Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:46:52
Deutsch
28.Mar 2024

Polynom = x^2+13x-3

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) = 3 = 3
f(1) = 11 = 11
f(2) = 27 = 3*3*3
f(3) = 45 = 3*3*5
f(4) = 65 = 5*13
f(5) = 87 = 3*29
f(6) = 111 = 3*37
f(7) = 137 = 137
f(8) = 165 = 3*5*11
f(9) = 195 = 3*5*13
f(10) = 227 = 227
f(11) = 261 = 3*3*29
f(12) = 297 = 3*3*3*11
f(13) = 335 = 5*67
f(14) = 375 = 3*5*5*5
f(15) = 417 = 3*139
f(16) = 461 = 461
f(17) = 507 = 3*13*13
f(18) = 555 = 3*5*37
f(19) = 605 = 5*11*11
f(20) = 657 = 3*3*73
f(21) = 711 = 3*3*79
f(22) = 767 = 13*59
f(23) = 825 = 3*5*5*11
f(24) = 885 = 3*5*59
f(25) = 947 = 947
f(26) = 1011 = 3*337
f(27) = 1077 = 3*359
f(28) = 1145 = 5*229
f(29) = 1215 = 3*3*3*3*3*5
f(30) = 1287 = 3*3*11*13
f(31) = 1361 = 1361
f(32) = 1437 = 3*479
f(33) = 1515 = 3*5*101
f(34) = 1595 = 5*11*29
f(35) = 1677 = 3*13*43
f(36) = 1761 = 3*587
f(37) = 1847 = 1847
f(38) = 1935 = 3*3*5*43
f(39) = 2025 = 3*3*3*3*5*5
f(40) = 2117 = 29*73
f(41) = 2211 = 3*11*67
f(42) = 2307 = 3*769
f(43) = 2405 = 5*13*37
f(44) = 2505 = 3*5*167
f(45) = 2607 = 3*11*79
f(46) = 2711 = 2711
f(47) = 2817 = 3*3*313
f(48) = 2925 = 3*3*5*5*13
f(49) = 3035 = 5*607
f(50) = 3147 = 3*1049
f(51) = 3261 = 3*1087
f(52) = 3377 = 11*307
f(53) = 3495 = 3*5*233
f(54) = 3615 = 3*5*241
f(55) = 3737 = 37*101
f(56) = 3861 = 3*3*3*11*13
f(57) = 3987 = 3*3*443
f(58) = 4115 = 5*823
f(59) = 4245 = 3*5*283
f(60) = 4377 = 3*1459
f(61) = 4511 = 13*347
f(62) = 4647 = 3*1549
f(63) = 4785 = 3*5*11*29
f(64) = 4925 = 5*5*197
f(65) = 5067 = 3*3*563
f(66) = 5211 = 3*3*3*193
f(67) = 5357 = 11*487
f(68) = 5505 = 3*5*367
f(69) = 5655 = 3*5*13*29
f(70) = 5807 = 5807
f(71) = 5961 = 3*1987
f(72) = 6117 = 3*2039
f(73) = 6275 = 5*5*251
f(74) = 6435 = 3*3*5*11*13
f(75) = 6597 = 3*3*733
f(76) = 6761 = 6761
f(77) = 6927 = 3*2309
f(78) = 7095 = 3*5*11*43
f(79) = 7265 = 5*1453
f(80) = 7437 = 3*37*67
f(81) = 7611 = 3*43*59
f(82) = 7787 = 13*599
f(83) = 7965 = 3*3*3*5*59
f(84) = 8145 = 3*3*5*181
f(85) = 8327 = 11*757
f(86) = 8511 = 3*2837
f(87) = 8697 = 3*13*223
f(88) = 8885 = 5*1777
f(89) = 9075 = 3*5*5*11*11
f(90) = 9267 = 3*3089
f(91) = 9461 = 9461
f(92) = 9657 = 3*3*29*37
f(93) = 9855 = 3*3*3*5*73
f(94) = 10055 = 5*2011
f(95) = 10257 = 3*13*263
f(96) = 10461 = 3*11*317
f(97) = 10667 = 10667
f(98) = 10875 = 3*5*5*5*29
f(99) = 11085 = 3*5*739
f(100) = 11297 = 11*13*79

