Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:19:46
Deutsch
29.Mar 2024

Polynom = x^2-104x+191

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) = 191 = 191
f(1) = 11 = 11
f(2) = 13 = 13
f(3) = 7 = 7
f(4) = 209 = 11*19
f(5) = 19 = 19
f(6) = 397 = 397
f(7) = 61 = 61
f(8) = 577 = 577
f(9) = 83 = 83
f(10) = 749 = 7*107
f(11) = 13 = 13
f(12) = 913 = 11*83
f(13) = 31 = 31
f(14) = 1069 = 1069
f(15) = 143 = 11*13
f(16) = 1217 = 1217
f(17) = 161 = 7*23
f(18) = 1357 = 23*59
f(19) = 89 = 89
f(20) = 1489 = 1489
f(21) = 97 = 97
f(22) = 1613 = 1613
f(23) = 209 = 11*19
f(24) = 1729 = 7*13*19
f(25) = 223 = 223
f(26) = 1837 = 11*167
f(27) = 59 = 59
f(28) = 1937 = 13*149
f(29) = 31 = 31
f(30) = 2029 = 2029
f(31) = 259 = 7*37
f(32) = 2113 = 2113
f(33) = 269 = 269
f(34) = 2189 = 11*199
f(35) = 139 = 139
f(36) = 2257 = 37*61
f(37) = 143 = 11*13
f(38) = 2317 = 7*331
f(39) = 293 = 293
f(40) = 2369 = 23*103
f(41) = 299 = 13*23
f(42) = 2413 = 19*127
f(43) = 19 = 19
f(44) = 2449 = 31*79
f(45) = 77 = 7*11
f(46) = 2477 = 2477
f(47) = 311 = 311
f(48) = 2497 = 11*227
f(49) = 313 = 313
f(50) = 2509 = 13*193
f(51) = 157 = 157
f(52) = 2513 = 7*359
f(53) = 157 = 157
f(54) = 2509 = 13*193
f(55) = 313 = 313
f(56) = 2497 = 11*227
f(57) = 311 = 311
f(58) = 2477 = 2477
f(59) = 77 = 7*11
f(60) = 2449 = 31*79
f(61) = 19 = 19
f(62) = 2413 = 19*127
f(63) = 299 = 13*23
f(64) = 2369 = 23*103
f(65) = 293 = 293
f(66) = 2317 = 7*331
f(67) = 143 = 11*13
f(68) = 2257 = 37*61
f(69) = 139 = 139
f(70) = 2189 = 11*199
f(71) = 269 = 269
f(72) = 2113 = 2113
f(73) = 259 = 7*37
f(74) = 2029 = 2029
f(75) = 31 = 31
f(76) = 1937 = 13*149
f(77) = 59 = 59
f(78) = 1837 = 11*167
f(79) = 223 = 223
f(80) = 1729 = 7*13*19
f(81) = 209 = 11*19
f(82) = 1613 = 1613
f(83) = 97 = 97
f(84) = 1489 = 1489
f(85) = 89 = 89
f(86) = 1357 = 23*59
f(87) = 161 = 7*23
f(88) = 1217 = 1217
f(89) = 143 = 11*13
f(90) = 1069 = 1069
f(91) = 31 = 31
f(92) = 913 = 11*83
f(93) = 13 = 13
f(94) = 749 = 7*107
f(95) = 83 = 83
f(96) = 577 = 577
f(97) = 61 = 61
f(98) = 397 = 397
f(99) = 19 = 19
f(100) = 209 = 11*19

