Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:41:49
Deutsch
29.Mar 2024

Polynom = x^2+8x-73

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) = 73 = 73
f(1) = 1 = 1
f(2) = 53 = 53
f(3) = 5 = 5
f(4) = 25 = 5*5
f(5) = 1 = 1
f(6) = 11 = 11
f(7) = 1 = 1
f(8) = 55 = 5*11
f(9) = 5 = 5
f(10) = 107 = 107
f(11) = 17 = 17
f(12) = 167 = 167
f(13) = 25 = 5*5
f(14) = 235 = 5*47
f(15) = 17 = 17
f(16) = 311 = 311
f(17) = 11 = 11
f(18) = 395 = 5*79
f(19) = 55 = 5*11
f(20) = 487 = 487
f(21) = 67 = 67
f(22) = 587 = 587
f(23) = 5 = 5
f(24) = 695 = 5*139
f(25) = 47 = 47
f(26) = 811 = 811
f(27) = 109 = 109
f(28) = 935 = 5*11*17
f(29) = 125 = 5*5*5
f(30) = 1067 = 11*97
f(31) = 71 = 71
f(32) = 1207 = 17*71
f(33) = 5 = 5
f(34) = 1355 = 5*271
f(35) = 179 = 179
f(36) = 1511 = 1511
f(37) = 199 = 199
f(38) = 1675 = 5*5*67
f(39) = 55 = 5*11
f(40) = 1847 = 1847
f(41) = 121 = 11*11
f(42) = 2027 = 2027
f(43) = 265 = 5*53
f(44) = 2215 = 5*443
f(45) = 289 = 17*17
f(46) = 2411 = 2411
f(47) = 157 = 157
f(48) = 2615 = 5*523
f(49) = 85 = 5*17
f(50) = 2827 = 11*257
f(51) = 367 = 367
f(52) = 3047 = 11*277
f(53) = 395 = 5*79
f(54) = 3275 = 5*5*131
f(55) = 53 = 53
f(56) = 3511 = 3511
f(57) = 227 = 227
f(58) = 3755 = 5*751
f(59) = 485 = 5*97
f(60) = 4007 = 4007
f(61) = 517 = 11*47
f(62) = 4267 = 17*251
f(63) = 275 = 5*5*11
f(64) = 4535 = 5*907
f(65) = 73 = 73
f(66) = 4811 = 17*283
f(67) = 619 = 619
f(68) = 5095 = 5*1019
f(69) = 655 = 5*131
f(70) = 5387 = 5387
f(71) = 173 = 173
f(72) = 5687 = 11*11*47
f(73) = 365 = 5*73
f(74) = 5995 = 5*11*109
f(75) = 769 = 769
f(76) = 6311 = 6311
f(77) = 809 = 809
f(78) = 6635 = 5*1327
f(79) = 425 = 5*5*17
f(80) = 6967 = 6967
f(81) = 223 = 223
f(82) = 7307 = 7307
f(83) = 935 = 5*11*17
f(84) = 7655 = 5*1531
f(85) = 979 = 11*89
f(86) = 8011 = 8011
f(87) = 1 = 1
f(88) = 8375 = 5*5*5*67
f(89) = 535 = 5*107
f(90) = 8747 = 8747
f(91) = 1117 = 1117
f(92) = 9127 = 9127
f(93) = 1165 = 5*233
f(94) = 9515 = 5*11*173
f(95) = 607 = 607
f(96) = 9911 = 11*17*53
f(97) = 79 = 79
f(98) = 10315 = 5*2063
f(99) = 1315 = 5*263
f(100) = 10727 = 17*631

