Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:04:35
Deutsch
29.Mar 2024

Polynom = x^2-64x-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) = 35 = 5*7
f(2) = 131 = 131
f(3) = 95 = 5*19
f(4) = 247 = 13*19
f(5) = 151 = 151
f(6) = 355 = 5*71
f(7) = 203 = 7*29
f(8) = 455 = 5*7*13
f(9) = 251 = 251
f(10) = 547 = 547
f(11) = 295 = 5*59
f(12) = 631 = 631
f(13) = 335 = 5*67
f(14) = 707 = 7*101
f(15) = 371 = 7*53
f(16) = 775 = 5*5*31
f(17) = 403 = 13*31
f(18) = 835 = 5*167
f(19) = 431 = 431
f(20) = 887 = 887
f(21) = 455 = 5*7*13
f(22) = 931 = 7*7*19
f(23) = 475 = 5*5*19
f(24) = 967 = 967
f(25) = 491 = 491
f(26) = 995 = 5*199
f(27) = 503 = 503
f(28) = 1015 = 5*7*29
f(29) = 511 = 7*73
f(30) = 1027 = 13*79
f(31) = 515 = 5*103
f(32) = 1031 = 1031
f(33) = 515 = 5*103
f(34) = 1027 = 13*79
f(35) = 511 = 7*73
f(36) = 1015 = 5*7*29
f(37) = 503 = 503
f(38) = 995 = 5*199
f(39) = 491 = 491
f(40) = 967 = 967
f(41) = 475 = 5*5*19
f(42) = 931 = 7*7*19
f(43) = 455 = 5*7*13
f(44) = 887 = 887
f(45) = 431 = 431
f(46) = 835 = 5*167
f(47) = 403 = 13*31
f(48) = 775 = 5*5*31
f(49) = 371 = 7*53
f(50) = 707 = 7*101
f(51) = 335 = 5*67
f(52) = 631 = 631
f(53) = 295 = 5*59
f(54) = 547 = 547
f(55) = 251 = 251
f(56) = 455 = 5*7*13
f(57) = 203 = 7*29
f(58) = 355 = 5*71
f(59) = 151 = 151
f(60) = 247 = 13*19
f(61) = 95 = 5*19
f(62) = 131 = 131
f(63) = 35 = 5*7
f(64) = 7 = 7
f(65) = 29 = 29
f(66) = 125 = 5*5*5
f(67) = 97 = 97
f(68) = 265 = 5*53
f(69) = 169 = 13*13
f(70) = 413 = 7*59
f(71) = 245 = 5*7*7
f(72) = 569 = 569
f(73) = 325 = 5*5*13
f(74) = 733 = 733
f(75) = 409 = 409
f(76) = 905 = 5*181
f(77) = 497 = 7*71
f(78) = 1085 = 5*7*31
f(79) = 589 = 19*31
f(80) = 1273 = 19*67
f(81) = 685 = 5*137
f(82) = 1469 = 13*113
f(83) = 785 = 5*157
f(84) = 1673 = 7*239
f(85) = 889 = 7*127
f(86) = 1885 = 5*13*29
f(87) = 997 = 997
f(88) = 2105 = 5*421
f(89) = 1109 = 1109
f(90) = 2333 = 2333
f(91) = 1225 = 5*5*7*7
f(92) = 2569 = 7*367
f(93) = 1345 = 5*269
f(94) = 2813 = 29*97
f(95) = 1469 = 13*113
f(96) = 3065 = 5*613
f(97) = 1597 = 1597
f(98) = 3325 = 5*5*7*19
f(99) = 1729 = 7*13*19
f(100) = 3593 = 3593

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^2-64x-7

f(0)=7
f(1)=5
f(2)=131
f(3)=19
f(4)=13
f(5)=151
f(6)=71
f(7)=29
f(8)=1
f(9)=251
f(10)=547
f(11)=59
f(12)=631
f(13)=67
f(14)=101
f(15)=53
f(16)=31
f(17)=1
f(18)=167
f(19)=431
f(20)=887
f(21)=1
f(22)=1
f(23)=1
f(24)=967
f(25)=491
f(26)=199
f(27)=503
f(28)=1
f(29)=73
f(30)=79
f(31)=103
f(32)=1031
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
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)=97
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=569
f(73)=1
f(74)=733
f(75)=409
f(76)=181
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=137
f(82)=113
f(83)=157
f(84)=239
f(85)=127
f(86)=1
f(87)=997
f(88)=421
