Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:58:54
Deutsch
28.Mar 2024

Polynom = x^2+48x-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) = 3 = 3
f(2) = 27 = 3*3*3
f(3) = 5 = 5
f(4) = 135 = 3*3*3*5
f(5) = 3 = 3
f(6) = 251 = 251
f(7) = 39 = 3*13
f(8) = 375 = 3*5*5*5
f(9) = 55 = 5*11
f(10) = 507 = 3*13*13
f(11) = 9 = 3*3
f(12) = 647 = 647
f(13) = 45 = 3*3*5
f(14) = 795 = 3*5*53
f(15) = 109 = 109
f(16) = 951 = 3*317
f(17) = 129 = 3*43
f(18) = 1115 = 5*223
f(19) = 75 = 3*5*5
f(20) = 1287 = 3*3*11*13
f(21) = 43 = 43
f(22) = 1467 = 3*3*163
f(23) = 195 = 3*5*13
f(24) = 1655 = 5*331
f(25) = 219 = 3*73
f(26) = 1851 = 3*617
f(27) = 61 = 61
f(28) = 2055 = 3*5*137
f(29) = 135 = 3*3*3*5
f(30) = 2267 = 2267
f(31) = 297 = 3*3*3*11
f(32) = 2487 = 3*829
f(33) = 325 = 5*5*13
f(34) = 2715 = 3*5*181
f(35) = 177 = 3*59
f(36) = 2951 = 13*227
f(37) = 3 = 3
f(38) = 3195 = 3*3*5*71
f(39) = 415 = 5*83
f(40) = 3447 = 3*3*383
f(41) = 447 = 3*149
f(42) = 3707 = 11*337
f(43) = 15 = 3*5
f(44) = 3975 = 3*5*5*53
f(45) = 257 = 257
f(46) = 4251 = 3*13*109
f(47) = 549 = 3*3*61
f(48) = 4535 = 5*907
f(49) = 585 = 3*3*5*13
f(50) = 4827 = 3*1609
f(51) = 311 = 311
f(52) = 5127 = 3*1709
f(53) = 165 = 3*5*11
f(54) = 5435 = 5*1087
f(55) = 699 = 3*233
f(56) = 5751 = 3*3*3*3*71
f(57) = 739 = 739
f(58) = 6075 = 3*3*3*3*3*5*5
f(59) = 195 = 3*5*13
f(60) = 6407 = 43*149
f(61) = 411 = 3*137
f(62) = 6747 = 3*13*173
f(63) = 865 = 5*173
f(64) = 7095 = 3*5*11*43
f(65) = 909 = 3*3*101
f(66) = 7451 = 7451
f(67) = 477 = 3*3*53
f(68) = 7815 = 3*5*521
f(69) = 125 = 5*5*5
f(70) = 8187 = 3*2729
f(71) = 1047 = 3*349
f(72) = 8567 = 13*659
f(73) = 1095 = 3*5*73
f(74) = 8955 = 3*3*5*199
f(75) = 143 = 11*13
f(76) = 9351 = 3*3*1039
f(77) = 597 = 3*199
f(78) = 9755 = 5*1951
f(79) = 1245 = 3*5*83
f(80) = 10167 = 3*3389
f(81) = 1297 = 1297
f(82) = 10587 = 3*3529
f(83) = 675 = 3*3*3*5*5
f(84) = 11015 = 5*2203
f(85) = 351 = 3*3*3*13
f(86) = 11451 = 3*11*347
f(87) = 1459 = 1459
f(88) = 11895 = 3*5*13*61
f(89) = 1515 = 3*5*101
f(90) = 12347 = 12347
f(91) = 393 = 3*131
f(92) = 12807 = 3*3*1423
f(93) = 815 = 5*163
f(94) = 13275 = 3*3*5*5*59
f(95) = 1689 = 3*563
f(96) = 13751 = 13751
f(97) = 1749 = 3*11*53
f(98) = 14235 = 3*5*13*73
f(99) = 905 = 5*181
f(100) = 14727 = 3*4909

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+48x-73

f(0)=73
f(1)=3
f(2)=1
f(3)=5
f(4)=1
f(5)=1
f(6)=251
f(7)=13
f(8)=1
f(9)=11
f(10)=1
f(11)=1
f(12)=647
f(13)=1
f(14)=53
f(15)=109
f(16)=317
f(17)=43
f(18)=223
f(19)=1
f(20)=1
f(21)=1
f(22)=163
f(23)=1
f(24)=331
f(25)=1
f(26)=617
f(27)=61
f(28)=137
f(29)=1
f(30)=2267
f(31)=1
f(32)=829
f(33)=1
f(34)=181
f(35)=59
f(36)=227
f(37)=1
f(38)=71
f(39)=83
f(40)=383
f(41)=149
f(42)=337
f(43)=1
f(44)=1
f(45)=257
f(46)=1
f(47)=1
f(48)=907