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+8x-73

f(0)=73
f(1)=1
f(2)=53
f(3)=5
f(4)=1
f(5)=1
f(6)=11
f(7)=1
f(8)=1
f(9)=1
f(10)=107
f(11)=17
f(12)=167
f(13)=1
f(14)=47
f(15)=1
f(16)=311
f(17)=1
f(18)=79
f(19)=1
f(20)=487
f(21)=67
f(22)=587
f(23)=1
f(24)=139
f(25)=1
f(26)=811
f(27)=109
f(28)=1
f(29)=1
f(30)=97
f(31)=71
f(32)=1
f(33)=1
f(34)=271
f(35)=179
f(36)=1511
f(37)=199
f(38)=1
f(39)=1
f(40)=1847
f(41)=1
f(42)=2027
f(43)=1
f(44)=443
f(45)=1
f(46)=2411
f(47)=157
f(48)=523
f(49)=1
f(50)=257
f(51)=367
f(52)=277
f(53)=1
f(54)=131
f(55)=1
f(56)=3511
f(57)=227
f(58)=751
f(59)=1
f(60)=4007
f(61)=1
f(62)=251
f(63)=1
f(64)=907
f(65)=1
f(66)=283
f(67)=619
f(68)=1019
f(69)=1
f(70)=5387
f(71)=173
f(72)=1
f(73)=1
f(74)=1
f(75)=769
f(76)=6311
f(77)=809
f(78)=1327
f(79)=1
f(80)=6967
f(81)=223
f(82)=7307
f(83)=1
f(84)=1531
f(85)=89
f(86)=8011
f(87)=1
f(88)=1
f(89)=1
f(90)=8747
f(91)=1117
f(92)=9127
f(93)=233
f(94)=1
f(95)=607
f(96)=1
f(97)=1
f(98)=2063
f(99)=263

b) Substitution of the polynom
The polynom f(x)=x^2+8x-73 could be written as f(y)= y^2-89 with x=y-4