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^2-104x+191

f(0)=191
f(1)=11
f(2)=13
f(3)=7
f(4)=19
f(5)=1
f(6)=397
f(7)=61
f(8)=577
f(9)=83
f(10)=107
f(11)=1
f(12)=1
f(13)=31
f(14)=1069
f(15)=1
f(16)=1217
f(17)=23
f(18)=59
f(19)=89
f(20)=1489
f(21)=97
f(22)=1613
f(23)=1
f(24)=1
f(25)=223
f(26)=167
f(27)=1
f(28)=149
f(29)=1
f(30)=2029
f(31)=37
f(32)=2113
f(33)=269
f(34)=199
f(35)=139
f(36)=1
f(37)=1
f(38)=331
f(39)=293
f(40)=103
f(41)=1
f(42)=127
f(43)=1
f(44)=79
f(45)=1
f(46)=2477
f(47)=311
f(48)=227
f(49)=313
f(50)=193
f(51)=157
f(52)=359
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)=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^2-104x+191 could be written as f(y)= y^2-2513 with x=y+52

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-52
f'(x)>2x-105 with x > 50

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

191, 11, 13, 7, 19, 1, 397, 61, 577, 83, 107, 1, 1, 31, 1069, 1, 1217, 23, 59, 89, 1489, 97, 1613, 1, 1, 223, 167, 1, 149, 1, 2029, 37, 2113, 269, 199, 139, 1, 1, 331, 293, 103, 1, 127, 1, 79, 1, 2477, 311, 227, 313, 193, 157, 359, 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, 1087, 151, 1, 1, 1583, 1, 1, 1, 2111, 281, 1, 1, 2671, 1, 2963, 389, 251, 1, 3571, 233, 1, 1, 4211, 547, 1, 1, 257, 1, 5231, 1, 1, 1, 541, 1, 6323, 1, 6703, 431, 1013, 911, 7487, 1, 607, 1, 1, 1, 1, 1117, 9151, 1171, 9587, 613, 1433, 641, 953, 1, 353, 1, 11411, 1, 11887, 379, 1, 1, 677, 1, 1, 1, 1, 883, 14387, 1831, 1, 271, 15443, 491, 1453, 1, 1, 1, 2441, 1, 929, 1, 18223, 1, 18803, 1, 19391, 1, 1, 317, 349, 653, 1, 2689, 1, 2767, 1, 1423, 23087, 1, 1249, 1, 659, 3089, 1, 1, 3673, 1, 26387, 1, 1, 1, 27763, 1, 28463, 1801, 941, 3691, 1, 1, 4373, 1, 2411, 991, 32083, 4057, 32831, 593, 33587, 1, 34351, 1, 1, 1, 1, 1, 36691, 1, 1973, 1, 1, 691, 39103, 449, 1, 2521, 40751, 1, 457, 1, 1, 487, 43283, 683, 4013, 1, 1, 1, 45887, 5791, 46771, 1, 619, 1, 48563, 557, 811, 1, 50387, 1, 3947, 809, 587, 599, 4091, 1, 1, 3413, 1777, 1, 2437, 1, 1, 1, 5273, 1, 58991, 1, 1, 7561, 8713, 7687, 62003, 3907, 1, 1, 1, 1153, 1, 1, 5087, 2083, 1, 1, 9749, 8597, 6301, 8731, 1193, 1, 71471, 643, 1, 1, 3877, 9277, 74771, 1, 1, 2389, 7001, 9697, 6011, 9839, 79283, 1, 1, 1, 7417, 10271, 997, 947, 631, 1, 85103, 1, 86291, 10861, 983, 1, 2861, 5581, 743, 5657, 1, 11467, 1, 11621, 1, 1, 1, 1, 1, 1, 1, 1, 3181, 6203, 1693, 1, 1, 1, 1297, 12889, 9433, 1, 105071, 1, 1, 13381, 5669, 1, 1, 6857, 1213, 1, 111731, 14051, 8699, 14221, 114451, 1, 115823, 1, 117203, 14737, 10781, 1, 1, 1, 6389, 1, 3319, 15439, 1, 1, 125651, 1, 761, 1997, 9887, 1, 1, 16339, 10111, 751, 132911, 8353, 1, 1, 1637, 17077, 137363, 1, 138863, 4363, 1823, 1, 1, 1621, 2351, 9011, 1, 1301, 1, 1, 1, 1, 7873, 1, 1, 1187, 152723, 1, 154303, 19387, 155891, 1399, 1, 1, 6917, 1, 1, 1553, 23189, 2549, 8629, 1, 165587, 1, 1879, 3001, 1181, 10607, 857, 