f(49)=1
f(50)=1609
f(51)=311
f(52)=1709
f(53)=1
f(54)=1087
f(55)=233
f(56)=1
f(57)=739
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=173
f(63)=1
f(64)=1
f(65)=101
f(66)=7451
f(67)=1
f(68)=521
f(69)=1
f(70)=2729
f(71)=349
f(72)=659
f(73)=1
f(74)=199
f(75)=1
f(76)=1039
f(77)=1
f(78)=1951
f(79)=1
f(80)=3389
f(81)=1297
f(82)=3529
f(83)=1
f(84)=2203
f(85)=1
f(86)=347
f(87)=1459
f(88)=1
f(89)=1
f(90)=12347
f(91)=131
f(92)=1423
f(93)=1
f(94)=1
f(95)=563
f(96)=13751
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+48x-73 could be written as f(y)= y^2-649 with x=y-24

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+24
f'(x)>2x+47

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, 3, 1, 5, 1, 1, 251, 13, 1, 11, 1, 1, 647, 1, 53, 109, 317, 43, 223, 1, 1, 1, 163, 1, 331, 1, 617, 61, 137, 1, 2267, 1, 829, 1, 181, 59, 227, 1, 71, 83, 383, 149, 337, 1, 1, 257, 1, 1, 907, 1, 1609, 311, 1709, 1, 1087, 233, 1, 739, 1, 1, 1, 1, 173, 1, 1, 101, 7451, 1, 521, 1, 2729, 349, 659, 1, 199, 1, 1039, 1, 1951, 1, 3389, 1297, 3529, 1, 2203, 1, 347, 1459, 1, 1, 12347, 131, 1423, 1, 1, 563, 13751, 1, 1, 1, 4909, 1, 15227, 1, 1049, 1999, 5417, 1, 1, 1, 641, 1, 661, 151, 283, 389, 6317, 601, 1301, 1, 379, 1, 1, 1, 1, 449, 21851, 1, 499, 569, 1, 487, 23687, 1, 1621, 3079, 8317, 1, 5119, 1, 673, 1, 8969, 1, 1103, 1163, 1, 1787, 643, 1, 1, 1249, 919, 1, 2069, 1, 31751, 1, 433, 821, 11069, 1399, 409, 1, 1, 1, 1, 1493, 7243, 1, 12329, 1, 12589, 1, 701, 541, 1009, 4969, 2677, 1, 577, 431, 4643, 211, 947, 1, 43451, 457, 2953, 1, 1, 1, 3539, 1, 1, 2957, 1447, 1, 1, 1, 5503, 6247, 1, 1, 10267, 1, 17417, 599, 709, 1, 887, 1, 1, 1, 3733, 1, 56951, 2393, 1, 1, 727, 1237, 1, 503, 4057, 7669, 1, 1, 967, 1, 1, 619, 1, 1, 13183, 1, 1, 4217, 1511, 571, 6277, 1, 23369, 883, 1, 1, 72251, 1, 4889, 1847, 24809, 1, 75527, 1, 1, 9649, 1, 1, 631, 1, 26669, 1259, 2459, 1, 1, 1151, 27817, 1, 5641, 1, 6599, 1, 3221, 1, 653, 1, 1, 937, 6037, 1, 1, 1283, 92987, 1, 1, 5927, 31817, 4003, 1, 811, 10883, 1, 1, 1, 4019, 1, 2609, 12799, 6869, 1, 104327, 1, 1, 2657, 7129, 4483, 757, 2269, 1, 1, 12323, 4649, 112247, 941, 7573, 3571, 38317, 1, 1, 1, 39229, 14797, 1, 1, 24091, 1, 4513, 15319, 1, 1033, 2113, 1, 1, 1, 8501, 1, 128951, 1801, 8693, 1, 1, 1381, 1223, 1117, 1, 1303, 15139, 1427, 1, 1, 4219, 17497, 769, 1, 28447, 1, 47917, 1129, 1, 1217, 146807, 1, 1, 1, 3331, 1, 151451, 6343, 1, 1, 1, 1, 14197, 1, 809, 19819, 53117, 6673, 32191, 1, 1, 10211, 2027, 1, 2551, 1, 55817, 1, 11273, 1, 2339, 2383, 1, 1, 1, 911, 971, 1, 3943, 4457, 1531, 7499, 857, 1, 1, 1, 61417, 1, 1, 1, 4813, 1, 63149, 1, 38239, 1, 1949, 24229, 4327, 1, 1, 1, 66089, 1, 13337, 2791, 15527, 1, 13577, 2557, 68489, 8599, 207287, 1, 1549, 13127, 1, 2207, 1, 1, 71549, 