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+4
f'(x)>2x+7

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

73, 1, 53, 5, 1, 1, 11, 1, 1, 1, 107, 17, 167, 1, 47, 1, 311, 1, 79, 1, 487, 67, 587, 1, 139, 1, 811, 109, 1, 1, 97, 71, 1, 1, 271, 179, 1511, 199, 1, 1, 1847, 1, 2027, 1, 443, 1, 2411, 157, 523, 1, 257, 367, 277, 1, 131, 1, 3511, 227, 751, 1, 4007, 1, 251, 1, 907, 1, 283, 619, 1019, 1, 5387, 173, 1, 1, 1, 769, 6311, 809, 1327, 1, 6967, 223, 7307, 1, 1531, 89, 8011, 1, 1, 1, 8747, 1117, 9127, 233, 1, 607, 1, 1, 2063, 263, 631, 1367, 1, 1, 463, 1, 12011, 1, 1, 317, 12907, 821, 13367, 1, 2767, 1759, 1301, 1, 269, 1, 15287, 971, 15787, 401, 3259, 2069, 16811, 1, 3467, 1, 1051, 2267, 1, 467, 1, 601, 1, 1237, 1, 509, 1877, 2617, 21227, 1, 4363, 691, 307, 1, 4603, 1, 23627, 1, 24247, 1, 1, 1, 1, 3229, 5231, 331, 2437, 1, 1, 1, 1, 3559, 613, 911, 347, 373, 30187, 1, 461, 1, 1, 1997, 409, 1021, 6607, 1, 33767, 1, 3137, 1, 641, 1, 36011, 4549, 1471, 929, 37547, 2371, 38327, 1, 7823, 449, 1, 5039, 479, 1, 1, 2621, 1, 1069, 1, 5449, 4001, 2777, 8971, 1, 45707, 1, 46567, 1, 1, 1, 48311, 1, 9839, 1, 50087, 6317, 761, 643, 1, 1, 4801, 6659, 977, 1, 54667, 1723, 3271, 701, 11311, 7129, 1, 659, 2339, 1, 59447, 1873, 60427, 1523, 1, 1, 1, 983, 1153, 1, 5857, 8117, 65447, 1, 2659, 1, 67511, 1063, 13711, 1, 1481, 797, 70667, 1, 14347, 4517, 4283, 1, 14779, 1861, 1, 4721, 6917, 1, 15439, 9719, 78311, 9859, 15887, 1, 80567, 1, 81707, 1, 1, 10429, 84011, 1, 3407, 1, 1093, 10867, 1, 2203, 1613, 2791, 1913, 5657, 18223, 2293, 5431, 11617, 673, 1, 1, 1, 1433, 1, 1, 2447, 98507, 1549, 99767, 1, 1, 1, 1, 757, 20719, 1303, 1979, 1, 106187, 2671, 21499, 1229, 1, 1, 22027, 1, 111467, 1, 112807, 2837, 1, 7177, 10501, 3631, 1, 2939, 118247, 14867, 119627, 1, 24203, 7607, 733, 1399, 24763, 1, 1291, 1, 1423, 1, 1, 947, 1933, 1, 2381, 823, 12037, 1, 1, 1, 27067, 1, 136811, 8597, 1627, 1, 139787, 1597, 8311, 1, 28559, 2243, 144311, 9067, 29167, 1, 13397, 18517, 13537, 1871, 30091, 1, 2141, 1, 6143, 1, 3301, 1, 156727, 1, 31663, 19889, 159911, 20089, 32303, 2029, 163127, 1, 881, 4139, 1, 20899, 9883, 1319, 33931, 2131, 1601, 21517, 1, 1, 1, 997, 1123, 1, 1, 1, 179687, 22567, 181387, 1, 3329, 11497, 1, 23209, 37307, 937, 2579, 11821, 1, 1193, 38351, 1, 11383, 1, 1, 1, 1, 1, 198827, 4993, 40123, 25189, 18401, 1, 1, 1, 206027, 25867, 207847, 1, 8387, 6581, 1, 1, 1, 1, 1, 27017, 1657, 1, 43787, 6871, 220811, 1, 4049, 5591, 1201, 1, 226487, 2843, 2687, 28669, 230311, 28909, 46447, 1, 234167, 1, 236107, 5927, 1013, 29879, 240011, 1, 9679, 3037, 1, 1801, 1, 6173, 1, 1, 249911, 7841, 50383, 6323, 1, 2897, 255947, 1, 1, 16187, 260011, 1, 3083, 6577, 2423, 1, 24197, 1, 4877, 1, 