f(89)=1109
f(90)=2333
f(91)=1
f(92)=367
f(93)=269
f(94)=1
f(95)=1
f(96)=613
f(97)=1597
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-64x-7 could be written as f(y)= y^2-1031 with x=y+32

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-32
f'(x)>2x-65 with x > 32

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, 131, 19, 13, 151, 71, 29, 1, 251, 547, 59, 631, 67, 101, 53, 31, 1, 167, 431, 887, 1, 1, 1, 967, 491, 199, 503, 1, 73, 79, 103, 1031, 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, 97, 1, 1, 1, 1, 569, 1, 733, 409, 181, 1, 1, 1, 1, 137, 113, 157, 239, 127, 1, 997, 421, 1109, 2333, 1, 367, 269, 1, 1, 613, 1597, 1, 1, 3593, 373, 1, 401, 4153, 307, 1, 2297, 1, 1, 163, 521, 1, 1, 5693, 1, 241, 1, 1, 467, 1, 1, 7069, 1, 7433, 293, 223, 571, 1637, 1, 8573, 877, 8969, 1, 1, 4789, 1, 263, 1, 5209, 1, 1, 11069, 1129, 397, 5869, 2393, 1, 1, 6329, 12893, 1, 461, 1361, 1979, 1, 1, 7297, 2969, 7549, 1181, 1, 2267, 1613, 1, 8329, 677, 8597, 499, 1, 18013, 1, 599, 1, 1, 1, 563, 769, 4057, 10289, 20873, 1, 3067, 311, 22073, 1, 349, 11497, 1, 1, 1, 1, 1, 1, 1327, 1, 739, 1871, 1061, 1033, 383, 2753, 1, 1, 4079, 14449, 5849, 14797, 1, 15149, 1, 443, 1, 1, 479, 16229, 1, 2371, 1, 1, 34313, 3469, 35069, 709, 5119, 1, 7321, 1, 7477, 1453, 647, 1, 1, 1, 1283, 20089, 8117, 1, 1, 1, 1, 853, 3313, 4349, 43913, 3167, 1279, 1, 1, 23029, 2447, 1, 1, 683, 661, 1873, 9829, 1, 10009, 3607, 1, 1, 51869, 5233, 1, 859, 1, 1, 1, 1451, 2927, 1, 56569, 1, 8219, 29009, 11701, 2269, 11897, 29989, 1, 1, 61469, 6197, 62473, 31489, 12697, 653, 1, 1, 1, 1321, 66569, 6709, 743, 1, 1, 1193, 2789, 35129, 70793, 1019, 10267, 557, 72953, 36749, 1, 1, 1, 5407, 76253, 7681, 77369, 7793, 1481, 5647, 1, 1, 1, 607, 6301, 1, 11867, 1, 643, 42409, 1, 1, 17317, 1, 12539, 8837, 88969, 1, 90173, 45389, 1, 6571, 18521, 1, 911, 1889, 1, 1367, 13759, 2551, 1, 1693, 1, 1, 1, 1439, 1, 1, 102653, 1, 20789, 1, 1, 4073, 839, 1, 1, 10853, 821, 1, 4421, 1049, 1721, 56269, 113213, 1627, 1259, 1, 115933, 58309, 809, 58997, 3391, 8527, 719, 929, 121469, 1, 1, 1, 1, 62497, 811, 2039, 127133, 2557, 18367, 1847, 1, 1, 26293, 1, 1, 9547, 1, 13513, 7151, 1, 137353, 2381, 3967, 1, 28069, 70549, 141853, 1097, 1, 1, 2957, 1087, 5857, 73597, 1, 2399, 1, 1, 7951, 3037, 1, 971, 30841, 11071, 4451, 991, 1, 15817, 2239, 1229, 1, 11527, 1, 6269, 1, 1553, 8707, 1, 823, 1, 168713, 1, 2621, 85597, 983, 12347, 1, 1, 1117, 1, 1753, 1, 5107, 89797, 1, 1, 9587, 18301, 26267, 1, 185593, 93229, 1, 1289, 37813, 13567, 27259, 1, 14813, 1, 1, 97609, 1, 14071, 2083, 5231, 199673, 1, 1, 1, 1, 7853, 41017, 1, 41381, 7993, 29819, 1, 1, 21149, 1, 106669, 42853, 1, 1, 108529, 7517, 21893, 1301, 1, 1, 15907, 44729, 112297, 1, 1, 227453, 1, 1, 1, 231293, 8933, 1, 6163, 6719, 1, 3539, 1, 239069, 4801, 18541, 