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+13x-3

f(0)=3
f(1)=11
f(2)=1
f(3)=5
f(4)=13
f(5)=29
f(6)=37
f(7)=137
f(8)=1
f(9)=1
f(10)=227
f(11)=1
f(12)=1
f(13)=67
f(14)=1
f(15)=139
f(16)=461
f(17)=1
f(18)=1
f(19)=1
f(20)=73
f(21)=79
f(22)=59
f(23)=1
f(24)=1
f(25)=947
f(26)=337
f(27)=359
f(28)=229
f(29)=1
f(30)=1
f(31)=1361
f(32)=479
f(33)=101
f(34)=1
f(35)=43
f(36)=587
f(37)=1847
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=769
f(43)=1
f(44)=167
f(45)=1
f(46)=2711
f(47)=313
f(48)=1
f(49)=607
f(50)=1049
f(51)=1087
f(52)=307
f(53)=233
f(54)=241
f(55)=1
f(56)=1
f(57)=443
f(58)=823
f(59)=283
f(60)=1459
f(61)=347
f(62)=1549
f(63)=1
f(64)=197
f(65)=563
f(66)=193
f(67)=487
f(68)=367
f(69)=1
f(70)=5807
f(71)=1987
f(72)=2039
f(73)=251
f(74)=1
f(75)=733
f(76)=6761
f(77)=2309
f(78)=1
f(79)=1453
f(80)=1
f(81)=1
f(82)=599
f(83)=1
f(84)=181
f(85)=757
f(86)=2837
f(87)=223
f(88)=1777
f(89)=1
f(90)=3089
f(91)=9461
f(92)=1
f(93)=1
f(94)=2011
f(95)=263
f(96)=317
f(97)=10667
f(98)=1
f(99)=739

b) Substitution of the polynom
The polynom f(x)=x^2+13x-3 could be written as f(y)= y^2-45.25 with x=y-6.5