270311, 1, 1, 1709, 5179, 1, 276587, 1, 1, 1, 280811, 1, 1, 1, 3203, 1, 5419, 7207, 5261, 2269, 26501, 18287, 1, 7369, 295847, 37117, 1, 3739, 1, 1, 302411, 3449, 60923, 7643, 306827, 9623, 1553, 3877, 12451, 2297, 2591, 1, 5741, 1, 318007, 9973, 320267, 1607, 1, 40459, 324811, 1, 65419, 1, 329387, 1, 1, 1, 1, 20947, 1, 5273, 1, 1699, 1, 42767, 343307, 2153, 1, 1, 348011, 43649, 2803, 1, 2039, 2011, 355127, 1, 71503, 44839, 359911, 45139, 1, 1, 1, 22871, 33377, 9209, 14783, 46349, 1, 23327, 74891, 1, 22171, 4297, 3911, 1, 76367, 11971, 1693, 24097, 77359, 1, 389287, 48817, 35617, 1, 1, 1, 7487, 2927, 79867, 2003, 401867, 6299, 1429, 1, 1, 4639, 8713, 51349, 1, 5167, 24391, 1, 417227, 10463, 1, 52639, 3491, 13241, 1, 1, 5413, 53617, 4021, 10789, 3463, 2467, 1619, 1, 87631, 10987, 440807, 3251, 443467, 1, 89227, 27967, 40801, 56269, 8209, 11321, 6397, 1, 9721, 1, 5407, 1, 462311, 1, 5471, 1, 2801, 1, 1789, 1, 94651, 1, 1, 29837, 1741, 3001, 43777, 1, 484327, 12143, 97423, 1, 489911, 30707, 98543, 1, 495527, 5647, 1, 6247, 20047, 7853, 504011, 63179, 1, 1, 46337, 15973, 2741, 1, 103087, 64609, 518311, 64969, 104239, 1, 524087, 1493, 7219, 1, 1, 3907, 532811, 2087, 1, 1, 1, 1, 1, 13577, 9901, 34127, 1, 4289, 1, 13799, 553447, 1, 1, 1, 2111, 3187, 33083, 70489, 113083, 14173, 2549, 1, 1, 3581, 2089, 71999, 52501, 72379, 1733, 1, 1901, 36571, 586667, 1, 1, 6719, 12613, 1, 119179, 1867, 598987, 1, 602087, 15091, 121039, 1, 3253, 1, 11117, 1, 36151, 77017, 617707, 7741, 124171, 1, 3607, 7109, 25087, 1, 5783, 4937, 2719, 1, 127343, 1, 639911, 1, 1, 8059, 58757, 2531, 4673, 1, 26111, 81799, 6763, 20551, 1669, 1, 38971, 7547, 9377, 1, 1, 1, 672311, 21061, 135119, 16931, 61717, 1, 62017, 2137, 2917, 1, 2081, 5077, 138427, 3469, 695467, 1, 4451, 1, 1, 1, 1, 1, 1, 4441, 5437, 44621, 1, 1, 1, 90089, 722411, 1, 8539, 4547, 13759, 91367, 4093, 1, 29443, 1, 739511, 46327, 1, 18617, 15881, 5501, 2161, 1879, 13697, 1, 1, 94819, 2083, 19051, 763787, 1, 767287, 9613, 154159, 8779, 10607, 8819, 9151, 1949, 781367, 24473, 46171, 1, 2221, 98779, 1, 1, 1, 9967, 799147, 1889, 11981, 20113, 161263, 2971, 1, 1, 162703, 1, 817127, 102367, 820747, 1, 1, 51637, 828011, 2207, 15121, 1, 75937, 52321, 2903, 1, 168527, 1, 49783, 106019, 3617, 1, 853687, 4861, 10853, 21481, 172219, 1, 864811, 54167, 173707, 1, 1, 109267, 1, 1, 1, 27551, 5107, 55337, 35491, 22229, 891047, 1, 2399, 1, 3391, 1, 1, 113039, 181243, 1, 1, 14249, 83077, 11447, 1, 1, 17387, 115429, 185071, 1, 929207, 14549, 933067, 1, 187387, 1, 11909, 1, 188939, 11833, 3691, 1, 952487, 1, 17389, 1, 1, 30071, 192847, 4831, 56951, 121267, 5821, 1, 11483, 5557, 7481, 11159, 39359, 1, 1, 61871, 991927, 1, 1, 124739, 90901, 1, 18253, 6287,

