Inhaltsverzeichnis

Development of
Algorithmic Constructions

08:36:19
Deutsch
29.Mar 2024

Polynom = x^2-118x+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) = 57 = 3*19
f(2) = 229 = 229
f(3) = 171 = 3*3*19
f(4) = 453 = 3*151
f(5) = 281 = 281
f(6) = 669 = 3*223
f(7) = 387 = 3*3*43
f(8) = 877 = 877
f(9) = 489 = 3*163
f(10) = 1077 = 3*359
f(11) = 587 = 587
f(12) = 1269 = 3*3*3*47
f(13) = 681 = 3*227
f(14) = 1453 = 1453
f(15) = 771 = 3*257
f(16) = 1629 = 3*3*181
f(17) = 857 = 857
f(18) = 1797 = 3*599
f(19) = 939 = 3*313
f(20) = 1957 = 19*103
f(21) = 1017 = 3*3*113
f(22) = 2109 = 3*19*37
f(23) = 1091 = 1091
f(24) = 2253 = 3*751
f(25) = 1161 = 3*3*3*43
f(26) = 2389 = 2389
f(27) = 1227 = 3*409
f(28) = 2517 = 3*839
f(29) = 1289 = 1289
f(30) = 2637 = 3*3*293
f(31) = 1347 = 3*449
f(32) = 2749 = 2749
f(33) = 1401 = 3*467
f(34) = 2853 = 3*3*317
f(35) = 1451 = 1451
f(36) = 2949 = 3*983
f(37) = 1497 = 3*499
f(38) = 3037 = 3037
f(39) = 1539 = 3*3*3*3*19
f(40) = 3117 = 3*1039
f(41) = 1577 = 19*83
f(42) = 3189 = 3*1063
f(43) = 1611 = 3*3*179
f(44) = 3253 = 3253
f(45) = 1641 = 3*547
f(46) = 3309 = 3*1103
f(47) = 1667 = 1667
f(48) = 3357 = 3*3*373
f(49) = 1689 = 3*563
f(50) = 3397 = 43*79
f(51) = 1707 = 3*569
f(52) = 3429 = 3*3*3*127
f(53) = 1721 = 1721
f(54) = 3453 = 3*1151
f(55) = 1731 = 3*577
f(56) = 3469 = 3469
f(57) = 1737 = 3*3*193
f(58) = 3477 = 3*19*61
f(59) = 1739 = 37*47
f(60) = 3477 = 3*19*61
f(61) = 1737 = 3*3*193
f(62) = 3469 = 3469
f(63) = 1731 = 3*577
f(64) = 3453 = 3*1151
f(65) = 1721 = 1721
f(66) = 3429 = 3*3*3*127
f(67) = 1707 = 3*569
f(68) = 3397 = 43*79
f(69) = 1689 = 3*563
f(70) = 3357 = 3*3*373
f(71) = 1667 = 1667
f(72) = 3309 = 3*1103
f(73) = 1641 = 3*547
f(74) = 3253 = 3253
f(75) = 1611 = 3*3*179
f(76) = 3189 = 3*1063
f(77) = 1577 = 19*83
f(78) = 3117 = 3*1039
f(79) = 1539 = 3*3*3*3*19
f(80) = 3037 = 3037
f(81) = 1497 = 3*499
f(82) = 2949 = 3*983
f(83) = 1451 = 1451
f(84) = 2853 = 3*3*317
f(85) = 1401 = 3*467
f(86) = 2749 = 2749
f(87) = 1347 = 3*449
f(88) = 2637 = 3*3*293
f(89) = 1289 = 1289
f(90) = 2517 = 3*839
f(91) = 1227 = 3*409
f(92) = 2389 = 2389
f(93) = 1161 = 3*3*3*43
f(94) = 2253 = 3*751
f(95) = 1091 = 1091
f(96) = 2109 = 3*19*37
f(97) = 1017 = 3*3*113
f(98) = 1957 = 19*103
f(99) = 939 = 3*313
f(100) = 1797 = 3*599