10711, 1019, 1, 24841, 21841, 1451, 1, 1721, 1, 1, 1, 1, 22691, 182387, 881, 184111, 1051, 1, 23339, 17053, 23557, 3209, 1, 14699, 1, 6221, 1, 1361, 24439, 10337, 1, 1231, 1, 1, 25111, 201791, 1949, 2099, 1, 205423, 1, 1, 26021, 5651, 26251, 30133, 13241, 1, 1, 1, 26947, 1, 1, 1, 1, 1, 1, 222163, 1, 32009, 2557, 2539, 1091, 20717, 14303, 2909, 1, 12197, 1531, 233683, 7333, 1, 3697, 33941, 2711, 18427, 1, 10501, 1, 18731, 1, 1933, 2801, 1, 1, 1, 1, 1, 1, 4297, 31817, 255551, 2467, 23417, 2309, 259631, 1481, 1733, 32839, 2719, 1439, 1, 1, 267887, 1, 1093, 1783, 24733, 4877, 274163, 1, 1, 17333, 278387, 2687, 3643, 1, 7639, 1, 284783, 8933, 3457, 1, 12569, 1, 15329, 1, 22571, 1, 1, 37087, 22907, 37361, 9677, 1, 302191, 1, 27673, 38189, 5197, 1, 308851, 19373, 2339, 1, 313331, 1709, 13721, 1, 10253, 1, 29101, 10039, 24799, 1, 2179, 1, 3593, 20507, 1, 1, 1321, 1, 17573, 1, 336211, 1, 338543, 1327, 14821, 1, 49033, 43051, 1, 21673, 347951, 21821, 5743, 6277, 2087, 1427, 1699, 1, 1, 1, 51413, 45137, 362303, 45439, 3541, 22871, 9923, 1, 16069, 46351, 1, 1, 374483, 1, 53849, 1, 1, 47581, 1, 1, 29567, 1, 1, 1, 1, 1, 12641, 1, 1, 1, 17257, 1, 36313, 3853, 21157, 1, 404531, 1951, 5153, 25523, 1, 1657, 58889, 1, 414803, 13003, 1, 3271, 11351, 7523, 32507, 4817, 1, 1, 1, 26821, 61493, 1741, 433087, 4177, 435731, 6829, 3623, 1, 441043, 1, 443711, 55631, 446387, 27983, 64153, 28151, 1, 1, 1373, 1, 1, 1, 1697, 7207, 12503, 1, 4349, 1, 6079, 1, 470831, 2683, 15277, 4567, 476351, 1, 1, 1877, 2161, 1, 484691, 60761, 1, 2657, 21317, 1, 1, 1, 495923, 1, 45341, 1, 5171, 1429, 1, 1, 72469, 63589, 8363, 4919, 1, 1, 515887, 1, 47161, 65027, 1, 2843, 40351, 8219, 75353, 16529, 3709, 3499, 28069, 1, 536243, 4801, 17393, 33791, 542131, 5227, 545087, 6211, 1, 1, 50093, 1, 554003, 3019, 1, 9973, 29473, 1847, 1, 35281, 1, 6449, 1, 71317, 572051, 4481, 575087, 1, 578131, 1, 581183, 1, 53113, 1, 6599, 1, 1, 3217, 4673, 74377, 1, 18691, 599663, 1, 1, 75541, 605887, 75931, 1, 1, 1, 1, 3919, 1, 1, 4079, 7489, 1, 10589, 1, 1, 6053, 57373, 79087, 1, 3613, 2011, 1, 640691, 80287, 49531, 11527, 1, 1, 2633, 1, 59417, 81901, 1, 82307, 1, 41357, 1, 1, 60601, 11933, 669887, 1, 1907, 1, 676463, 21191, 1, 4483, 683071, 1, 1, 43003, 62701, 6173, 1, 2347, 6761, 87257, 11471, 1, 1, 2753, 7283, 1, 709823, 1, 37537, 1, 716591, 44893, 2063, 1, 723391, 1, 9439, 11383, 56171, 1, 733651, 91921, 1, 1, 2927, 2017, 39157, 1, 2551, 2531, 107273, 7237, 1, 23629, 20483, 1, 761363, 13627, 2243, 1, 768371, 1, 771887, 48353, 8521, 5113, 3727, 4243, 2617, 1, 786031, 3517, 789587, 3191, 2707, 99367, 1, 1, 114329, 50131, 1, 1, 807487, 101161, 42689, 1, 1, 6379, 6763, 3307, 2749, 1, 1999, 51713, 1, 51941, 832883, 1, 1, 1, 5639, 1, 76717, 1, 1, 1, 121609, 1, 854963, 53551, 1, 53783, 862387, 1, 37657, 108497, 5147, 27241, 2791, 13679, 1, 1, 1, 1, 1, 1, 888623, 7951, 1, 8599,