6. Sequence of the polynom (only primes)

73, 53, 5, 11, 107, 17, 167, 47, 311, 79, 487, 67, 587, 139, 811, 109, 97, 71, 271, 179, 1511, 199, 1847, 2027, 443, 2411, 157, 523, 257, 367, 277, 131, 3511, 227, 751, 4007, 251, 907, 283, 619, 1019, 5387, 173, 769, 6311, 809, 1327, 6967, 223, 7307, 1531, 89, 8011, 8747, 1117, 9127, 233, 607, 2063, 263, 631, 1367, 463, 12011, 317, 12907, 821, 13367, 2767, 1759, 1301, 269, 15287, 971, 15787, 401, 3259, 2069, 16811, 3467, 1051, 2267, 467, 601, 1237, 509, 1877, 2617, 21227, 4363, 691, 307, 4603, 23627, 24247, 3229, 5231, 331, 2437, 3559, 613, 911, 347, 373, 30187, 461, 1997, 409, 1021, 6607, 33767, 3137, 641, 36011, 4549, 1471, 929, 37547, 2371, 38327, 7823, 449, 5039, 479, 2621, 1069, 5449, 4001, 2777, 8971, 45707, 46567, 48311, 9839, 50087, 6317, 761, 643, 4801, 6659, 977, 54667, 1723, 3271, 701, 11311, 7129, 659, 2339, 59447, 1873, 60427, 1523, 983, 1153, 5857, 8117, 65447, 2659, 67511, 1063, 13711, 1481, 797, 70667, 14347, 4517, 4283, 14779, 1861, 4721, 6917, 15439, 9719, 78311, 9859, 15887, 80567, 81707, 10429, 84011, 3407, 1093, 10867, 2203, 1613, 2791, 1913, 5657, 18223, 2293, 5431, 11617, 673, 1433, 2447, 98507, 1549, 99767, 757, 20719, 1303, 1979, 106187, 2671, 21499, 1229, 22027, 111467, 112807, 2837, 7177, 10501, 3631, 2939, 118247, 14867, 119627, 24203, 7607, 733, 1399, 24763, 1291, 1423, 947, 1933, 2381, 823, 12037, 27067, 136811, 8597, 1627, 139787, 1597, 8311, 28559, 2243, 144311, 9067, 29167, 13397, 18517, 13537, 1871, 30091, 2141, 6143, 3301, 156727, 31663, 19889, 159911, 20089, 32303, 2029, 163127, 881, 4139, 20899, 9883, 1319, 33931, 2131, 1601, 21517, 997, 1123, 179687, 22567, 181387, 3329, 11497, 23209, 37307, 937, 2579, 11821, 1193, 38351, 11383, 198827, 4993, 40123, 25189, 18401, 206027, 25867, 207847, 8387, 6581, 27017, 1657, 43787, 6871, 220811, 4049, 5591, 1201, 226487, 2843, 2687, 28669, 230311, 28909, 46447, 234167, 236107, 5927, 1013, 29879, 240011, 9679, 3037, 1801, 6173, 249911, 7841, 50383, 6323, 2897, 255947, 16187, 260011, 3083, 6577, 2423, 24197, 4877, 270311, 1709, 5179, 276587, 280811, 3203, 5419, 7207, 5261, 2269, 26501, 18287, 7369, 295847, 37117, 3739, 302411, 3449, 60923, 7643, 306827, 9623, 1553, 3877, 12451, 2297, 2591, 5741, 318007, 9973, 320267, 1607, 40459, 324811, 65419, 329387, 20947, 5273, 1699, 42767, 343307, 2153, 348011, 43649, 2803, 2039, 2011, 355127, 71503, 44839, 359911, 45139, 22871, 33377, 9209, 14783, 46349, 23327, 74891, 22171, 4297, 3911, 76367, 11971, 1693, 24097, 77359, 389287, 48817, 35617, 7487, 2927, 79867, 2003, 401867, 6299, 1429, 4639, 8713, 51349, 5167, 24391, 417227, 10463, 52639, 3491, 13241, 5413, 53617, 4021, 10789, 3463, 2467, 1619, 87631, 10987, 440807, 3251, 443467, 89227, 27967, 40801, 56269, 8209, 11321, 6397, 9721, 5407, 462311, 5471, 2801, 1789, 94651, 29837, 1741, 3001, 43777, 484327, 12143, 97423, 489911, 30707, 98543, 495527, 5647, 6247, 20047, 7853, 504011, 63179, 46337, 15973, 2741, 103087, 64609, 518311, 64969, 104239, 524087, 1493, 7219, 3907, 532811, 2087, 13577, 9901, 34127, 4289, 13799, 553447, 2111, 3187, 33083, 70489, 113083, 14173, 2549, 3581, 2089, 71999, 52501, 72379, 1733, 1901, 36571, 586667, 6719, 12613, 119179, 1867, 598987, 602087, 15091, 121039, 3253, 11117, 36151, 77017, 617707, 7741, 124171, 3607, 7109, 25087, 5783, 4937, 2719, 127343, 639911, 8059, 58757, 2531, 4673, 26111, 81799, 6763, 20551, 1669, 38971, 7547, 9377, 672311, 21061, 135119, 16931, 61717, 62017, 2137, 2917, 2081, 5077, 138427, 3469, 695467, 4451, 4441, 5437, 44621, 90089, 722411, 8539, 4547, 13759, 91367, 4093, 29443, 739511, 46327, 18617, 15881, 5501, 2161, 1879, 13697, 94819, 2083, 19051, 763787, 767287, 9613, 154159, 8779, 10607, 8819, 9151, 1949, 781367, 24473, 46171, 2221, 98779, 9967, 799147, 1889, 11981, 20113, 161263, 2971, 162703, 817127, 102367, 820747, 51637, 828011, 2207, 15121, 75937, 52321, 2903, 168527, 49783, 106019, 3617, 853687, 4861, 10853, 21481, 172219, 864811, 54167, 173707, 109267, 27551, 5107, 55337, 35491, 22229, 891047, 2399, 3391, 113039, 181243, 14249, 83077, 11447, 17387, 115429, 185071, 929207, 14549, 933067, 187387, 11909, 188939, 11833, 3691, 952487, 17389, 30071, 192847, 4831, 56951, 121267, 5821, 11483, 5557, 7481, 11159, 39359, 61871, 991927, 124739, 90901, 18253, 6287,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+8x-73 and
the reducible primes which appear as divisor for the first time
p | x^2+8x-73 and p < x^2+8x-73

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)
1103300.3000000.3000000.0000000.0000000.0000000.000000
21005736210.5700000.3600000.21000019.00000012.000000inf
31.0006522763760.6520000.2760000.37600011.4385977.66666717.904762
410.0006.6491.9904.6590.6649000.1990000.46590010.1978537.21014512.390958
5100.00067.16115.20251.9590.6716100.1520200.51959010.1009187.63919611.152393
61.000.000675.035123.265551.7700.6750350.1232650.55177010.0509978.10847310.619334
710.000.0006.777.3851.028.3845.749.0010.6777380.1028380.57490010.0400508.34287110.419198
8100.000.00067.959.5738.845.23959.114.3340.6795960.0884520.59114310.0274048.60110610.282540
91.000.000.000681.045.23377.628.582603.416.6510.6810450.0776290.60341710.0213298.77631310.207620
1010.000.000.0006.822.063.357691.788.0356.130.275.3220.6822060.0691790.61302810.0170488.91151110.159274