26947, 72169, 1, 1747, 1523, 73417, 1, 1, 1, 224027, 1, 1931, 1, 1, 9533, 229751, 9613, 3089, 1, 7079, 1, 18119, 1, 1, 2293, 79817, 5009, 1, 1, 27043, 2777, 1, 2053, 1, 1, 83117, 15647, 1289, 1, 253367, 3533, 1, 1, 1, 1, 259451, 10853, 1, 6563, 1, 1, 265607, 1, 1, 33589, 89917, 3761, 54367, 1, 91309, 1, 92009, 2309, 1, 11633, 31139, 1, 1, 1181, 284507, 1, 95549, 7193, 1481, 1, 2221, 1, 19541, 1471, 98429, 1, 297467, 1, 6659, 18797, 3049, 1, 60811, 2543, 7853, 19211, 102829, 1, 62143, 1, 1429, 1, 21013, 1, 24419, 6637, 3947, 1, 1, 13463, 324251, 6779, 1979, 1, 109609, 4583, 331127, 1, 22229, 1307, 8609, 7019, 13523, 1, 1, 42697, 38083, 1433, 69019, 3607, 1, 43579, 23321, 1, 1, 1, 118189, 4447, 1, 14923, 1, 1, 8039, 1, 1, 1, 1, 613, 1, 46279, 123817, 1, 74779, 1, 1, 1, 1, 3167, 1, 1, 1, 1, 2861, 3229, 388727, 16249, 130409, 1, 26249, 1, 5581, 5521, 1, 1, 133769, 8387, 1, 1, 1, 1, 45439, 17093, 1, 1, 1663, 1997, 138889, 1, 1, 1, 140617, 1, 28297, 1, 427067, 1373, 47743, 1, 1, 4517, 39541, 1, 29173, 10973, 146749, 6133, 2447, 1, 1, 6983, 149417, 1, 90187, 3769, 1, 7109, 16901, 1907, 91807, 19183, 1, 1, 1, 1, 1, 1627, 3643, 11783, 31513, 19753, 11057, 4967, 10627, 1, 4111, 1, 483767, 1, 1, 30497, 163117, 1, 1, 1, 1, 62047, 2273, 1, 1697, 10459, 1, 1, 11251, 4231, 509147, 1, 170669, 3209, 1, 1, 39827, 7211, 1, 1, 1601, 10937, 47857, 1, 1, 1543, 6571, 11119, 21407, 1, 179369, 67447, 13873, 1, 108799, 1, 1, 1, 1, 1, 552887, 23099, 4751, 6967, 1, 1, 1, 23473, 37657, 14159, 2593, 1, 570887, 1, 38261, 1, 192317, 24103, 8923, 2423, 64783, 4567, 65123, 1, 23567, 1, 1, 4639, 1, 1, 598427, 1, 4663, 15077, 1, 1, 11467, 1, 1, 1, 22741, 1973, 617147, 1, 3181, 38867, 2927, 8681, 11393, 1, 209929, 39461, 2089, 1, 9787, 26573, 71039, 6163, 1, 1, 1861, 13487, 216329, 3253, 43481, 1, 1, 1, 43913, 4127, 1, 1, 665207, 5557, 2971, 1, 1, 14029, 135007, 5639, 17393, 7727, 3851, 1, 136987, 1, 1, 86239, 9221, 1, 1, 1, 2351, 1, 5197, 29303, 1, 29443, 47221, 1, 3889, 2477, 1, 1, 1, 90019, 1, 7537, 1987, 1, 80963, 91297, 81343, 1, 1, 15359, 246317, 1, 1, 1, 12227, 3461, 19213, 2347, 10037, 1, 756251, 31583, 16883, 19037, 1, 15937, 58979, 1, 51349, 1, 23447, 10771, 31091, 1, 260269, 1, 4933, 6551, 157579, 2531, 3257, 4507, 1, 1, 4013, 33349, 267389, 1, 53717, 1, 11087, 1, 1, 20369, 272189, 1, 820187, 1, 18307, 25801, 91939, 1, 166219, 1, 3919, 2011, 5273, 1, 2591, 3907, 1867, 1, 56629, 3547, 77557, 1, 95203, 21467, 1, 35933, 864251, 1, 1, 1, 1, 1, 875447, 2437, 58613, 1, 294317, 1, 1, 1, 1, 1, 3011, 1, 4177, 18749, 1, 1, 1, 2521, 69959, 6329, 304429, 1, 61141, 1, 920951, 38453, 20551, 1, 103183, 19387, 15287, 1, 1, 2213, 24109, 1, 188827, 1, 1, 118747, 3823, 7949, 1, 1, 1, 60107, 4283, 1, 967607, 1, 5309, 1, 65033, 1, 89041, 1, 1, 1, 1, 5153, 76259, 4139, 1, 124669, 37013, 3793, 200671, 1, 1, 31543, 25933, 1, 1, 1, 339817, 1, 6203, 4273, 1027547, 42899, 1, 25841, 4603, 1, 9539, 2713, 1, 2377,