1, 1, 121997, 3769, 4241, 1, 1, 5081, 3571, 2221, 1, 2663, 9769, 51001, 18287, 1, 1, 4391, 1, 8423, 2473, 1, 1, 1, 1987, 1, 26833, 2777, 3863, 1, 1, 54709, 1, 1901, 138349, 39679, 1, 1171, 2161, 281993, 141529, 2273, 1567, 8179, 143669, 288413, 28949, 1303, 1, 1, 1, 1, 147997, 59417, 1, 23021, 1, 43067, 1, 5147, 152389, 1, 1, 8803, 1699, 310333, 6229, 16451, 1, 314813, 22567, 9059, 159097, 1, 1, 321593, 1, 3559, 4643, 326153, 5279, 1, 1, 2281, 1, 1, 1759, 1, 1, 2659, 13033, 1, 24371, 68473, 1, 3413, 34589, 347069, 1, 49919, 175309, 1, 176497, 5449, 1297, 1, 1, 1, 36017, 361373, 1, 2347, 1, 10463, 183709, 368633, 1, 371069, 1, 53359, 1, 75193, 1669, 15137, 1, 20047, 1, 54767, 38461, 1, 193549, 1, 1487, 1, 4001, 1, 39461, 5011, 1, 398393, 28547, 1, 1, 1, 10651, 406013, 1, 58367, 1, 14177, 206209, 1, 207497, 83257, 4261, 4603, 42017, 421469, 1, 32621, 4013, 1, 1609, 4519, 215309, 431933, 1733, 5953, 1, 62459, 7561, 87973, 1, 1, 7159, 63599, 6379, 7591, 1549, 8501, 225949, 1, 1709, 1861, 1, 35281, 1, 2549, 46273, 1, 33247, 1, 1787, 93913, 1, 3719, 1, 67867, 1, 16477, 12611, 5059, 240997, 1973, 1, 15683, 48757, 1, 49037, 491773, 35227, 1, 247997, 1877, 1, 500233, 1, 71867, 7207, 26627, 1, 1, 255097, 4093, 2819, 1, 51593, 5023, 1789, 5051, 260849, 14947, 1, 8093, 3613, 1, 1, 1, 1, 4021, 3673, 1, 269597, 1523, 1663, 77659, 1, 546569, 1, 549533, 21193, 110501, 5653, 1, 278489, 558473, 55997, 1, 2963, 6203, 40427, 113497, 4007, 1, 286009, 19777, 1, 2657, 57809, 579613, 1, 116533, 22469, 3347, 41947, 3253, 1, 31151, 59341, 594953, 1, 2441, 1, 1, 301349, 1, 2089, 86767, 8699, 1, 306029, 4909, 4591, 1, 1, 1, 3271, 1, 1, 7927, 313909, 1, 3467, 126517, 317089, 2851, 63737, 8753, 9151, 1, 321889, 9929, 5483, 1777, 1, 1, 1867, 655069, 1, 12421, 329969, 132313, 1, 1, 25633, 668093, 66973, 671369, 1, 3109, 1, 135589, 3299, 1, 17971, 1, 9803, 7559, 1, 691193, 346429, 2137, 348097, 1, 1723, 7229, 70289, 704569, 1, 707933, 1, 20323, 1, 7523, 1, 2339, 1, 103067, 10331, 1, 11719, 1, 364997, 11257, 52387, 105019, 14737, 56813, 1, 1, 371869, 1, 1, 1, 4751, 2797, 5801, 755869, 1, 15497, 1, 5261, 1, 153269, 384049, 1, 1, 3209, 77513, 4597, 1, 1, 55871, 1, 2999, 787513, 78929, 11807, 1, 16217, 1, 1, 1, 160357, 5659, 1913, 1, 1, 81077, 42767, 1, 1, 408997, 1, 58687, 1, 3301, 6037, 82889, 63901, 59467, 1, 13487, 33521, 419929, 15881, 1, 120767, 1, 44687, 7211, 5881, 32869, 1, 1, 1, 86201, 863869, 86573, 29917, 2879, 1, 1, 175013, 8273, 67601, 88069, 46451, 1, 1, 444109, 178021, 14387, 1, 1, 128239, 1, 11411, 6949, 12401, 15641, 1, 65071, 26083, 6827, 916733, 1, 3727, 3181, 132059, 1, 14281, 7883, 7457, 4133,

6. Sequence of the polynom (only primes)