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+6.5
f'(x)>2x+12

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

3, 11, 1, 5, 13, 29, 37, 137, 1, 1, 227, 1, 1, 67, 1, 139, 461, 1, 1, 1, 73, 79, 59, 1, 1, 947, 337, 359, 229, 1, 1, 1361, 479, 101, 1, 43, 587, 1847, 1, 1, 1, 1, 769, 1, 167, 1, 2711, 313, 1, 607, 1049, 1087, 307, 233, 241, 1, 1, 443, 823, 283, 1459, 347, 1549, 1, 197, 563, 193, 487, 367, 1, 5807, 1987, 2039, 251, 1, 733, 6761, 2309, 1, 1453, 1, 1, 599, 1, 181, 757, 2837, 223, 1777, 1, 3089, 9461, 1, 1, 2011, 263, 317, 10667, 1, 739, 1, 1279, 1303, 2389, 811, 4129, 12611, 389, 1, 2659, 1, 1, 13997, 1, 1, 14717, 4987, 1, 281, 349, 1, 1, 499, 1, 1, 5749, 449, 613, 401, 1, 18587, 6287, 6379, 353, 1, 6659, 20261, 761, 463, 1, 1, 7237, 373, 1487, 1, 22907, 2579, 1, 953, 1609, 1, 2251, 643, 1693, 1, 1, 1, 26687, 1801, 1823, 2129, 9337, 859, 5737, 1, 1, 1, 1, 2027, 6151, 10369, 10487, 31817, 1, 1, 491, 11087, 1019, 523, 1, 11579, 35111, 3943, 797, 659, 421, 1, 37397, 1, 509, 38567, 1, 4373, 7951, 2677, 1229, 40961, 13789, 557, 1, 4733, 1, 43427, 1, 1, 44687, 1367, 1, 1, 1031, 1, 1, 1223, 3209, 1, 16339, 16487, 1, 1, 1129, 51257, 1567, 17389, 1, 3539, 1373, 54011, 6053, 1, 11083, 1433, 18787, 5167, 3821, 3853, 577, 6529, 1, 919, 1, 547, 61211, 1, 1, 12541, 2341, 7079, 64217, 863, 4349, 1, 1699, 22259, 1, 1, 2531, 5297, 1, 1, 1, 1, 1, 72047, 1613, 1, 1, 853, 24919, 15061, 1, 25469, 76961, 1, 1, 15727, 26399, 2417, 1, 5393, 5431, 82037, 1, 1, 1523, 5623, 28309, 1, 2609, 5779, 3491, 751, 9829, 89057, 1, 1, 1, 709, 30689, 1, 691, 10433, 3259, 857, 1, 19273, 2939, 32537, 1, 1, 1, 1, 33587, 1, 20407, 1, 34439, 727, 1, 2339, 1, 3229, 2749, 107867, 1447, 1, 109847, 4093, 1123, 22369, 1, 1301, 941, 1, 7681, 1783, 12953, 1, 1493, 719, 1, 120017, 40237, 1, 24421, 2729, 13723, 1, 3203, 1, 2297, 42349, 1151, 128477, 1, 2887, 773, 43787, 44029, 1, 1, 1, 1709, 15083, 1, 1, 45989, 46237, 139457, 1, 9397, 141707, 1439, 15913, 1, 9649, 4409, 1, 49009, 1, 1, 5531, 1283, 1, 10111, 10163, 2099, 1, 51599, 1, 1, 5821, 1153, 52919, 967, 2467, 53719, 53987, 14797, 1, 1, 1, 55337, 55609, 33529, 1021, 1, 170111, 1, 1, 34519, 57809, 2003, 13469, 1, 2357, 1, 1, 1, 1, 1097, 4663, 182711, 1423, 2459, 37057, 1, 1889, 187877, 12583, 12641, 17317, 1, 1, 2971, 1, 1, 195761, 1, 1013, 7937, 1, 6067, 201107, 1, 1, 203807, 1, 68539, 1, 13829, 1877, 16097, 1, 4691, 42403, 70979, 71287, 214787, 1, 1, 217577, 8093, 1, 4007, 1, 1723, 223211, 74719, 1, 1559, 2293, 8443, 17609, 1, 1399, 7993, 77587, 1, 1, 5237, 26293, 21601, 1187, 15971, 48109, 1, 1, 6581, 1811, 1091, 18959, 82487, 7529, 49891, 1, 1061, 1, 28163, 1, 1381, 1, 85837, 258527, 1, 1, 261587, 29179, 2663, 1231, 17713, 1327, 1, 89599, 17989, 