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^2-118x+3

f(0)=3
f(1)=19
f(2)=229
f(3)=1
f(4)=151
f(5)=281
f(6)=223
f(7)=43
f(8)=877
f(9)=163
f(10)=359
f(11)=587
f(12)=47
f(13)=227
f(14)=1453
f(15)=257
f(16)=181
f(17)=857
f(18)=599
f(19)=313
f(20)=103
f(21)=113
f(22)=37
f(23)=1091
f(24)=751
f(25)=1
f(26)=2389
f(27)=409
f(28)=839
f(29)=1289
f(30)=293
f(31)=449
f(32)=2749
f(33)=467
f(34)=317
f(35)=1451
f(36)=983
f(37)=499
f(38)=3037
f(39)=1
f(40)=1039
f(41)=83
f(42)=1063
f(43)=179
f(44)=3253
f(45)=547
f(46)=1103
f(47)=1667
f(48)=373
f(49)=563
f(50)=79
f(51)=569
f(52)=127
f(53)=1721
f(54)=1151
f(55)=577
f(56)=3469
f(57)=193
f(58)=61
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-118x+3 could be written as f(y)= y^2-3478 with x=y+59

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-59
f'(x)>2x-119 with x > 59

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, 19, 229, 1, 151, 281, 223, 43, 877, 163, 359, 587, 47, 227, 1453, 257, 181, 857, 599, 313, 103, 113, 37, 1091, 751, 1, 2389, 409, 839, 1289, 293, 449, 2749, 467, 317, 1451, 983, 499, 3037, 1, 1039, 83, 1063, 179, 3253, 547, 1103, 1667, 373, 563, 79, 569, 127, 1721, 1151, 577, 3469, 193, 61, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 491, 1, 1, 439, 337, 191, 1283, 1, 521, 853, 617, 1, 1, 383, 1, 1303, 307, 487, 3083, 541, 379, 1789, 1249, 653, 4091, 1, 1481, 2311, 1601, 277, 5171, 1, 1, 1, 659, 1021, 6323, 1087, 1, 3463, 2377, 1223, 7547, 431, 2657, 4093, 2801, 479, 239, 1511, 1, 4759, 1, 1663, 10211, 1741, 1187, 1, 1, 1901, 1, 661, 4049, 6199, 4217, 1, 13163, 2237, 4561, 367, 1579, 1, 14747, 2503, 1699, 1, 5281, 2687, 349, 1, 5657, 8629, 5849, 991, 18131, 1, 1, 9511, 1, 3271, 1049, 3373, 761, 10429, 7057, 3581, 21803, 1229, 7481, 11383, 1, 1301, 23747, 4013, 1, 12373, 929, 1, 25763, 1, 2939, 13399, 9049, 4583, 27851, 1567, 9521, 14461, 1, 1, 30011, 1, 1, 15559, 3499, 1, 1697, 5437, 1, 16693, 11257, 5693, 1, 647, 11777, 17863, 12041, 2029, 36923, 6221, 12577, 19069, 4283, 1, 39371, 1, 1489, 1069, 13681, 6911, 1, 2351, 14249, 21589, 14537, 2447, 44483, 7487, 15121, 619, 571, 1, 47147, 7933, 1, 1, 859, 8237, 601, 2797, 16937, 25639, 1, 967, 52691, 8861, 17881, 27061, 6067, 9181, 911, 9343, 1, 1, 1, 509, 1361, 1093, 19841, 30013, 20177, 3391, 61547, 10343, 20857, 31543, 1, 10687, 1, 10861, 811, 1, 1171, 11213, 1, 3797, 22961, 1, 23321, 3917, 1, 11933, 24049, 1, 2713, 12301, 74363, 12487, 8387, 809, 25537, 677, 77747, 1, 26297, 39733, 26681, 1493, 81203, 1, 27457, 41479, 9283, 1, 84731, 14221, 9547, 43261, 1, 14621, 4649, 1, 1571, 739, 1, 5077, 92003, 1, 31081, 46933, 10499, 1, 95747, 16063, 3593, 48823, 1, 16487, 99563, 1, 33617, 2671, 1, 5711, 103451, 17351, 743, 52711, 3929, 17791, 2903, 18013, 1, 54709, 36697, 18461, 111443, 6229, 1979, 1, 2003, 709, 115547, 19373, 38977, 1, 13147, 1, 119723, 20071, 1, 60919, 40849, 20543, 821, 2309, 41801, 3319, 42281, 1, 128291, 21503, 1, 65239, 1, 21991, 1, 1, 4969, 67453, 1, 1, 137147, 1, 1, 1621, 2459, 7829, 7457, 23741, 47737, 1, 1787, 1, 146291, 1, 16427, 74311, 49801, 25031, 1, 8431, 50849, 1783, 1, 1, 155723, 1373, 1, 79063, 17659, 1, 887, 26893, 1, 1, 54601, 27437, 165443, 3079, 55697, 1, 56249, 9421, 8969, 