7, 5, 131, 19, 13, 151, 71, 29, 251, 547, 59, 631, 67, 101, 53, 31, 167, 431, 887, 967, 491, 199, 503, 73, 79, 103, 1031, 97, 569, 733, 409, 181, 137, 113, 157, 239, 127, 997, 421, 1109, 2333, 367, 269, 613, 1597, 3593, 373, 401, 4153, 307, 2297, 163, 521, 5693, 241, 467, 7069, 7433, 293, 223, 571, 1637, 8573, 877, 8969, 4789, 263, 5209, 11069, 1129, 397, 5869, 2393, 6329, 12893, 461, 1361, 1979, 7297, 2969, 7549, 1181, 2267, 1613, 8329, 677, 8597, 499, 18013, 599, 563, 769, 4057, 10289, 20873, 3067, 311, 22073, 349, 11497, 1327, 739, 1871, 1061, 1033, 383, 2753, 4079, 14449, 5849, 14797, 15149, 443, 479, 16229, 2371, 34313, 3469, 35069, 709, 5119, 7321, 7477, 1453, 647, 1283, 20089, 8117, 853, 3313, 4349, 43913, 3167, 1279, 23029, 2447, 683, 661, 1873, 9829, 10009, 3607, 51869, 5233, 859, 1451, 2927, 56569, 8219, 29009, 11701, 2269, 11897, 29989, 61469, 6197, 62473, 31489, 12697, 653, 1321, 66569, 6709, 743, 1193, 2789, 35129, 70793, 1019, 10267, 557, 72953, 36749, 5407, 76253, 7681, 77369, 7793, 1481, 5647, 607, 6301, 11867, 643, 42409, 17317, 12539, 8837, 88969, 90173, 45389, 6571, 18521, 911, 1889, 1367, 13759, 2551, 1693, 1439, 102653, 20789, 4073, 839, 10853, 821, 4421, 1049, 1721, 56269, 113213, 1627, 1259, 115933, 58309, 809, 58997, 3391, 8527, 719, 929, 121469, 62497, 811, 2039, 127133, 2557, 18367, 1847, 26293, 9547, 13513, 7151, 137353, 2381, 3967, 28069, 70549, 141853, 1097, 2957, 1087, 5857, 73597, 2399, 7951, 3037, 971, 30841, 11071, 4451, 991, 15817, 2239, 1229, 11527, 6269, 1553, 8707, 823, 168713, 2621, 85597, 983, 12347, 1117, 1753, 5107, 89797, 9587, 18301, 26267, 185593, 93229, 1289, 37813, 13567, 27259, 14813, 97609, 14071, 2083, 5231, 199673, 7853, 41017, 41381, 7993, 29819, 21149, 106669, 42853, 108529, 7517, 21893, 1301, 15907, 44729, 112297, 227453, 231293, 8933, 6163, 6719, 3539, 239069, 4801, 18541, 121997, 3769, 4241, 5081, 3571, 2221, 2663, 9769, 51001, 18287, 4391, 8423, 2473, 1987, 26833, 2777, 3863, 54709, 1901, 138349, 39679, 1171, 2161, 281993, 141529, 2273, 1567, 8179, 143669, 288413, 28949, 1303, 147997, 59417, 23021, 43067, 5147, 152389, 8803, 1699, 310333, 6229, 16451, 314813, 22567, 9059, 159097, 321593, 3559, 4643, 326153, 5279, 2281, 1759, 2659, 13033, 24371, 68473, 3413, 34589, 347069, 49919, 175309, 176497, 5449, 1297, 36017, 361373, 2347, 10463, 183709, 368633, 371069, 53359, 75193, 1669, 15137, 20047, 54767, 38461, 193549, 1487, 4001, 39461, 5011, 398393, 28547, 10651, 406013, 58367, 14177, 206209, 207497, 83257, 4261, 4603, 42017, 421469, 32621, 4013, 1609, 4519, 215309, 431933, 1733, 5953, 62459, 7561, 87973, 7159, 63599, 6379, 7591, 1549, 8501, 225949, 1709, 1861, 35281, 2549, 46273, 33247, 1787, 93913, 3719, 67867, 16477, 12611, 5059, 240997, 1973, 15683, 48757, 49037, 491773, 35227, 247997, 1877, 500233, 71867, 7207, 26627, 255097, 4093, 2819, 51593, 5023, 1789, 5051, 260849, 14947, 8093, 3613, 4021, 3673, 