1, 1, 2333, 274007, 1667, 1, 277157, 92737, 93089, 11213, 1, 1, 283511, 3271, 1, 1, 95929, 1319, 289937, 1, 6491, 293177, 1, 7573, 1, 3967, 9049, 299711, 1, 6709, 60601, 1, 101737, 27847, 4099, 1, 23819, 1, 11551, 1, 1, 1, 10909, 105829, 1931, 63949, 1, 1, 1, 1663, 4339, 326567, 1, 109619, 5077, 1, 36923, 2399, 111539, 1, 1, 1, 8699, 5081, 7591, 2539, 31267, 8849, 115429, 69493, 2113, 4021, 351011, 39133, 7853, 1, 118589, 1, 358157, 23957, 1, 32887, 4481, 1, 1, 24439, 122599, 1, 1, 24763, 1, 41543, 1, 28949, 25171, 25253, 10271, 1, 2161, 6977, 1, 3301, 5783, 11779, 25997, 78241, 1, 1, 394967, 1, 1, 1409, 133337, 1, 1, 26921, 135029, 406361, 45293, 1, 82039, 1, 137587, 9629, 27689, 1, 9719, 3583, 46723, 2909, 1, 1, 1, 3847, 5711, 2963, 1451, 48029, 433577, 1, 2237, 2221, 1, 13339, 1, 1, 16451, 1, 148949, 29879, 17981, 1, 1, 1, 1, 10139, 457607, 2593, 1, 1, 30869, 154799, 465761, 1, 1, 8543, 1, 1, 473987, 1, 1, 36779, 53279, 1979, 96457, 6449, 14699, 2143, 1, 1, 1, 54673, 54829, 6779, 1, 33179, 1481, 1, 167309, 100669, 3739, 5113, 39047, 5851, 34031, 2767, 1, 1, 1, 11503, 1, 2273, 15817, 174469, 104971, 2699, 175919, 14303, 58963, 1, 8209, 178349, 4159, 48907, 35963, 36061, 1, 20143, 1, 109363, 3323, 183259, 19009, 14173, 3359, 22229, 1, 20693, 560207, 37447, 37547, 1, 5101, 189239, 22769, 1, 1, 573761, 1, 38453, 115663, 1, 1, 582887, 1, 1, 53407, 196337, 1949, 9109, 1583, 6841, 13877, 6043, 1, 2797, 2753, 1, 606017, 8101, 40609, 1, 5233, 1, 1, 41131, 206179, 620111, 18839, 1, 9613, 23201, 69779, 1, 1, 1, 634367, 3593, 1, 11621, 1, 1, 643961, 16553, 8629, 129757, 19709, 217337, 1, 14561, 1, 658487, 1, 220579, 1, 1, 17093, 1, 1, 14923, 26927, 1, 1, 1, 45317, 45427, 18461, 76079, 2311, 1, 45979, 230449, 1, 231559, 3571, 1, 77743, 77929, 19001, 4271, 47093, 8963, 18199, 21559, 1, 1, 79613, 1, 1, 1, 13151, 3607, 8353, 728417, 1, 1, 12433, 245087, 1, 147739, 3797, 1, 2423, 2239, 16607, 13619, 250259, 250837, 4463, 1, 10103, 759467, 1, 6521, 152941, 1, 23279, 769961, 257239, 10313, 155047, 1, 28843, 70957, 52153, 1, 60449, 1, 263129, 158233, 1, 1, 4127, 266099, 1, 1, 3391, 268487, 73387, 1, 1, 18899, 3719, 1, 1, 4969, 273899, 63347, 91703, 1, 5717, 276929, 1, 14143, 1, 1, 2633, 93529, 1, 1, 1, 1, 11657, 284269, 1, 13177, 10597, 8689, 862067, 11519, 57719, 78877, 289837, 22343, 174649, 19447, 1, 23753, 2053, 2029, 176899, 295459, 2447, 890147, 1, 1, 68909, 6959, 2969, 1, 1, 1, 907211, 3061, 4049, 182587, 304949, 4561, 918677, 4721, 5591, 924437, 102929, 34381, 1, 62143, 311359, 3331, 10781, 62659, 1, 9533, 1, 2731, 1, 1, 953507, 24499, 1, 1, 7121, 107033, 87751, 322409, 64613, 1, 1, 7559, 977087, 21757, 1, 4219, 1, 2719, 1, 1, 1, 3119, 8521, 1, 200191, 334319, 334987, 2671, 1, 67399,