28541, 3019, 86461, 1, 29101, 3733, 29383, 1, 1459, 59617, 1, 180563, 10079, 60761, 91573, 1, 10271, 185747, 1637, 62497, 4957, 1, 31687, 1, 31981, 21419, 2617, 64849, 32573, 196331, 10957, 66041, 1, 1, 3719, 201731, 33773, 3571, 102229, 1201, 34381, 1, 34687, 23227, 1, 70297, 1, 2693, 1319, 71537, 107773, 72161, 1, 218363, 36551, 73417, 5821, 24683, 1, 1, 37501, 8369, 1367, 2053, 1, 2909, 12821, 77249, 116359, 1277, 13037, 3863, 1, 79201, 119293, 1, 40093, 12713, 40423, 27059, 122263, 1, 1, 1949, 13807, 83177, 125269, 1, 1559, 5897, 1, 85201, 128311, 28627, 2269, 259691, 2287, 1, 1583, 1871, 1193, 265883, 4943, 1, 134503, 90017, 15061, 6329, 45533, 1, 137653, 30707, 46237, 14657, 46591, 1, 140839, 1, 47303, 1, 15887, 1, 144061, 96401, 16127, 291371, 1, 97849, 147319, 1217, 1, 1973, 1, 33347, 7927, 1, 1, 8231, 16981, 1, 1, 103001, 5743, 1, 1, 1259, 3347, 35083, 52813, 1, 1237, 1873, 160711, 5659, 53951, 1, 6037, 109049, 164149, 1, 18367, 3221, 55487, 1, 167623, 37379, 56263, 338747, 1, 4211, 9007, 1, 3023, 345803, 1, 2699, 174679, 116849, 19541, 352931, 59021, 1427, 178261, 13249, 1, 3187, 60223, 40283, 1, 1, 61031, 1, 20479, 1, 185533, 124097, 6917, 1279, 1, 125737, 189223, 42187, 63487, 382163, 63901, 1, 192949, 129049, 3407, 389651, 1, 3533, 1, 131561, 21997, 397211, 66413, 1, 4663, 44699, 67261, 404843, 1, 1, 1609, 136657, 68543, 21713, 1, 7283, 208213, 139241, 23279, 420323, 70271, 140977, 4933, 15761, 71143, 4157, 1523, 1, 216061, 144481, 72461, 436091, 1, 146249, 1, 147137, 1, 1, 1, 148921, 3673, 49939, 75133, 1, 75583, 50539, 228103, 3547, 76487, 2543, 1, 8123, 232189, 8171, 25951, 468491, 78311, 157081, 1, 52667, 79231, 1553, 79693, 17761, 1, 3739, 80621, 485123, 1, 162641, 1, 4421, 1439, 10501, 82493, 165457, 248893, 6163, 83437, 502043, 83911, 56099, 253159, 169249, 1, 510611, 28447, 171161, 3259, 9059, 1, 27329, 86783, 174049, 261799, 1, 87751, 6361, 88237, 1, 1487, 2917, 89213, 1, 9967, 179897, 270583, 1, 1, 1619, 4799, 182857, 1, 61283, 92173, 554531, 92671, 6883, 279511, 186841, 1993, 4987, 31391, 1, 1, 1, 31727, 30137, 1, 10099, 288583, 21433, 96703, 1, 97213, 1, 293173, 1543, 1, 590963, 32917, 198017, 1, 199049, 11087, 16223, 5279, 201121, 15919, 1, 101341, 14177, 101863, 1, 1, 205297, 102911, 2203, 1, 207401, 5113, 208457, 1, 628547, 105023, 11083, 316663, 1, 106087, 638123, 106621, 23753, 321469, 214849, 107693, 647771, 1, 217001, 2161, 218081, 1, 657491, 2969, 1, 17431, 24593, 5839, 667283, 111487, 74507, 336103, 224617, 112583, 677147, 1, 226817, 1, 2017, 4231, 687083, 1, 230137, 346039, 4057, 115903, 1931, 116461, 1, 351061, 1, 1, 1, 13127, 1, 7577, 237977, 1, 717323, 1, 1, 361213, 1, 6367, 11927, 6397, 27073, 366343, 1, 1, 737843, 1, 2399, 371509, 1, 1, 748211, 124991, 5827, 1, 9323, 126151, 39929, 126733, 4457, 381949, 255217, 1, 4297, 42829, 257561, 3049, 258737, 14407, 779747, 130253, 1, 1, 87427, 131437, 790403, 6949, 1877, 1, 3203, 1, 2851, 14869, 1, 403261, 269441, 45007, 811931, 135623, 271849, 1, 91019, 1, 822803, 137437, 1, 9631, 14563, 2273, 833747, 1, 279137, 419623, 1, 46829, 844763, 141101, 1, 425149, 1, 1, 855851, 1, 95507, 22669, 3467, 7589, 867011, 48271, 290249, 436309, 6779, 16229, 11117, 3121,