6. Sequence of the polynom (only primes)

191, 11, 13, 7, 19, 397, 61, 577, 83, 107, 31, 1069, 1217, 23, 59, 89, 1489, 97, 1613, 223, 167, 149, 2029, 37, 2113, 269, 199, 139, 331, 293, 103, 127, 79, 2477, 311, 227, 313, 193, 157, 359, 1087, 151, 1583, 2111, 281, 2671, 2963, 389, 251, 3571, 233, 4211, 547, 257, 5231, 541, 6323, 6703, 431, 1013, 911, 7487, 607, 1117, 9151, 1171, 9587, 613, 1433, 641, 953, 353, 11411, 11887, 379, 677, 883, 14387, 1831, 271, 15443, 491, 1453, 2441, 929, 18223, 18803, 19391, 317, 349, 653, 2689, 2767, 1423, 23087, 1249, 659, 3089, 3673, 26387, 27763, 28463, 1801, 941, 3691, 4373, 2411, 991, 32083, 4057, 32831, 593, 33587, 34351, 36691, 1973, 691, 39103, 449, 2521, 40751, 457, 487, 43283, 683, 4013, 45887, 5791, 46771, 619, 48563, 557, 811, 50387, 3947, 809, 587, 599, 4091, 3413, 1777, 2437, 5273, 58991, 7561, 8713, 7687, 62003, 3907, 1153, 5087, 2083, 9749, 8597, 6301, 8731, 1193, 71471, 643, 3877, 9277, 74771, 2389, 7001, 9697, 6011, 9839, 79283, 7417, 10271, 997, 947, 631, 85103, 86291, 10861, 983, 2861, 5581, 743, 5657, 11467, 11621, 3181, 6203, 1693, 1297, 12889, 9433, 105071, 13381, 5669, 6857, 1213, 111731, 14051, 8699, 14221, 114451, 115823, 117203, 14737, 10781, 6389, 3319, 15439, 125651, 761, 1997, 9887, 16339, 10111, 751, 132911, 8353, 1637, 17077, 137363, 138863, 4363, 1823, 1621, 2351, 9011, 1301, 7873, 1187, 152723, 154303, 19387, 155891, 1399, 6917, 1553, 23189, 2549, 8629, 165587, 1879, 3001, 1181, 10607, 857, 10711, 1019, 24841, 21841, 1451, 1721, 22691, 182387, 881, 184111, 1051, 23339, 17053, 23557, 3209, 14699, 6221, 1361, 24439, 10337, 1231, 25111, 201791, 1949, 2099, 205423, 26021, 5651, 26251, 30133, 13241, 26947, 222163, 32009, 2557, 2539, 1091, 20717, 14303, 2909, 12197, 1531, 233683, 7333, 3697, 33941, 2711, 18427, 10501, 18731, 1933, 2801, 4297, 31817, 255551, 2467, 23417, 2309, 259631, 1481, 1733, 32839, 2719, 1439, 267887, 1093, 1783, 24733, 4877, 274163, 17333, 278387, 2687, 3643, 7639, 284783, 8933, 3457, 12569, 15329, 22571, 37087, 22907, 37361, 9677, 302191, 27673, 38189, 5197, 308851, 19373, 2339, 313331, 1709, 13721, 10253, 29101, 10039, 24799, 2179, 3593, 20507, 1321, 17573, 336211, 338543, 1327, 14821, 49033, 43051, 21673, 347951, 21821, 5743, 6277, 2087, 1427, 1699, 51413, 45137, 362303, 45439, 3541, 22871, 9923, 16069, 46351, 374483, 53849, 47581, 29567, 12641, 17257, 