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
242200.5000000.5000000.0000001.0000001.000000-nan
382200.2500000.2500000.0000001.0000001.000000-nan
4166510.3750000.3125000.0625003.0000002.500000inf
532151140.4687500.3437500.1250002.5000002.2000004.000000
6643522130.5468750.3437500.2031252.3333332.0000003.250000
71287647290.5937500.3671880.2265622.1714292.1363642.230769
825615687690.6093750.3398440.2695312.0526321.8510642.379310
95123251571680.6347660.3066410.3281252.0833331.8045982.434783
101.0246712853860.6552730.2783200.3769532.0646151.8152872.297619
112.0481.3485078410.6582030.2475590.4106452.0089421.7789472.178756
124.0962.7119221.7890.6618650.2250980.4367682.0111281.8185402.127229
138.1925.4351.6873.7480.6634520.2059330.4575202.0047951.8297182.095025
1416.38410.9283.0637.8650.6669920.1869510.4800422.0106721.8156492.098453
1532.76821.9355.61416.3210.6694030.1713260.4980772.0072291.8328442.075143
1665.53644.00510.38033.6250.6714630.1583860.5130772.0061551.8489492.060229
17131.07288.13419.37268.7620.6724090.1477970.5246122.0028181.8662812.044966
18262.144176.42136.320140.1010.6729930.1385500.5344432.0017361.8748712.037477
19524.288353.54868.202285.3460.6743390.1300850.5442542.0040021.8778082.036716
201.048.576708.005128.722579.2830.6752060.1227590.5524472.0025711.8873642.030107
212.097.1521.417.987242.7761.175.2110.6761490.1157650.5603842.0027921.8860492.028734
224.194.3042.839.341460.0062.379.3350.6769520.1096740.5672782.0023741.8947752.024602
238.388.6085.684.024873.2194.810.8050.6775880.1040960.5734932.0018811.8982772.021912
2416.777.21611.378.5491.664.3789.714.1710.6782140.0992050.5790102.0018481.9060262.019240
2533.554.43222.775.8673.179.69519.596.1720.6787740.0947620.5840112.0016501.9104402.017277
2667.108.86445.587.3176.084.00739.503.3100.6793040.0906590.5886452.0015621.9133932.015869
27134.217.72891.240.99011.664.72579.576.2650.6797980.0869090.5928892.0014561.9172772.014420
28268.435.456182.607.06022.403.102160.203.9580.6802640.0834580.5968062.0013711.9205862.013213
29536.870.912365.439.37843.101.664322.337.7140.6806840.0802830.6004012.0012341.9239152.012046
301.073.741.824731.306.05583.040.006648.266.0490.6810820.0773370.6037452.0011691.9266082.011139
312.147.483.6481.463.417.804160.211.0741.303.206.7300.6814570.0746040.6068532.0011021.9293242.010296
324.294.967.2962.928.328.123309.492.0852.618.836.0380.6818050.0720590.6097452.0010201.9317772.009532
338.589.934.5925.859.508.293598.548.5635.260.959.7300.6821370.0696800.6124562.0009741.9339702.008893
3417.179.869.18411.724.361.3691.158.869.48910.565.491.8800.6824480.0674550.6149932.0009121.9361332.008282
3534.359.738.36823.458.795.0372.246.020.76721.212.774.2700.6827410.0653680.6173732.0008591.9381142.007741
3668.719.476.73646.936.620.7144.357.319.67142.579.301.0430.6830180.0634070.6196102.0008111.9400172.007248