6. Sequence of the polynom (only primes)

3, 19, 229, 151, 281, 223, 43, 877, 163, 359, 587, 47, 227, 1453, 257, 181, 857, 599, 313, 103, 113, 37, 1091, 751, 2389, 409, 839, 1289, 293, 449, 2749, 467, 317, 1451, 983, 499, 3037, 1039, 83, 1063, 179, 3253, 547, 1103, 1667, 373, 563, 79, 569, 127, 1721, 1151, 577, 3469, 193, 61, 491, 439, 337, 191, 1283, 521, 853, 617, 383, 1303, 307, 487, 3083, 541, 379, 1789, 1249, 653, 4091, 1481, 2311, 1601, 277, 5171, 659, 1021, 6323, 1087, 3463, 2377, 1223, 7547, 431, 2657, 4093, 2801, 479, 239, 1511, 4759, 1663, 10211, 1741, 1187, 1901, 661, 4049, 6199, 4217, 13163, 2237, 4561, 367, 1579, 14747, 2503, 1699, 5281, 2687, 349, 5657, 8629, 5849, 991, 18131, 9511, 3271, 1049, 3373, 761, 10429, 7057, 3581, 21803, 1229, 7481, 11383, 1301, 23747, 4013, 12373, 929, 25763, 2939, 13399, 9049, 4583, 27851, 1567, 9521, 14461, 30011, 15559, 3499, 1697, 5437, 16693, 11257, 5693, 647, 11777, 17863, 12041, 2029, 36923, 6221, 12577, 19069, 4283, 39371, 1489, 1069, 13681, 6911, 2351, 14249, 21589, 14537, 2447, 44483, 7487, 15121, 619, 571, 47147, 7933, 859, 8237, 601, 2797, 16937, 25639, 967, 52691, 8861, 17881, 27061, 6067, 9181, 911, 9343, 509, 1361, 1093, 19841, 30013, 20177, 3391, 61547, 10343, 20857, 31543, 10687, 10861, 811, 1171, 11213, 3797, 22961, 23321, 3917, 11933, 24049, 2713, 12301, 74363, 12487, 8387, 809, 25537, 677, 77747, 26297, 39733, 26681, 1493, 81203, 27457, 41479, 9283, 84731, 14221, 9547, 43261, 14621, 4649, 1571, 739, 5077, 92003, 31081, 46933, 10499, 95747, 16063, 3593, 48823, 16487, 99563, 33617, 2671, 5711, 103451, 17351, 743, 52711, 3929, 17791, 2903, 18013, 54709, 36697, 18461, 111443, 6229, 1979, 2003, 709, 115547, 19373, 38977, 13147, 119723, 20071, 60919, 40849, 20543, 821, 2309, 41801, 3319, 42281, 128291, 21503, 65239, 21991, 4969, 67453, 137147, 1621, 2459, 7829, 7457, 23741, 47737, 1787, 146291, 16427, 74311, 49801, 25031, 8431, 50849, 1783, 155723, 1373, 79063, 17659, 887, 26893, 54601, 27437, 165443, 3079, 55697, 56249, 9421, 8969, 28541, 3019, 86461, 29101, 3733, 29383, 1459, 59617, 180563, 10079, 60761, 91573, 10271, 185747, 1637, 62497, 4957, 31687, 31981, 21419, 2617, 64849, 32573, 196331, 10957, 66041, 3719, 201731, 33773, 3571, 102229, 1201, 34381, 34687, 23227, 70297, 2693, 1319, 71537, 107773, 72161, 218363, 36551, 73417, 5821, 24683, 37501, 8369, 1367, 2053, 2909, 12821, 77249, 116359, 1277, 13037, 3863, 79201, 119293, 40093, 12713, 40423, 27059, 122263, 1949, 13807, 83177, 125269, 1559, 5897, 85201, 128311, 28627, 2269, 259691, 2287, 1583, 1871, 1193, 265883, 4943, 134503, 90017, 15061, 6329, 45533, 137653, 30707, 46237, 14657, 46591, 140839, 47303, 15887, 144061, 96401, 16127, 291371, 97849, 147319, 1217, 1973, 33347, 7927, 8231, 16981, 103001, 5743, 1259, 3347, 35083, 52813, 1237, 1873, 160711, 5659, 53951, 6037, 109049, 164149, 18367, 3221, 55487, 167623, 37379, 56263, 338747, 4211, 9007, 3023, 