3547, 108079, 5437, 15629, 3931, 22147, 27851, 4483, 1123, 863, 29207, 1237, 1019, 4271, 120283, 123083, 4421, 3557, 25183, 31657, 127343, 18397, 130223, 1723, 26627, 3319, 6841, 5981, 4993, 28111, 35327, 7477, 4951, 1031, 1091, 36467, 29327, 857, 5927, 1619, 151279, 7603, 152843, 4457, 2063, 4259, 7919, 159179, 421, 32479, 40801, 32803, 2333, 1039, 42437, 6823, 883, 7561, 8737, 9241, 8821, 1721, 6361, 5113, 36131, 45377, 4241, 9161, 26297, 1321, 2029, 37507, 6791, 941, 194543, 48857, 7043, 39619, 49747, 1459, 10039, 4919, 1447, 1381, 2161, 51551, 1801, 1571, 1499, 1609, 1427, 1873, 1931, 10957, 2179, 44771, 1307, 227663, 32797, 11527, 231503, 46687, 58601, 3251, 2383, 12697, 6949, 61051, 61547, 247183, 12409, 35597, 1787, 51043, 9151, 36749, 11273, 261323, 65587, 53087, 66617, 6221, 269579, 1933, 54751, 55171, 1871, 39709, 1993, 14741, 14057, 6883, 3079, 10193, 1637, 71887, 2897, 2879, 14593, 41849, 10501, 59023, 74051, 1607, 74597, 15139, 1327, 4013, 2111, 76801, 15583, 312779, 1493, 1613, 79601, 63907, 80167, 16937, 1997, 65731, 2011, 66191, 47609, 335579, 337903, 9721, 4523, 2633, 3461, 1291, 3061, 88301, 70879, 88897, 50969, 2557, 8353, 90697, 1693, 13043, 1259, 3581, 18503, 19541, 1913, 94351, 1847, 4129, 381103, 2731, 1481, 2459, 96827, 77711, 11173, 14011, 3821, 396079, 19867, 1439, 14281, 3469, 1511, 10979, 2539, 2399, 4357, 103801, 83299, 21019, 21149, 106397, 12197, 85903, 432143, 21673, 2207, 677, 88547, 5843, 63629, 3191, 15451, 4463, 113027, 13033, 114377, 458863, 23011, 6323, 3491, 16631, 93407, 117101, 117787, 472523, 1579, 23833, 6733, 4133, 120551, 17321, 11863, 489179, 491983, 6529, 99523, 1709, 500443, 10271, 13679, 20359, 127601, 102367, 73529, 27241, 25951, 130477, 18743, 105251, 131927, 3571, 3323, 3119, 134851, 4327, 3307, 77689, 23773, 5443, 7253, 110543, 19793, 15877, 139297, 558683, 757, 6311, 80669, 113539, 114143, 4933, 30197, 5783, 30517, 4663, 16741, 20981, 15919, 29527, 8837, 13841, 17093, 2797, 7933, 610703, 153067, 22091, 88589, 6217, 623279, 626443, 157007, 17989, 22543, 6661, 158597, 635983, 33641, 160997, 129119, 4373, 129763, 1901, 2719, 6569, 165037, 19001, 166667, 668303, 8501, 821, 5897, 169951, 1867, 170777, 4201, 4903, 98297, 34487, 173267, 174101, 3989, 704779, 35323, 19139, 178301, 142979, 7789, 3221, 35999, 5167, 8147, 7937, 146383, 185987, 4261, 26693, 149827, 6473, 752603, 37717, 5413, 5711, 6563, 8069, 4783, 773579, 38767, 777103, 194717, 156127, 27943, 4481, 6971, 2087, 159683, 10529, 5531, 5431, 805583, 115597, 28027, 4967, 2437, 3347, 823643, 43541, 8291, 830923, 29741, 2039, 10657, 42187, 6053, 6199, 212777, 7417, 7369, 2347, 30661, 2269, 864079, 867803, 11443, 31193, 2129, 879023, 882779, 3089, 222107, 223051, 178819, 9739, 6427, 901679, 45179, 47657, 6131, 6271, 4649, 3727, 228757, 4751, 45943, 31751, 9227, 5743, 33091, 37139, 936203, 6701, 5839, 236477, 2351, 34061, 12097, 47881, 959579, 2081, 5659, 975323, 983243, 10709, 247301, 142169, 5021, 251297, 4111, 202243, 253307, 2677, 4451, 256337, 8219, 257351, 8969, 2903, 5101, 7411, 10093, 52081, 1043663, 261427, 37493, 7121,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+56x-37 and
the reducible primes which appear as divisor for the first time
p | x^2+56x-37 and p < x^2+56x-37