36313, 3853, 21157, 404531, 1951, 5153, 25523, 1657, 58889, 414803, 13003, 3271, 11351, 7523, 32507, 4817, 26821, 61493, 1741, 433087, 4177, 435731, 6829, 3623, 441043, 443711, 55631, 446387, 27983, 64153, 28151, 1373, 1697, 7207, 12503, 4349, 6079, 470831, 2683, 15277, 4567, 476351, 1877, 2161, 484691, 60761, 2657, 21317, 495923, 45341, 5171, 1429, 72469, 63589, 8363, 4919, 515887, 47161, 65027, 2843, 40351, 8219, 75353, 16529, 3709, 3499, 28069, 536243, 4801, 17393, 33791, 542131, 5227, 545087, 6211, 50093, 554003, 3019, 9973, 29473, 1847, 35281, 6449, 71317, 572051, 4481, 575087, 578131, 581183, 53113, 6599, 3217, 4673, 74377, 18691, 599663, 75541, 605887, 75931, 3919, 4079, 7489, 10589, 6053, 57373, 79087, 3613, 2011, 640691, 80287, 49531, 11527, 2633, 59417, 81901, 82307, 41357, 60601, 11933, 669887, 1907, 676463, 21191, 4483, 683071, 43003, 62701, 6173, 2347, 6761, 87257, 11471, 2753, 7283, 709823, 37537, 716591, 44893, 2063, 723391, 9439, 11383, 56171, 733651, 91921, 2927, 2017, 39157, 2551, 2531, 107273, 7237, 23629, 20483, 761363, 13627, 2243, 768371, 771887, 48353, 8521, 5113, 3727, 4243, 2617, 786031, 3517, 789587, 3191, 2707, 99367, 114329, 50131, 807487, 101161, 42689, 6379, 6763, 3307, 2749, 1999, 51713, 51941, 832883, 5639, 76717, 121609, 854963, 53551, 53783, 862387, 37657, 108497, 5147, 27241, 2791, 13679, 888623, 7951, 8599,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-104x+191 and
the reducible primes which appear as divisor for the first time
p | x^2-104x+191 and p < x^2-104x+191

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)
11010461.0000000.4000001.0000000.0000000.0000000.000000
21003711260.3700000.1100000.3700003.7000002.7500004.333333
31.0005541284260.5540000.1280000.55400014.97297311.63636416.384615
410.0006.3389635.3750.6338000.0963000.63380011.4404347.52343812.617371
5100.00065.3057.37257.9330.6530500.0737200.65305010.3037237.65524410.778233
61.000.000660.53460.222600.3120.6605340.0602220.66053410.1146018.16901810.362177
710.000.0006.653.385511.7386.141.6470.6653390.0511740.66533910.0727378.49752610.230759
8100.000.00066.892.1834.437.34262.454.8410.6689220.0443730.66892210.0538588.67112110.169070
91.000.000.000671.651.09039.156.382632.494.7080.6716510.0391560.67165110.0408008.82428810.127234
1010.000.000.0006.738.268.597350.422.6686.387.845.9290.6738270.0350420.67382710.0323948.94931110.099445