6. Sequence of the polynom (only primes)

73, 3, 5, 251, 13, 11, 647, 53, 109, 317, 43, 223, 163, 331, 617, 61, 137, 2267, 829, 181, 59, 227, 71, 83, 383, 149, 337, 257, 907, 1609, 311, 1709, 1087, 233, 739, 173, 101, 7451, 521, 2729, 349, 659, 199, 1039, 1951, 3389, 1297, 3529, 2203, 347, 1459, 12347, 131, 1423, 563, 13751, 4909, 15227, 1049, 1999, 5417, 641, 661, 151, 283, 389, 6317, 601, 1301, 379, 449, 21851, 499, 569, 487, 23687, 1621, 3079, 8317, 5119, 673, 8969, 1103, 1163, 1787, 643, 1249, 919, 2069, 31751, 433, 821, 11069, 1399, 409, 1493, 7243, 12329, 12589, 701, 541, 1009, 4969, 2677, 577, 431, 4643, 211, 947, 43451, 457, 2953, 3539, 2957, 1447, 5503, 6247, 10267, 17417, 599, 709, 887, 3733, 56951, 2393, 727, 1237, 503, 4057, 7669, 967, 619, 13183, 4217, 1511, 571, 6277, 23369, 883, 72251, 4889, 1847, 24809, 75527, 9649, 631, 26669, 1259, 2459, 1151, 27817, 5641, 6599, 3221, 653, 937, 6037, 1283, 92987, 5927, 31817, 4003, 811, 10883, 4019, 2609, 12799, 6869, 104327, 2657, 7129, 4483, 757, 2269, 12323, 4649, 112247, 941, 7573, 3571, 38317, 39229, 14797, 24091, 4513, 15319, 1033, 2113, 8501, 128951, 1801, 8693, 1381, 1223, 1117, 1303, 15139, 1427, 4219, 17497, 769, 28447, 47917, 1129, 1217, 146807, 3331, 151451, 6343, 14197, 809, 19819, 53117, 6673, 32191, 10211, 2027, 2551, 55817, 11273, 2339, 2383, 911, 971, 3943, 4457, 1531, 7499, 857, 61417, 4813, 63149, 38239, 1949, 24229, 4327, 66089, 13337, 2791, 15527, 13577, 2557, 68489, 8599, 207287, 1549, 13127, 2207, 71549, 26947, 72169, 1747, 1523, 73417, 224027, 1931, 9533, 229751, 9613, 3089, 7079, 18119, 2293, 79817, 5009, 27043, 2777, 2053, 83117, 15647, 1289, 253367, 3533, 259451, 10853, 6563, 265607, 33589, 89917, 3761, 54367, 91309, 92009, 2309, 11633, 31139, 1181, 284507, 95549, 7193, 1481, 2221, 19541, 1471, 98429, 297467, 6659, 18797, 3049, 60811, 2543, 7853, 19211, 102829, 62143, 1429, 21013, 24419, 6637, 3947, 13463, 324251, 6779, 1979, 109609, 4583, 331127, 22229, 1307, 8609, 7019, 13523, 42697, 38083, 1433, 69019, 3607, 43579, 23321, 118189, 4447, 14923, 8039, 613, 46279, 123817, 74779, 3167, 2861, 3229, 388727, 16249, 130409, 26249, 5581, 5521, 133769, 8387, 45439, 17093, 1663, 1997, 138889, 140617, 28297, 427067, 1373, 47743, 4517, 39541, 29173, 10973, 146749, 6133, 2447, 6983, 149417, 90187, 3769, 7109, 16901, 1907, 91807, 19183, 1627, 3643, 11783, 31513, 19753, 11057, 4967, 10627, 4111, 483767, 30497, 163117, 62047, 2273, 1697, 10459, 11251, 4231, 509147, 170669, 3209, 39827, 7211, 1601, 10937, 47857, 1543, 6571, 11119, 21407, 179369, 67447, 13873, 