6. Sequence of the polynom (only primes)

3, 11, 5, 13, 29, 37, 137, 227, 67, 139, 461, 73, 79, 59, 947, 337, 359, 229, 1361, 479, 101, 43, 587, 1847, 769, 167, 2711, 313, 607, 1049, 1087, 307, 233, 241, 443, 823, 283, 1459, 347, 1549, 197, 563, 193, 487, 367, 5807, 1987, 2039, 251, 733, 6761, 2309, 1453, 599, 181, 757, 2837, 223, 1777, 3089, 9461, 2011, 263, 317, 10667, 739, 1279, 1303, 2389, 811, 4129, 12611, 389, 2659, 13997, 14717, 4987, 281, 349, 499, 5749, 449, 613, 401, 18587, 6287, 6379, 353, 6659, 20261, 761, 463, 7237, 373, 1487, 22907, 2579, 953, 1609, 2251, 643, 1693, 26687, 1801, 1823, 2129, 9337, 859, 5737, 2027, 6151, 10369, 10487, 31817, 491, 11087, 1019, 523, 11579, 35111, 3943, 797, 659, 421, 37397, 509, 38567, 4373, 7951, 2677, 1229, 40961, 13789, 557, 4733, 43427, 44687, 1367, 1031, 1223, 3209, 16339, 16487, 1129, 51257, 1567, 17389, 3539, 1373, 54011, 6053, 11083, 1433, 18787, 5167, 3821, 3853, 577, 6529, 919, 547, 61211, 12541, 2341, 7079, 64217, 863, 4349, 1699, 22259, 2531, 5297, 72047, 1613, 853, 24919, 15061, 25469, 76961, 15727, 26399, 2417, 5393, 5431, 82037, 1523, 5623, 28309, 2609, 5779, 3491, 751, 9829, 89057, 709, 30689, 691, 10433, 3259, 857, 19273, 2939, 32537, 33587, 20407, 34439, 727, 2339, 3229, 2749, 107867, 1447, 109847, 4093, 1123, 22369, 1301, 941, 7681, 1783, 12953, 1493, 719, 120017, 40237, 24421, 2729, 13723, 3203, 2297, 42349, 1151, 128477, 2887, 773, 43787, 44029, 1709, 15083, 45989, 46237, 139457, 9397, 141707, 1439, 15913, 9649, 4409, 49009, 5531, 1283, 10111, 10163, 2099, 51599, 5821, 1153, 52919, 967, 2467, 53719, 53987, 14797, 55337, 55609, 33529, 1021, 170111, 34519, 57809, 2003, 13469, 2357, 1097, 4663, 182711, 1423, 2459, 37057, 1889, 187877, 12583, 12641, 17317, 2971, 195761, 1013, 7937, 6067, 201107, 203807, 68539, 13829, 1877, 16097, 4691, 42403, 70979, 71287, 214787, 217577, 8093, 4007, 1723, 223211, 74719, 1559, 2293, 8443, 17609, 1399, 7993, 77587, 5237, 26293, 21601, 1187, 15971, 48109, 6581, 1811, 1091, 18959, 82487, 7529, 49891, 1061, 28163, 1381, 85837, 258527, 261587, 29179, 2663, 1231, 17713, 1327, 89599, 17989, 2333, 274007, 1667, 277157, 92737, 93089, 11213, 283511, 3271, 95929, 1319, 289937, 6491, 293177, 7573, 3967, 9049, 299711, 6709, 60601, 