345803, 2699, 174679, 116849, 19541, 352931, 59021, 1427, 178261, 13249, 3187, 60223, 40283, 61031, 20479, 185533, 124097, 6917, 1279, 125737, 189223, 42187, 63487, 382163, 63901, 192949, 129049, 3407, 389651, 3533, 131561, 21997, 397211, 66413, 4663, 44699, 67261, 404843, 1609, 136657, 68543, 21713, 7283, 208213, 139241, 23279, 420323, 70271, 140977, 4933, 15761, 71143, 4157, 1523, 216061, 144481, 72461, 436091, 146249, 147137, 148921, 3673, 49939, 75133, 75583, 50539, 228103, 3547, 76487, 2543, 8123, 232189, 8171, 25951, 468491, 78311, 157081, 52667, 79231, 1553, 79693, 17761, 3739, 80621, 485123, 162641, 4421, 1439, 10501, 82493, 165457, 248893, 6163, 83437, 502043, 83911, 56099, 253159, 169249, 510611, 28447, 171161, 3259, 9059, 27329, 86783, 174049, 261799, 87751, 6361, 88237, 1487, 2917, 89213, 9967, 179897, 270583, 1619, 4799, 182857, 61283, 92173, 554531, 92671, 6883, 279511, 186841, 1993, 4987, 31391, 31727, 30137, 10099, 288583, 21433, 96703, 97213, 293173, 1543, 590963, 32917, 198017, 199049, 11087, 16223, 5279, 201121, 15919, 101341, 14177, 101863, 205297, 102911, 2203, 207401, 5113, 208457, 628547, 105023, 11083, 316663, 106087, 638123, 106621, 23753, 321469, 214849, 107693, 647771, 217001, 2161, 218081, 657491, 2969, 17431, 24593, 5839, 667283, 111487, 74507, 336103, 224617, 112583, 677147, 226817, 2017, 4231, 687083, 230137, 346039, 4057, 115903, 1931, 116461, 351061, 13127, 7577, 237977, 717323, 361213, 6367, 11927, 6397, 27073, 366343, 737843, 2399, 371509, 748211, 124991, 5827, 9323, 126151, 39929, 126733, 4457, 381949, 255217, 4297, 42829, 257561, 3049, 258737, 14407, 779747, 130253, 87427, 131437, 790403, 6949, 1877, 3203, 2851, 14869, 403261, 269441, 45007, 811931, 135623, 271849, 91019, 822803, 137437, 9631, 14563, 2273, 833747, 279137, 419623, 46829, 844763, 141101, 425149, 855851, 95507, 22669, 3467, 7589, 867011, 48271, 290249, 436309, 6779, 16229, 11117, 3121,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-118x+3 and
the reducible primes which appear as divisor for the first time
p | x^2-118x+3 and p < x^2-118x+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)
11010371.0000000.3000001.0000000.0000000.0000000.000000
21005411430.5400000.1100000.5400005.4000003.6666676.142857
31.000702906120.7020000.0900000.70200013.0000008.18181814.232558
410.0007.3856586.7270.7385000.0658000.73850010.5199437.31111110.991830
5100.00073.2875.00568.2820.7328700.0500500.7328709.9237647.60638310.150438
61.000.000725.77240.718685.0540.7257720.0407180.7257729.9031488.13546510.032718
710.000.0007.204.703344.8876.859.8160.7204700.0344890.7204709.9269508.47013610.013540
8100.000.00071.677.7492.984.61568.693.1340.7167780.0298460.7167789.9487458.65389310.013845
91.000.000.000713.919.80126.347.074687.572.7270.7139200.0263470.7139209.9601318.82762910.009336
1010.000.000.0007.116.721.696235.789.0186.880.932.6780.7116720.0235790.7116729.9685178.94934410.007571