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)
11010641.0000000.6000000.4000000.0000000.0000000.000000
21006529360.6500000.2900000.3600006.5000004.8333339.000000
31.0006321804520.6320000.1800000.4520009.7230776.20689712.555555
410.0006.4941.3465.1480.6494000.1346000.51480010.2753167.47777811.389380
5100.00066.08310.20855.8750.6608300.1020800.55875010.1760087.58395210.853729
61.000.000666.18682.516583.6700.6661860.0825160.58367010.0810508.08346410.445995
710.000.0006.700.300695.0216.005.2790.6700300.0695020.60052810.0577028.42286310.288826
8100.000.00067.287.0066.010.60661.276.4000.6728700.0601060.61276410.0423878.64809210.203755
91.000.000.000675.088.38552.893.969622.194.4160.6750880.0528940.62219410.0329688.80010610.153900
1010.000.000.0006.768.797.007472.462.7026.296.334.3050.6768800.0472460.62963310.0265358.93226110.119561


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
245321.2500000.7500000.5000001.6666671.000000inf
388531.0000000.6250000.3750001.6000001.6666671.500000
41614770.8750000.4375000.4375001.7500001.4000002.333333
5322411130.7500000.3437500.4062501.7142861.5714291.857143
6644521240.7031250.3281250.3750001.8750001.9090911.846154
71288237450.6406250.2890620.3515621.8222221.7619051.875000
825615864940.6171880.2500000.3671881.9268291.7297302.088889
95123171002170.6191410.1953120.4238282.0063291.5625002.308511
101.0246471854620.6318360.1806640.4511722.0410091.8500002.129032
112.0481.3103379730.6396480.1645510.4750982.0247291.8216222.106061
124.0962.6346242.0100.6430660.1523440.4907232.0106871.8516322.065776
138.1925.2941.1304.1640.6462400.1379390.5083012.0098711.8108972.071642
1416.38410.6982.0378.6610.6529540.1243290.5286252.0207781.8026552.079971
1532.76821.5253.78917.7360.6568910.1156310.5412602.0120581.8600882.047801
1665.53643.2406.99136.2490.6597900.1066740.5531162.0088271.8450782.043809
17131.07286.73613.00873.7280.6617430.0992430.5625002.0059201.8606782.033932
18262.144173.86624.273149.5930.6632460.0925940.5706522.0045431.8660062.028985
19524.288348.59645.576303.0200.6648940.0869290.5779652.0049691.8776422.025630
201.048.576698.60386.210612.3930.6662400.0822160.5840232.0040481.8915662.020966
212.097.1521.399.907163.2521.236.6550.6675280.0778450.5896832.0038661.8936552.019381
224.194.3042.804.256310.2282.494.0280.6685870.0739640.5946232.0031731.9003012.016753
238.388.6085.618.315590.0035.028.3120.6697550.0703340.5994222.0034961.9018372.016141
2416.777.21611.252.2891.126.81310.125.4760.6706890.0671630.6035252.0027871.9098432.013693
2533.554.43222.535.9782.155.45020.380.5280.6716240.0642370.6073872.0027901.9128732.012797
2667.108.86445.125.3764.131.02340.994.3530.6724210.0615570.6108632.0023701.9165482.011447
27134.217.72890.354.4797.927.92082.426.5590.6731930.0590680.6141262.0022991.9191182.010681
28268.435.456180.894.33615.241.389165.652.9470.6738840.0567790.6171052.0020521.9224952.009703
29536.870.912362.139.83529.347.764332.792.0710.6745380.0546640.6198742.0019411.9255312.008972
301.073.741.824724.931.85256.586.330668.345.5220.6751450.0527000.6224452.0018011.9281312.008298
312.147.483.6481.451.101.770109.256.0871.341.845.6830.6757220.0508760.6248462.0017081.9307862.007712
324.294.967.2962.904.520.400211.205.7772.693.314.6230.6762610.0491750.6270862.0015971.9331262.007172
338.589.934.5925.813.422.235408.722.2465.404.699.9890.6767710.0475820.6291902.0015081.9351852.006710
3417.179.869.18411.635.075.755791.791.53310.843.284.2220.6772510.0460880.6311622.0014161.9372362.006269