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
388441.0000000.5000000.5000001.6000002.0000001.333333
41613670.8125000.3750000.4375001.6250001.5000001.750000
5322310130.7187500.3125000.4062501.7692311.6666671.857143
6643711260.5781250.1718750.4062501.6086961.1000002.000000
71284416280.3437500.1250000.2187501.1891891.4545451.076923
825610841670.4218750.1601560.2617192.4545452.5625002.392857
9512251711800.4902340.1386720.3515622.3240741.7317072.686567
101.0245691314380.5556640.1279300.4277342.2669321.8450702.433333
112.0481.2152349810.5932620.1142580.4790042.1353251.7862602.239726
124.0962.5324372.0950.6181640.1066890.5114752.0839511.8675212.135576
138.1925.1688154.3530.6308590.0994870.5313722.0410741.8649892.077804
1416.38410.4801.5068.9740.6396480.0919190.5477292.0278641.8478532.061567
1532.76821.1742.70418.4700.6461790.0825200.5636602.0204201.7954852.058168
1665.53642.6825.02237.6600.6512760.0766300.5746462.0157741.8572492.038982
17131.07285.7199.43276.2870.6539840.0719600.5820242.0083171.8781362.025677
18262.144172.16717.571154.5960.6567650.0670280.5897372.0085051.8629132.026505
19524.288345.38333.205312.1780.6587660.0633340.5954322.0060931.8897622.019315
201.048.576692.72162.913629.8080.6606300.0599990.6006322.0056601.8946852.017464
212.097.1521.388.664119.8071.268.8570.6621670.0571280.6050382.0046511.9043282.014673
224.194.3042.783.787227.7062.556.0810.6637070.0542890.6094172.0046511.9006072.014475
238.388.6085.578.740434.3105.144.4300.6650380.0517740.6132642.0040111.9073282.012624
2416.777.21611.177.099830.58510.346.5140.6662070.0495070.6167002.0035171.9124242.011207
2533.554.43222.392.3901.589.36620.803.0240.6673450.0473670.6199782.0034171.9135502.010631
2667.108.86444.853.3903.048.18641.805.2040.6683680.0454220.6229462.0030641.9178632.009573
27134.217.72889.832.8875.855.22583.977.6620.6693070.0436250.6256822.0028111.9208882.008785
28268.435.456179.895.45711.264.683168.630.7740.6701630.0419640.6281992.0025571.9238692.008043
29536.870.912360.226.40021.708.247338.518.1530.6709740.0404350.6305392.0024211.9271072.007452
301.073.741.824721.258.98341.891.214679.367.7690.6717250.0390140.6327112.0022381.9297372.006887
312.147.483.6481.444.020.60180.932.6061.363.087.9950.6724240.0376870.6347372.0020831.9319712.006407
324.294.967.2962.890.865.124156.545.9442.734.319.1800.6730820.0364490.6366332.0019561.9342752.005974
338.589.934.5925.787.009.150303.110.6055.483.898.5450.6736970.0352870.6384102.0018261.9362402.005581
3417.179.869.18411.584.015.940587.497.33310.996.518.6070.6742780.0341970.6400822.0017281.9382282.005238