101737, 27847, 4099, 23819, 11551, 10909, 105829, 1931, 63949, 1663, 4339, 326567, 109619, 5077, 36923, 2399, 111539, 8699, 5081, 7591, 2539, 31267, 8849, 115429, 69493, 2113, 4021, 351011, 39133, 7853, 118589, 358157, 23957, 32887, 4481, 24439, 122599, 24763, 41543, 28949, 25171, 25253, 10271, 2161, 6977, 3301, 5783, 11779, 25997, 78241, 394967, 1409, 133337, 26921, 135029, 406361, 45293, 82039, 137587, 9629, 27689, 9719, 3583, 46723, 2909, 3847, 5711, 2963, 1451, 48029, 433577, 2237, 2221, 13339, 16451, 148949, 29879, 17981, 10139, 457607, 2593, 30869, 154799, 465761, 8543, 473987, 36779, 53279, 1979, 96457, 6449, 14699, 2143, 54673, 54829, 6779, 33179, 1481, 167309, 100669, 3739, 5113, 39047, 5851, 34031, 2767, 11503, 2273, 15817, 174469, 104971, 2699, 175919, 14303, 58963, 8209, 178349, 4159, 48907, 35963, 36061, 20143, 109363, 3323, 183259, 19009, 14173, 3359, 22229, 20693, 560207, 37447, 37547, 5101, 189239, 22769, 573761, 38453, 115663, 582887, 53407, 196337, 1949, 9109, 1583, 6841, 13877, 6043, 2797, 2753, 606017, 8101, 40609, 5233, 41131, 206179, 620111, 18839, 9613, 23201, 69779, 634367, 3593, 11621, 643961, 16553, 8629, 129757, 19709, 217337, 14561, 658487, 220579, 17093, 14923, 26927, 45317, 45427, 18461, 76079, 2311, 45979, 230449, 231559, 3571, 77743, 77929, 19001, 4271, 47093, 8963, 18199, 21559, 79613, 13151, 3607, 8353, 728417, 12433, 245087, 147739, 3797, 2423, 2239, 16607, 13619, 250259, 250837, 4463, 10103, 759467, 6521, 152941, 23279, 769961, 257239, 10313, 155047, 28843, 70957, 52153, 60449, 263129, 158233, 4127, 266099, 3391, 268487, 73387, 18899, 3719, 4969, 273899, 63347, 91703, 5717, 276929, 14143, 2633, 93529, 11657, 284269, 13177, 10597, 8689, 862067, 11519, 57719, 78877, 289837, 22343, 174649, 19447, 23753, 2053, 2029, 176899, 295459, 2447, 890147, 68909, 6959, 2969, 907211, 3061, 4049, 182587, 304949, 4561, 918677, 4721, 5591, 924437, 102929, 34381, 62143, 311359, 3331, 10781, 62659, 9533, 2731, 953507, 24499, 7121, 107033, 87751, 322409, 64613, 7559, 977087, 21757, 4219, 2719, 3119, 8521, 200191, 334319, 334987, 2671, 67399,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+13x-3 and
the reducible primes which appear as divisor for the first time
p | x^2+13x-3 and p < x^2+13x-3