269597, 1523, 1663, 77659, 546569, 549533, 21193, 110501, 5653, 278489, 558473, 55997, 2963, 6203, 40427, 113497, 4007, 286009, 19777, 2657, 57809, 579613, 116533, 22469, 3347, 41947, 3253, 31151, 59341, 594953, 2441, 301349, 2089, 86767, 8699, 306029, 4909, 4591, 3271, 7927, 313909, 3467, 126517, 317089, 2851, 63737, 8753, 9151, 321889, 9929, 5483, 1777, 1867, 655069, 12421, 329969, 132313, 25633, 668093, 66973, 671369, 3109, 135589, 3299, 17971, 9803, 7559, 691193, 346429, 2137, 348097, 1723, 7229, 70289, 704569, 707933, 20323, 7523, 2339, 103067, 10331, 11719, 364997, 11257, 52387, 105019, 14737, 56813, 371869, 4751, 2797, 5801, 755869, 15497, 5261, 153269, 384049, 3209, 77513, 4597, 55871, 2999, 787513, 78929, 11807, 16217, 160357, 5659, 1913, 81077, 42767, 408997, 58687, 3301, 6037, 82889, 63901, 59467, 13487, 33521, 419929, 15881, 120767, 44687, 7211, 5881, 32869, 86201, 863869, 86573, 29917, 2879, 175013, 8273, 67601, 88069, 46451, 444109, 178021, 14387, 128239, 11411, 6949, 12401, 15641, 65071, 26083, 6827, 916733, 3727, 3181, 132059, 14281, 7883, 7457, 4133,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-64x-7 and
the reducible primes which appear as divisor for the first time
p | x^2-64x-7 and p < x^2-64x-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)
11010551.0000000.5000000.5000000.0000000.0000000.000000
21004020200.4000000.2000000.2000004.0000004.0000004.000000
31.0006011624390.6010000.1620000.43900015.0250008.10000021.950001
410.0006.5611.1825.3790.6561000.1182000.53790010.9168057.29629612.252848
5100.00066.8549.20957.6450.6685400.0920900.57645010.1896067.79103210.716676
61.000.000672.71475.578597.1360.6727140.0755780.59713610.0624348.20697110.358851
710.000.0006.758.071637.5276.120.5440.6758070.0637530.61205410.0459798.43535110.249832
8100.000.00067.799.8785.515.96262.283.9160.6779990.0551600.62283910.0324318.65212210.176206
91.000.000.000679.630.69048.652.312630.978.3780.6796310.0486520.63097810.0240708.82027710.130679
1010.000.000.0006.809.537.393434.939.3516.374.598.0420.6809540.0434940.63746010.0194678.93974710.102720


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
245231.2500000.5000000.7500001.6666671.0000003.000000
388351.0000000.3750000.6250001.6000001.5000001.666667
41615690.9375000.3750000.5625001.8750002.0000001.800000
5322612140.8125000.3750000.4375001.7333332.0000001.555556
6642612140.4062500.1875000.2187501.0000001.0000001.000000
71285425290.4218750.1953120.2265622.0769232.0833332.071429
825612851770.5000000.1992190.3007812.3703702.0400002.655172
9512283931900.5527340.1816410.3710942.2109381.8235292.467532
101.0246161654510.6015620.1611330.4404302.1766781.7741942.373684
112.0481.2813049770.6254880.1484380.4770512.0795451.8424242.166297
124.0962.6425492.0930.6450200.1340330.5109862.0624511.8059212.142272