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
122111010
242111010
382111010
4165141112
53211561424
6642291318310
7128472027318719
82568738499311532
9512157688919562656
101.0242851251603710141106
112.0485072302777018372182
124.096922413509124326136336
138.1921.687743944226601242618
1416.3843.0631.3961.6674231.1054181.117
1532.7685.6142.5323.0827672.0187592.070
1665.53610.3804.6845.6961.3913.7491.4083.832
17131.07219.3728.78710.5852.5657.0402.6607.107
18262.14436.32016.41719.9034.80613.2524.94713.315
19524.28868.20230.80937.3939.06724.9449.18025.011
201.048.576128.72258.04670.67617.02847.35017.09047.254
212.097.152242.776109.200133.57632.05589.17232.26389.286
224.194.304460.006206.754253.25260.380169.36960.861169.396
238.388.608873.219392.680480.539114.331322.029114.909321.950
2416.777.2161.664.378747.607916.771217.874614.354218.581613.569
2533.554.4323.179.6951.426.7851.752.910414.6931.174.332416.4121.174.258
2667.108.8646.084.0072.727.9963.356.011792.8302.248.483793.9992.248.695
27134.217.72811.664.7255.228.4896.436.2361.517.9474.312.7061.519.6744.314.398
28268.435.45622.403.10210.038.01512.365.0872.910.5618.289.0722.913.7208.289.749
29536.870.91243.101.66419.305.30223.796.3625.591.76815.957.2115.595.16115.957.524
301.073.741.82483.040.00637.186.29945.853.70710.761.91330.756.79410.764.09330.757.206
312.147.483.648160.211.07471.729.94288.481.13220.735.78459.365.53020.741.28859.368.472
324.294.967.296309.492.085138.524.467170.967.61840.012.295114.725.00840.017.284114.737.498
338.589.934.592598.548.563267.832.656330.715.90777.295.792221.969.40577.304.749221.978.617
3417.179.869.1841.158.869.489518.450.710640.418.779149.503.108429.914.367149.508.265429.943.749
3534.359.738.3682.246.020.7671.004.591.5411.241.429.226289.479.067833.518.753289.486.002833.536.945
3668.719.476.7364.357.319.6711.948.576.9962.408.742.675561.088.6251.617.537.403561.095.6611.617.597.982


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
4161010001
5324311102
66413852614
7128291514411410
825669353415241218
9512168888033523647
101.0243862171697311289112
112.048841450391180236195230
124.0961.789934855396487415491
138.1923.7481.9001.8488521.0168591.021
1416.3847.8654.0373.8281.7982.1001.8222.145
1532.76816.3218.4157.9063.7564.3413.8244.400
1665.53633.62517.07816.5477.8648.9727.8498.940
17131.07268.76235.03233.73016.17218.28616.21118.093
18262.144140.10171.32768.77433.21636.71533.41336.757
19524.288285.346144.873140.47367.99174.48368.25174.621
201.048.576579.283293.961285.322138.451150.893138.969150.970
212.097.1521.175.211595.308579.903282.489305.089282.495305.138
224.194.3042.379.3351.204.1491.175.186572.770616.387573.520616.658
238.388.6084.810.8052.433.0932.377.7121.160.9211.243.0831.162.3301.244.471
2416.777.2169.714.1714.910.9844.803.1872.349.5912.506.5432.350.2952.507.742
2533.554.43219.596.1729.903.4339.692.7394.749.3055.047.0074.748.4245.051.436
2667.108.86439.503.31019.951.28619.552.0249.591.85510.160.4279.587.39510.163.633
27134.217.72879.576.26540.172.62339.403.64219.346.96520.446.09619.343.69720.439.507
28268.435.456160.203.95880.846.94979.357.00938.995.01141.112.76338.989.56041.106.624
29536.870.912322.337.714162.600.650159.737.06478.540.46582.623.94778.539.21782.634.085
301.073.741.824648.266.049326.898.296321.367.753158.113.856166.006.871158.123.464166.021.858
312.147.483.6481.303.206.730656.921.913646.284.817318.175.957333.414.826318.194.835333.421.112
324.294.967.2962.618.836.0381.319.715.3751.299.120.663639.995.639669.433.577639.986.540669.420.282
338.589.934.5925.260.959.7302.650.440.7312.610.518.9991.286.740.8541.343.772.4631.286.706.1051.343.740.308
3417.179.869.18410.565.491.8805.321.344.5145.244.147.3662.586.139.3522.696.646.7382.586.099.9122.696.605.878
3534.359.738.36821.212.774.27010.681.183.66010.531.590.6105.196.044.2885.410.518.0115.195.986.7505.410.225.221
3668.719.476.73642.579.301.04321.434.848.49721.144.452.54610.436.757.60810.853.080.74810.436.651.89410.852.810.793


8. Check for existing Integer Sequences by OEIS

Found in Database : 73, 1, 53, 5, 1, 1, 11, 1, 1, 1, 107, 17, 167, 1, 47, 1, 311, 1, 79, 1,
Found in Database : 73, 53, 5, 11, 107, 17, 167, 47, 311, 79, 487, 67, 587, 139, 811, 109, 97, 71, 271, 179, 1511, 199,
Found in Database : 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139,