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)
123211.5000001.0000000.5000000.0000000.0000000.000000
244221.0000000.5000000.5000001.3333331.0000002.000000
388351.0000000.3750000.6250002.0000001.5000002.500000
416165111.0000000.3125000.6875002.0000001.6666672.200000
532318230.9687500.2500000.7187501.9375001.6000002.090909
6645411430.8437500.1718750.6718751.7419351.3750001.869565
71285813450.4531250.1015620.3515621.0740741.1818181.046512
8256150271230.5859380.1054690.4804692.5862072.0769232.733333
9512339532860.6621090.1035160.5585942.2600001.9629632.325203
101.024722906320.7050780.0878910.6171882.1297931.6981132.209790
112.0481.4761681.3080.7207030.0820310.6386722.0443211.8666672.069620
124.0962.9952972.6980.7312010.0725100.6586912.0291331.7678572.062691
138.1926.0385455.4930.7370610.0665280.6705322.0160271.8350172.035953
1416.38412.0951.01011.0850.7382200.0616460.6765752.0031471.8532112.018023
1532.76824.1371.85622.2810.7366030.0566410.6799621.9956181.8376242.010014
1665.53648.1283.45844.6700.7343750.0527650.6816101.9939511.8631472.004847
17131.07295.8986.45189.4470.7316440.0492170.6824261.9925611.8655292.002395
18262.144191.23911.992179.2470.7295190.0457460.6837731.9941921.8589372.003947
19524.288381.52622.500359.0260.7277030.0429150.6847881.9950221.8762512.002968
201.048.576760.80542.584718.2210.7255600.0406110.6849491.9941101.8926222.000471
212.097.1521.517.72780.7721.436.9550.7237090.0385150.6851941.9948961.8967692.000714
224.194.3043.029.448153.4862.875.9620.7222770.0365940.6856831.9960431.9002382.001428
238.388.6086.046.200292.8055.753.3950.7207630.0349050.6858581.9958091.9076982.000511
2416.777.21612.072.053558.85311.513.2000.7195500.0333100.6862401.9966351.9086182.001114
2533.554.43224.105.7121.069.49123.036.2210.7184060.0318730.6865331.9968201.9137252.000853
2667.108.86448.141.4512.050.57646.090.8750.7173640.0305560.6868081.9970971.9173382.000800
27134.217.72896.152.7253.938.80592.213.9200.7163940.0293460.6870471.9972961.9208292.000698
28268.435.456192.056.9247.580.767184.476.1570.7154680.0282410.6872271.9974151.9246362.000524
29536.870.912383.661.77314.605.635369.056.1380.7146260.0272050.6874211.9976461.9266702.000563
301.073.741.824766.481.73428.187.287738.294.4470.7138420.0262510.6875901.9978061.9298912.000494
312.147.483.6481.531.398.46854.459.5611.476.938.9070.7131130.0253600.6877531.9979581.9320612.000474
324.294.967.2963.059.903.426105.336.9972.954.566.4290.7124390.0245260.6879141.9981111.9342242.000466
338.589.934.5926.114.376.542203.950.3365.910.426.2060.7118070.0237430.6880641.9982251.9361702.000438
3417.179.869.18412.218.533.975395.321.69411.823.212.2810.7112120.0230110.6882021.9983291.9383232.000399