108799, 552887, 23099, 4751, 6967, 23473, 37657, 14159, 2593, 570887, 38261, 192317, 24103, 8923, 2423, 64783, 4567, 65123, 23567, 4639, 598427, 4663, 15077, 11467, 22741, 1973, 617147, 3181, 38867, 2927, 8681, 11393, 209929, 39461, 2089, 9787, 26573, 71039, 6163, 1861, 13487, 216329, 3253, 43481, 43913, 4127, 665207, 5557, 2971, 14029, 135007, 5639, 17393, 7727, 3851, 136987, 86239, 9221, 2351, 5197, 29303, 29443, 47221, 3889, 2477, 90019, 7537, 1987, 80963, 91297, 81343, 15359, 246317, 12227, 3461, 19213, 2347, 10037, 756251, 31583, 16883, 19037, 15937, 58979, 51349, 23447, 10771, 31091, 260269, 4933, 6551, 157579, 2531, 3257, 4507, 4013, 33349, 267389, 53717, 11087, 20369, 272189, 820187, 18307, 25801, 91939, 166219, 3919, 2011, 5273, 2591, 3907, 1867, 56629, 3547, 77557, 95203, 21467, 35933, 864251, 875447, 2437, 58613, 294317, 3011, 4177, 18749, 2521, 69959, 6329, 304429, 61141, 920951, 38453, 20551, 103183, 19387, 15287, 2213, 24109, 188827, 118747, 3823, 7949, 60107, 4283, 967607, 5309, 65033, 89041, 5153, 76259, 4139, 124669, 37013, 3793, 200671, 31543, 25933, 339817, 6203, 4273, 1027547, 42899, 25841, 4603, 9539, 2713, 2377,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+48x-73 and
the reducible primes which appear as divisor for the first time
p | x^2+48x-73 and p < x^2+48x-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)
1104220.4000000.2000000.4000000.0000000.0000000.000000
2100527450.5200000.0700000.52000013.0000003.50000022.500000
31.000569475220.5690000.0470000.56900010.9423076.71428611.600000
410.0006.0943255.7690.6094000.0325000.60940010.7100176.91489411.051724
5100.00062.7862.51260.2740.6278600.0251200.62786010.3029217.72923110.447911
61.000.000639.16620.395618.7710.6391660.0203950.63916610.1800728.11902910.265968
710.000.0006.472.453172.6696.299.7840.6472450.0172670.64724510.1264048.46624210.181124
8100.000.00065.308.5541.497.48663.811.0680.6530860.0149750.65308610.0902328.67258210.129088
91.000.000.000657.621.37913.208.431644.412.9480.6576210.0132080.65762110.0694528.82040410.098764
1010.000.000.0006.612.325.516118.183.4486.494.142.0680.6612330.0118180.66123310.0549138.94757710.077610


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)
122111.0000000.5000000.5000000.0000000.0000000.000000
242110.5000000.2500000.2500001.0000001.0000001.000000
383210.3750000.2500000.1250001.5000002.0000001.000000
4168350.5000000.1875000.3125002.6666671.5000005.000000
532174130.5312500.1250000.4062502.1250001.3333332.600000
664334290.5156250.0625000.4531251.9411761.0000002.230769
7128679580.5234380.0703120.4531252.0303032.2500002.000000
8256138151230.5390620.0585940.4804692.0597011.6666672.120690
9512282282540.5507810.0546880.4960942.0434781.8666672.065041