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
122110011
242110011
384311021
4166422031
53210734051
66411744061
712816974165
8256411922414617
9512712447431630
101.0241314289462659
112.0482347216241116113
124.09643715428342136214
138.19281528553044066399
1416.3841.50650799947696727
1532.7682.7049161.78841.37461.320
1665.5365.0221.6883.33442.50862.504
17131.0729.4323.1686.26444.72864.694
18262.14417.5715.86511.70648.80868.753
19524.28833.20511.04722.158416.555616.640
201.048.57662.91320.86042.053431.388631.515
212.097.152119.80739.97979.828459.707660.090
224.194.304227.70675.844151.8624113.6896114.007
238.388.608434.310144.626289.6844216.8396217.461
2416.777.216830.585276.815553.7704415.0006415.575
2533.554.4321.589.366530.0221.059.3444794.6056794.751
2667.108.8643.048.1861.016.4532.031.73341.523.99461.524.182
27134.217.7285.855.2251.951.3583.903.86742.927.39562.927.820
28268.435.45611.264.6833.754.5057.510.17845.633.11565.631.558
29536.870.91221.708.2477.237.29714.470.950410.855.227610.853.010
301.073.741.82441.891.21413.962.50327.928.711420.945.893620.945.311
312.147.483.64880.932.60626.977.97553.954.631440.467.684640.464.912
324.294.967.296156.545.94452.177.651104.368.293478.275.026678.270.908
338.589.934.592303.110.605101.035.153202.075.4524151.560.9216151.549.674
3417.179.869.184587.497.333195.825.309391.672.0244293.758.6486293.738.675


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
121010100
243210201
384310211
4167430412
53213672524
6642614124859
71282814145869
825667343318171616
9512180899145455337
101.0244382262121199912496
112.048981504477284206276215
124.0962.0951.0481.047611457580447
138.1924.3532.1682.1851.1869781.1891.000
1416.3848.9744.4764.4982.3632.1052.4182.088
1532.76818.4709.2899.1814.8264.3684.9634.313
1665.53637.66018.99418.6669.9348.87310.0468.807
17131.07276.28738.38437.90319.97218.18120.23117.903
18262.144154.59677.77476.82240.35736.85740.78036.602
19524.288312.178156.780155.39881.27774.56681.99374.342
201.048.576629.808316.538313.270163.717150.537164.771150.783
212.097.1521.268.857637.366631.491328.985304.278330.626304.968
224.194.3042.556.0811.284.0141.272.067661.962615.259664.011614.849
238.388.6085.144.4302.583.7582.560.6721.330.5921.240.9251.332.1001.240.813
2416.777.21610.346.5145.196.7795.149.7352.671.1972.499.9982.674.4592.500.860
2533.554.43220.803.02410.443.95910.359.0655.363.4015.036.1085.368.6475.034.868
2667.108.86441.805.20420.982.54520.822.65910.766.95310.132.20610.771.71310.134.332
27134.217.72883.977.66242.145.31641.832.34621.602.47520.384.86421.604.61020.385.713
28268.435.456168.630.77484.614.43284.016.34243.331.50440.991.09643.330.10140.978.073
29536.870.912338.518.153169.832.956168.685.19786.885.10982.382.92086.880.64882.369.476
301.073.741.824679.367.769340.786.239338.581.530174.181.876165.508.585174.181.264165.496.044
312.147.483.6481.363.087.995683.671.000679.416.995349.146.997332.412.043349.131.444332.397.511
324.294.967.2962.734.319.1801.371.250.3931.363.068.787699.744.625667.418.115699.762.323667.394.117
338.589.934.5925.483.898.5452.749.862.1302.734.036.4151.402.212.0871.339.736.0721.402.270.5461.339.679.840
3417.179.869.18410.996.518.6075.513.556.8705.482.961.7372.809.625.5572.688.645.0322.809.701.0982.688.546.920


8. Check for existing Integer Sequences by OEIS

Found in Database : 191, 11, 13, 7, 19, 1, 397, 61, 577, 83, 107, 1, 1, 31, 1069, 1, 1217, 23, 59, 89,
Found in Database : 191, 11, 13, 7, 19, 397, 61, 577, 83, 107, 31, 1069, 1217, 23, 59, 89, 1489, 97, 1613, 223, 167, 149, 2029, 37, 2113, 269, 199, 139, 331, 293,
Found in Database : 7, 11, 13, 19, 23, 31, 37, 59, 61, 79, 83, 89, 97, 103, 107, 127, 139, 149,