138.1925.3589824.3760.6540530.1198730.5341802.0280091.7887072.090779
1416.38410.8271.8259.0020.6608280.1113890.5494382.0207171.8584522.057130
1532.76821.7753.37218.4030.6645200.1029050.5616152.0111761.8476712.044323
1665.53643.7436.27437.4690.6674650.0957340.5717322.0088631.8606172.036027
17131.07287.70111.78375.9180.6691060.0898970.5792082.0049151.8780682.026155
18262.144175.82922.174153.6550.6707340.0845870.5861472.0048691.8818642.023960
19524.288352.16841.866310.3020.6717070.0798530.5918542.0029011.8880672.019472
201.048.576705.50278.953626.5490.6728190.0752950.5975242.0033111.8858502.019159
212.097.1521.412.912149.6721.263.2400.6737290.0713690.6023602.0027041.8957102.016187
224.194.3042.830.258283.9662.546.2920.6747860.0677030.6070832.0031381.8972552.015683
238.388.6085.667.503541.0305.126.4730.6756190.0644960.6111232.0024691.9052632.013309
2416.777.21611.347.7151.033.52010.314.1950.6763770.0616030.6147742.0022421.9102822.011948
2533.554.43222.717.1071.977.27620.739.8310.6770230.0589270.6180952.0019101.9131472.010805
2667.108.86445.476.3353.790.07941.686.2560.6776500.0564770.6211742.0018541.9168182.009961
27134.217.72891.030.4977.279.07983.751.4180.6782300.0542330.6239972.0017111.9205612.009089
28268.435.456182.199.51214.005.455168.194.0570.6787460.0521740.6265722.0015221.9240702.008253
29536.870.912364.652.72726.984.196337.668.5310.6792190.0502620.6289572.0013921.9266922.007613
301.073.741.824729.797.19752.049.256677.747.9410.6796770.0484750.6312022.0013481.9288792.007140
312.147.483.6481.460.492.341100.525.3041.359.967.0370.6800950.0468110.6332842.0012301.9313502.006597
324.294.967.2962.922.719.742194.366.7762.728.352.9660.6804990.0452550.6352442.0011881.9335112.006191
338.589.934.5925.848.672.096376.237.9785.472.434.1180.6808750.0438000.6370752.0011061.9357112.005765
3417.179.869.18411.703.412.247729.083.47810.974.328.7690.6812280.0424380.6387902.0010381.9378252.005383


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
122110101
242110101
383210102
4166420303
53212570408
66412570408
71282511146478
8256512328194208
9512934548384438
101.0241657986684858
112.04830415614814041528
124.09654926928026042778
138.19298249948348144898
1416.3841.82591990692448898
1532.7683.3721.6711.7011.68241.6788
1665.5366.2743.1513.1233.11943.1438
17131.07211.7835.9585.8255.89945.8728
18262.14422.17411.19010.98411.095411.0678
19524.28841.86621.11020.75620.903420.9518
201.048.57678.95339.81639.13739.435439.5068
212.097.152149.67275.51774.15574.876474.7848
224.194.304283.966142.963141.003141.8784142.0768
238.388.608541.030271.907269.123270.2204270.7988
2416.777.2161.033.520519.349514.171517.0044516.5048
2533.554.4321.977.276992.516984.760988.8304988.4348
2667.108.8643.790.0791.901.7341.888.3451.895.56241.894.5058