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)
1107430.7000000.4000000.3000000.0000000.0000000.000000
21006414500.6400000.1400000.5000009.1428583.50000016.666666
31.000659875720.6590000.0870000.57200010.2968756.21428611.440000
410.0006.7155716.1440.6715000.0571000.61440010.1896816.56321910.741259
5100.00067.6344.27963.3550.6763400.0427900.63355010.0720787.49387010.311687
61.000.000678.96234.918644.0440.6789620.0349180.64404410.0387688.16031710.165638
710.000.0006.806.274295.8596.510.4150.6806270.0295860.65104210.0245298.47296510.108649
8100.000.00068.201.4832.565.19565.636.2880.6820150.0256520.65636310.0203848.67032910.081737
91.000.000.000683.137.29822.621.103660.516.1950.6831370.0226210.66051610.0164598.81847410.063278
1010.000.000.0006.840.595.009202.403.8486.638.191.1610.6840600.0202400.66381910.0134998.94756710.050005


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
243210.7500000.5000000.2500001.5000001.000000inf
386330.7500000.3750000.3750002.0000001.5000003.000000
41610550.6250000.3125000.3125001.6666671.6666671.666667
532197120.5937500.2187500.3750001.9000001.4000002.400000
664399300.6093750.1406250.4687502.0526321.2857142.500000
71288217650.6406250.1328120.5078122.1025641.8888892.166667
8256170321380.6640620.1250000.5390622.0731711.8823532.123077
9512336532830.6562500.1035160.5527341.9764711.6562502.050725
101.024677895880.6611330.0869140.5742192.0148811.6792452.077739
112.0481.3551501.2050.6616210.0732420.5883792.0014771.6853932.049320
124.0962.7302732.4570.6665040.0666500.5998542.0147601.8200002.039004
138.1925.4884815.0070.6699220.0587160.6112062.0102571.7619052.037851
1416.38411.04585210.1930.6741330.0520020.6221312.0125731.7713102.035750
1532.76822.0951.55420.5410.6742860.0474240.6268622.0004531.8239442.015207
1665.53644.2782.90441.3740.6756290.0443120.6313172.0039831.8687262.014215
17131.07288.6595.46983.1900.6764140.0417250.6346892.0023261.8832642.010683
18262.144177.56110.306167.2550.6773410.0393140.6380272.0027411.8844402.010518
19524.288355.45419.343336.1110.6779750.0368940.6410812.0018701.8768682.009572
201.048.576711.92736.525675.4020.6789460.0348330.6441142.0028671.8882802.009461
212.097.1521.425.00069.2351.355.7650.6794930.0330140.6464792.0016101.8955512.007345
224.194.3042.852.122131.9682.720.1540.6799990.0314640.6485352.0014891.9060882.006361
238.388.6085.708.588251.1715.457.4170.6805170.0299420.6505752.0015231.9032722.006290
2416.777.21611.424.681479.47510.945.2060.6809640.0285790.6523852.0013151.9089582.005565
2533.554.43222.863.524919.19021.944.3340.6813860.0273940.6539922.0012401.9170762.004926
2667.108.86445.755.1301.763.12743.992.0030.6818050.0262730.6555322.0012281.9181312.004709
27134.217.72891.559.5153.384.58588.174.9300.6821720.0252170.6569542.0010761.9196492.004340
28268.435.456183.214.4016.509.683176.704.7180.6825270.0242500.6582762.0010421.9233332.004025
29536.870.912366.603.55312.541.507354.062.0460.6828520.0233600.6594922.0009541.9265932.003693
301.073.741.824733.545.23824.199.843709.345.3950.6831670.0225380.6606292.0009221.9295802.003449
312.147.483.6481.467.718.60046.746.5341.420.972.0660.6834600.0217680.6616922.0008561.9316872.003216
324.294.967.2962.936.647.23590.421.4192.846.225.8160.6837410.0210530.6626892.0008241.9342912.003013
338.589.934.5925.875.536.919175.081.5845.700.455.3350.6840030.0203820.6636202.0007641.9362842.002812
3417.179.869.18411.755.345.528339.347.16811.415.998.3600.6842510.0197530.6644992.0007271.9382232.002647
3534.359.738.36823.518.756.231658.383.72622.860.372.5050.6844860.0191610.6653242.0006861.9401482.002486
3668.719.476.73647.052.896.0581.278.493.75245.774.402.3060.6847100.0186050.6661052.0006541.9418672.002347


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
122010200
242010200
383021200
4165041310
5327062410
6649082412
7128170154643
82563203081167
9512530511417913
101.0248908725241426
112.048150014835423043
124.096273027165715978
138.1924810479121122113125
1416.3848520850222211202217
1532.7681.55401.552397387397373
1665.5362.90402.902747728745684
17131.0725.46905.4671.4091.3441.3841.332
18262.14410.306010.3042.6172.5252.5902.574
19524.28819.343019.3414.8894.7434.8484.863
201.048.57636.525036.5239.1519.0309.1309.214
212.097.15269.235069.23317.33217.21217.34217.349
224.194.304131.9680131.96633.05532.94132.96733.005
238.388.608251.1710251.16962.82462.79162.71462.842
2416.777.216479.4750479.473119.913119.925119.778119.859
2533.554.432919.1900919.188229.713229.857229.607230.013
2667.108.8641.763.12701.763.125440.376441.458440.506440.787
27134.217.7283.384.58503.384.583846.126847.034845.811845.614
28268.435.4566.509.68306.509.6811.627.3781.627.7821.627.2061.627.317
29536.870.91212.541.507012.541.5053.136.0463.135.2893.134.9273.135.245
301.073.741.82424.199.843024.199.8416.050.3726.050.3306.049.6186.049.523
312.147.483.64846.746.534046.746.53211.688.71511.685.78411.685.53211.686.503
324.294.967.29690.421.419090.421.41722.609.73222.603.99122.604.18722.603.509
338.589.934.592175.081.5840175.081.58243.774.36243.773.49043.763.91343.769.819
3417.179.869.184339.347.1680339.347.16684.837.25984.849.02984.826.09284.834.788
3534.359.738.368658.383.7260658.383.724164.601.619164.606.019164.579.457164.596.631
3668.719.476.7361.278.493.75201.278.493.750319.619.688319.631.005319.608.591319.634.468