101.024584485360.5703120.0468750.5234382.0709221.7142862.110236
112.0481.212791.1330.5917970.0385740.5532232.0753421.6458332.113806
124.0962.4521482.3040.5986330.0361330.5625002.0231021.8734182.033539
138.1924.9832634.7200.6082760.0321040.5761722.0322191.7770272.048611
1416.38410.0294919.5380.6121220.0299680.5821532.0126431.8669202.020763
1532.76820.28792019.3670.6191100.0280760.5910342.0228341.8737272.030509
1665.53640.9531.72439.2290.6248930.0263060.5985872.0186821.8739132.025559
17131.07282.4783.21179.2670.6292570.0244980.6047592.0139671.8625292.020622
18262.144166.0545.961160.0930.6334460.0227390.6107062.0133131.8564312.019668
19524.288333.69711.276322.4210.6364770.0215070.6149692.0095691.8916292.013961
201.048.576670.37721.295649.0820.6393210.0203080.6190132.0089391.8885242.013150
212.097.1521.346.28440.3291.305.9550.6419580.0192300.6227282.0082491.8938252.012003
224.194.3042.702.99376.7252.626.2680.6444440.0182930.6261512.0077441.9024772.010994
238.388.6085.425.313146.5505.278.7630.6467480.0174700.6292782.0071501.9100682.009986
2416.777.21610.883.110280.38510.602.7250.6486840.0167120.6319722.0059871.9132382.008562
2533.554.43221.827.336535.71821.291.6180.6505050.0159660.6345402.0056161.9106512.008127
2667.108.86443.766.6131.028.41942.738.1940.6521730.0153250.6368492.0051281.9197022.007278
27134.217.72887.742.8501.976.70185.766.1490.6537350.0147280.6390082.0047901.9220782.006780
28268.435.456175.869.9363.800.413172.069.5230.6551670.0141580.6410092.0043791.9226042.006264
29536.870.912352.455.4807.325.456345.130.0240.6565000.0136450.6428552.0040691.9275422.005759
301.073.741.824706.249.14914.131.989692.117.1600.6577460.0131610.6445842.0037971.9291622.005381
312.147.483.6481.414.998.84227.297.3711.387.701.4710.6589100.0127110.6461992.0035411.9316022.005009
324.294.967.2962.834.668.95852.791.4692.781.877.4890.6599980.0122910.6477062.0033011.9339402.004666
338.589.934.5925.678.082.992102.228.3775.575.854.6150.6610160.0119010.6491152.0030851.9364562.004349
3417.179.869.18411.372.688.030198.143.93411.174.544.0960.6619780.0115330.6504442.0029101.9382482.004095


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
121101000
241101000
382111100
4163121101
5324131201
6644131201
71289181602
8256151141806
951228127113014
101.02448147124023
112.04879178138040
124.0961481147169078
138.192263126211240138
1416.384491149012360254
1532.768920191914530466
1665.5361.72411.72318520871
17131.0723.21113.21011.63201.578
18262.1445.96115.96012.98902.971
19524.28811.276111.27515.63705.638
201.048.57621.295121.294110.632010.662
212.097.15240.329140.328120.155020.173
224.194.30476.725176.724138.424038.300
238.388.608146.5501146.549173.488073.061
2416.777.216280.3851280.3841140.3960139.988
2533.554.432535.7181535.7171268.1360267.581