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
123210021
243210021
385320131
4167342131
53211562333
664218134746
7128371423810613
825664224210191025
9512100336716321438
101.0241856212329652962
112.0483371142234413143119
124.0966241944308222886228
138.1921.130373757152395149434
1416.3842.0376711.366272725283757
1532.7683.7891.2602.5294961.3585151.420
1665.5366.9912.3044.6878912.5859222.593
17131.07213.0084.3398.6691.6874.8351.7054.781
18262.14424.2738.10616.1673.1519.0403.1278.955
19524.28845.57615.25330.3235.85516.9825.83916.900
201.048.57686.21028.87757.33311.03832.09111.11231.969
212.097.152163.25254.682108.57020.91760.67121.02360.641
224.194.304310.228103.786206.44239.792115.27039.772115.394
238.388.608590.003197.162392.84175.370219.23675.622219.775
2416.777.2161.126.813375.919750.894143.888418.917144.332419.676
2533.554.4322.155.450718.3151.437.135275.436802.007275.307802.700
2667.108.8644.131.0231.375.9342.755.089527.0731.539.486527.1301.537.334
27134.217.7287.927.9202.641.5705.286.3501.010.6552.953.9701.010.7392.952.556
28268.435.45615.241.3895.080.34610.161.0431.940.9525.680.6581.943.2765.676.503
29536.870.91229.347.7649.782.18319.565.5813.735.99210.936.8463.738.27910.936.647
301.073.741.82456.586.33018.861.56637.724.7647.199.04321.092.6077.203.01621.091.664
312.147.483.648109.256.08736.418.59272.837.49513.895.28840.730.86713.896.73040.733.202
324.294.967.296211.205.77770.399.644140.806.13326.843.97778.761.81226.839.82978.760.159
338.589.934.592408.722.246136.236.597272.485.64951.915.869152.442.88551.909.066152.454.426
3417.179.869.184791.791.533263.929.911527.861.622100.536.177295.369.023100.522.806295.363.527


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
242110011
383210111
4167432212
53213763334
6642413114677
712845232210111212
825694474721242623
951221711110652595254
101.024462227235101122120119
112.048973501472223257254239
124.0962.0101.022988477538500495
138.1924.1642.1002.0641.0361.0781.0241.026
1416.3848.6614.3854.2762.1152.2282.1482.170
1532.76817.7368.9938.7434.3804.5014.3734.482
1665.53636.24918.34217.9078.9709.1408.9589.181
17131.07273.72837.35036.37818.29018.49318.36818.577
18262.144149.59375.64973.94437.33237.45937.28537.517
19524.288303.020153.237149.78375.47575.62875.67376.244
201.048.576612.393309.697302.696152.454153.192152.667154.080
212.097.1521.236.655625.123611.532307.563309.929308.609310.554
224.194.3042.494.0281.259.7831.234.245619.931626.163621.480626.454
238.388.6085.028.3122.537.2442.491.0681.251.2121.261.7281.252.5921.262.780
2416.777.21610.125.4765.107.3075.018.1692.519.7032.541.4352.523.2072.541.131
2533.554.43220.380.52810.275.04110.105.4875.074.0625.114.8255.076.9495.114.692
2667.108.86440.994.35320.657.50820.336.84510.207.85510.286.90510.215.52810.284.065
27134.217.72882.426.55941.525.09440.901.46520.528.98920.678.16720.538.16520.681.238
28268.435.456165.652.94783.413.63682.239.31141.273.36741.550.73641.277.12541.551.719
29536.870.912332.792.071167.539.315165.252.75682.935.53783.464.84482.933.97883.457.712
301.073.741.824668.345.522336.369.403331.976.119166.586.089167.590.496166.573.331167.595.606
312.147.483.6481.341.845.683675.147.736666.697.947334.483.739336.435.465334.490.151336.436.328
324.294.967.2962.693.314.6231.354.796.1661.338.518.457671.422.793675.222.661671.445.591675.223.578
338.589.934.5925.404.699.9892.718.069.6072.686.630.3821.347.475.1881.354.860.8161.347.475.3461.354.888.639
3417.179.869.18410.843.284.2225.452.061.4865.391.222.7362.703.638.1962.717.966.9862.703.654.9402.718.024.100


8. Check for existing Integer Sequences by OEIS

Found in Database : 37, 5, 79, 7, 29, 67, 1, 101, 19, 137, 89, 1, 41, 43, 23, 257, 223, 1, 1, 347,
Found in Database : 37, 5, 79, 7, 29, 67, 101, 19, 137, 89, 41, 43, 23, 257, 223, 347, 1483, 73, 269, 71, 419, 463, 607, 2543, 397, 3023, 787, 131,
Found in Database : 5, 7, 19, 23, 29, 37, 41, 43, 67, 71, 73, 79, 89, 101, 103, 113, 131, 137, 149,