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
241100010
383210030
4165410230
53212842343
6643018127977
712865402514181815
8256138756329383833
951228314913462757868
101.024588301287134144157153
112.0481.205623582285306301313
124.0962.4571.2891.168583614625635
138.1925.0072.5992.4081.2181.2671.2421.280
1416.38410.1935.2814.9122.5052.5512.5422.595
1532.76820.54110.6189.9235.1295.0965.1335.183
1665.53641.37421.29920.07510.32710.26710.33610.444
17131.07283.19042.76640.42420.71020.66720.88720.926
18262.144167.25586.08981.16641.70341.77241.84741.933
19524.288336.111172.424163.68783.97483.95183.95884.228
201.048.576675.402346.126329.276169.044168.524168.852168.982
212.097.1521.355.765694.087661.678339.160338.465339.240338.900
224.194.3042.720.1541.390.3791.329.775680.847679.467679.814680.026
238.388.6085.457.4172.787.2012.670.2161.364.6941.364.1031.364.0811.364.539
2416.777.21610.945.2065.583.3705.361.8362.736.1352.736.5312.736.7352.735.805
2533.554.43221.944.33411.185.79910.758.5355.485.6185.486.6185.488.0625.484.036
2667.108.86443.992.00322.405.71721.586.28610.998.58910.999.71510.999.42810.994.271
27134.217.72888.174.93044.871.51243.303.41822.043.01722.048.76922.044.49222.038.652
28268.435.456176.704.71889.861.80086.842.91844.173.77244.181.27844.181.57544.168.093
29536.870.912354.062.046179.941.248174.120.79888.516.19588.517.78688.511.38188.516.684
301.073.741.824709.345.395360.272.552349.072.843177.320.884177.345.681177.339.889177.338.941
312.147.483.6481.420.972.066721.292.012699.680.054355.238.280355.248.435355.251.394355.233.957
324.294.967.2962.846.225.8161.444.017.2681.402.208.548711.552.265711.585.260711.542.959711.545.332
338.589.934.5925.700.455.3352.890.670.9392.809.784.3961.425.071.4661.425.165.9511.425.137.3881.425.080.530
3417.179.869.18411.415.998.3605.786.259.1465.629.739.2142.853.976.8102.854.002.7402.854.027.9782.853.990.832
3534.359.738.36822.860.372.50511.581.886.04211.278.486.4635.715.062.3725.715.067.9405.715.180.2815.715.061.912
3668.719.476.73645.774.402.30623.181.515.34722.592.886.95911.443.588.48811.443.654.47411.443.613.80011.443.545.544


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 11, 1, 5, 13, 29, 37, 137, 1, 1, 227, 1, 1, 67, 1, 139, 461, 1, 1, 1,
Found in Database : 3, 11, 5, 13, 29, 37, 137, 227, 67, 139, 461, 73, 79, 59, 947, 337, 359, 229, 1361, 479, 101, 43, 587, 1847,
Found in Database : 3, 5, 11, 13, 29, 37, 43, 59, 67, 73, 79, 101, 137, 139,