27134.217.7287.279.0793.650.7253.628.3543.640.32543.638.7428
28268.435.45614.005.4557.025.3556.980.1007.005.74946.999.6948
29536.870.91226.984.19613.533.97613.450.22013.495.900413.488.2848
301.073.741.82452.049.25626.102.11725.947.13926.030.025426.019.2198
312.147.483.648100.525.30450.406.43350.118.87150.266.088450.259.2048
324.294.967.296194.366.77697.449.72296.917.05497.188.350497.178.4148
338.589.934.592376.237.978188.613.350187.624.628188.121.2834188.116.6838
3417.179.869.184729.083.478365.491.594363.591.884364.546.4774364.536.9898


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
121010010
243210120
385230131
4169360351
53214771355
66414771355
712829151446127
825677383916162817
9512190929843396246
101.024451226225111105132103
112.048977484493234232274237
124.0962.0931.0341.059527477555534
138.1924.3762.1612.2151.1141.0421.1441.076
1416.3849.0024.4994.5032.2872.1742.3212.220
1532.76818.4039.1669.2374.6314.4684.7814.523
1665.53637.46918.64018.8299.5069.0659.6189.280
17131.07275.91837.81438.10419.24018.44919.48718.742
18262.144153.65576.64777.00839.10637.44739.20437.898
19524.288310.302155.300155.00278.98076.09178.67176.560
201.048.576626.549313.587312.962158.875154.084159.134154.456
212.097.1521.263.240632.174631.066320.195311.201320.397311.447
224.194.3042.546.2921.273.7861.272.506645.498628.063645.137627.594
238.388.6085.126.4732.563.5902.562.8831.298.0071.265.0701.298.5521.264.844
2416.777.21610.314.1955.157.3835.156.8122.611.0592.547.1652.609.3682.546.603
2533.554.43220.739.83110.369.18410.370.6475.245.5565.124.9605.244.1455.125.170
2667.108.86441.686.25620.842.37720.843.87910.539.61810.302.40810.537.84310.306.387
27134.217.72883.751.41841.871.32741.880.09121.164.07120.711.23621.163.66920.712.442
28268.435.456168.194.05784.098.39084.095.66742.488.97441.607.53842.490.24141.607.304
29536.870.912337.668.531168.827.978168.840.55385.271.42383.563.86485.266.46783.566.777
301.073.741.824677.747.941338.868.776338.879.165171.092.688167.776.995171.094.207167.784.051
312.147.483.6481.359.967.037679.977.647679.989.390343.198.636336.799.449343.188.119336.780.833
324.294.967.2962.728.352.9661.364.160.9861.364.191.980688.305.817675.889.429688.286.661675.871.059
338.589.934.5925.472.434.1182.736.211.0572.736.223.0611.380.194.4241.356.049.9511.380.164.1661.356.025.577
3417.179.869.18410.974.328.7695.487.161.2065.487.167.5632.767.045.1782.720.164.7432.766.983.5242.720.135.324


8. Check for existing Integer Sequences by OEIS

Found in Database : 7, 5, 131, 19, 13, 151, 71, 29, 1, 251, 547, 59, 631, 67, 101, 53, 31, 1, 167, 431,
Found in Database : 7, 5, 131, 19, 13, 151, 71, 29, 251, 547, 59, 631, 67, 101, 53, 31, 167, 431, 887, 967, 491, 199, 503, 73, 79, 103, 1031,
Found in Database : 5, 7, 13, 19, 29, 31, 53, 59, 67, 71, 73, 79, 97, 101, 103, 113, 127, 131, 137,