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
122100110
242100110
383200120
4165300140
5328500170
664118001100
7128138203100
825627816017100
951253842043100
101.02490879080100
112.04816881570158100
124.09629782860287100
138.19254585340535100
1416.3841.010899901.000100
1532.7681.85681.84501.846100
1665.5363.45883.44703.448100
17131.0726.45186.44006.441100
18262.14411.992811.981011.982100
19524.28822.500822.489022.490100
201.048.57642.584842.573042.574100
212.097.15280.772880.761080.762100
224.194.304153.4868153.4750153.476100
238.388.608292.8058292.7940292.795100
2416.777.216558.8538558.8420558.843100
2533.554.4321.069.49181.069.48001.069.481100
2667.108.8642.050.57682.050.56502.050.566100
27134.217.7283.938.80583.938.79403.938.795100
28268.435.4567.580.76787.580.75607.580.757100
29536.870.91214.605.635814.605.624014.605.625100
301.073.741.82428.187.287828.187.276028.187.277100
312.147.483.64854.459.561854.459.550054.459.551100
324.294.967.296105.336.9978105.336.9860105.336.987100
338.589.934.592203.950.3368203.950.3250203.950.326100
3417.179.869.184395.321.6948395.321.6830395.321.684100


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
121100100
242200101
385411202
41611652513
5322310138627
6644318251214413
71284520251314414
8256123626137222638
951228615513185457680
101.024632345287183101169179
112.0481.308730578368210340390
124.0962.6981.4821.216753465716764
138.1925.4932.9812.5121.5019671.4981.527
1416.38411.0856.0155.0702.9692.0383.0383.040
1532.76822.28112.05910.2225.9954.1496.0706.067
1665.53644.67024.16520.50511.8858.51412.10912.162
17131.07289.44747.97141.47623.70317.39924.16024.185
18262.144179.24795.64783.60047.63035.28548.17148.161
19524.288359.026191.266167.76095.14771.99695.79596.088
201.048.576718.221381.398336.823189.444146.336191.601190.840
212.097.1521.436.955760.365676.590377.898296.444381.291381.322
224.194.3042.875.9621.517.1771.358.785755.663599.217760.731760.351
238.388.6085.753.3953.028.1992.725.1961.508.1721.210.6271.516.8681.517.728
2416.777.21611.513.2006.045.4935.467.7073.012.5372.443.2863.028.4323.028.945
2533.554.43223.036.22112.070.74410.965.4776.016.4914.928.4906.045.4456.045.795
2667.108.86446.090.87524.101.54021.989.33512.020.6379.929.56212.071.94412.068.732
27134.217.72892.213.92048.130.96844.082.95224.010.97219.998.60124.102.45124.101.896
28268.435.456184.476.15796.120.38388.355.77447.962.43440.251.47448.134.51348.127.736
29536.870.912369.056.138192.002.953177.053.18595.822.66980.963.75296.132.47296.137.245
301.073.741.824738.294.447383.555.131354.739.316191.456.545162.789.312192.010.863192.037.727
312.147.483.6481.476.938.907766.312.561710.626.346382.539.938327.177.427383.581.143383.640.399
324.294.967.2962.954.566.4291.531.077.3421.423.489.087764.384.435657.375.556766.364.916766.441.522
338.589.934.5925.910.426.2063.059.296.5422.851.129.6641.527.542.4421.320.379.4771.531.171.4081.531.332.879
3417.179.869.18411.823.212.2816.113.199.1245.710.013.1573.052.645.4192.651.343.9013.059.512.8723.059.710.089


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 19, 229, 1, 151, 281, 223, 43, 877, 163, 359, 587, 47, 227, 1453, 257, 181, 857, 599, 313,
Found in Database : 3, 19, 229, 151, 281, 223, 43, 877, 163, 359, 587, 47, 227, 1453, 257, 181, 857, 599, 313, 103, 113, 37, 1091, 751, 2389, 409, 839, 1289, 293, 449, 2749, 467, 317, 1451, 983, 499, 3037,
Found in Database : 3, 19, 37, 43, 47, 61, 79, 83, 103, 113, 127,