2667.108.8641.028.41911.028.4181514.1240514.294
27134.217.7281.976.70111.976.7001988.8270987.873
28268.435.4563.800.41313.800.41211.900.68901.899.723
29536.870.9127.325.45617.325.45513.662.43503.663.020
301.073.741.82414.131.989114.131.98817.064.14707.067.841
312.147.483.64827.297.371127.297.370113.645.861013.651.509
324.294.967.29652.791.469152.791.468126.395.005026.396.463
338.589.934.592102.228.3771102.228.376151.109.064051.119.312
3417.179.869.184198.143.9341198.143.933199.064.920099.079.013


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
121000100
241000100
381000100
4165220131
53213842452
6642914137895
712858302616161610
8256123685336273228
951225413212073567154
101.024536277257138128147123
112.0481.133586545293273298269
124.0962.3041.2021.100596567605536
138.1924.7202.4332.2851.2611.1391.2221.098
1416.3849.5384.9124.6242.5382.2472.4962.257
1532.76819.3679.9099.4565.1454.5635.0264.633
1665.53639.22920.11819.10910.2799.35410.2349.362
17131.07279.26740.58038.68520.65818.91620.64119.052
18262.144160.09382.07478.01741.65538.54241.51738.379
19524.288322.421164.681157.73883.63677.71783.48877.580
201.048.576649.082331.356317.724167.737156.713168.300156.332
212.097.1521.305.955665.600640.353337.627315.520337.278315.530
224.194.3042.626.2681.336.9071.289.359677.273635.655677.250636.090
238.388.6085.278.7632.685.6412.593.1201.357.2761.280.4901.359.1431.281.854
2416.777.21610.602.7255.389.8245.212.8992.722.9812.576.6062.724.0282.579.110
2533.554.43221.291.61810.811.83110.479.7855.462.3395.183.3025.462.7115.183.266
2667.108.86442.738.19421.691.73121.046.46110.954.12210.414.59210.953.58410.415.896
27134.217.72885.766.14943.503.44242.262.70521.959.21220.923.74621.958.88820.924.303
28268.435.456172.069.52387.225.24384.844.27844.001.86342.028.77644.018.24442.020.640
29536.870.912345.130.024174.860.054170.269.96888.185.60284.386.25588.188.07684.370.091
301.073.741.824692.117.160350.477.766341.639.392176.686.699169.372.587176.702.349169.355.525
312.147.483.6481.387.701.471702.377.240685.324.229353.992.344339.860.875354.008.719339.839.533
324.294.967.2962.781.877.4891.407.416.1241.374.461.363709.134.120681.816.287709.134.028681.793.054
338.589.934.5925.575.854.6152.819.812.6572.756.041.9561.420.371.3731.367.543.8151.420.412.2241.367.527.203
3417.179.869.18411.174.544.0965.648.970.5315.525.573.5632.844.786.1932.742.461.6192.844.849.4752.742.446.809


8. Check for existing Integer Sequences by OEIS

Found in Database : 73, 3, 1, 5, 1, 1, 251, 13, 1, 11, 1, 1, 647, 1, 53, 109, 317, 43, 223, 1,
Found in Database : 73, 3, 5, 251, 13, 11, 647, 53, 109, 317, 43, 223, 163, 331, 617, 61, 137, 2267, 829, 181, 59, 227, 71, 83,
Found in Database : 3, 5, 11, 13, 43, 53, 59, 61, 71, 73, 83, 101, 109, 131, 137, 149,