Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:06:11
Deutsch
29.Mar 2024

Polynom = x^2+32x-71

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) = 71 = 71
f(1) = 19 = 19
f(2) = 3 = 3
f(3) = 17 = 17
f(4) = 73 = 73
f(5) = 57 = 3*19
f(6) = 157 = 157
f(7) = 101 = 101
f(8) = 249 = 3*83
f(9) = 149 = 149
f(10) = 349 = 349
f(11) = 201 = 3*67
f(12) = 457 = 457
f(13) = 257 = 257
f(14) = 573 = 3*191
f(15) = 317 = 317
f(16) = 697 = 17*41
f(17) = 381 = 3*127
f(18) = 829 = 829
f(19) = 449 = 449
f(20) = 969 = 3*17*19
f(21) = 521 = 521
f(22) = 1117 = 1117
f(23) = 597 = 3*199
f(24) = 1273 = 19*67
f(25) = 677 = 677
f(26) = 1437 = 3*479
f(27) = 761 = 761
f(28) = 1609 = 1609
f(29) = 849 = 3*283
f(30) = 1789 = 1789
f(31) = 941 = 941
f(32) = 1977 = 3*659
f(33) = 1037 = 17*61
f(34) = 2173 = 41*53
f(35) = 1137 = 3*379
f(36) = 2377 = 2377
f(37) = 1241 = 17*73
f(38) = 2589 = 3*863
f(39) = 1349 = 19*71
f(40) = 2809 = 53*53
f(41) = 1461 = 3*487
f(42) = 3037 = 3037
f(43) = 1577 = 19*83
f(44) = 3273 = 3*1091
f(45) = 1697 = 1697
f(46) = 3517 = 3517
f(47) = 1821 = 3*607
f(48) = 3769 = 3769
f(49) = 1949 = 1949
f(50) = 4029 = 3*17*79
f(51) = 2081 = 2081
f(52) = 4297 = 4297
f(53) = 2217 = 3*739
f(54) = 4573 = 17*269
f(55) = 2357 = 2357
f(56) = 4857 = 3*1619
f(57) = 2501 = 41*61
f(58) = 5149 = 19*271
f(59) = 2649 = 3*883
f(60) = 5449 = 5449
f(61) = 2801 = 2801
f(62) = 5757 = 3*19*101
f(63) = 2957 = 2957
f(64) = 6073 = 6073
f(65) = 3117 = 3*1039
f(66) = 6397 = 6397
f(67) = 3281 = 17*193
f(68) = 6729 = 3*2243
f(69) = 3449 = 3449
f(70) = 7069 = 7069
f(71) = 3621 = 3*17*71
f(72) = 7417 = 7417
f(73) = 3797 = 3797
f(74) = 7773 = 3*2591
f(75) = 3977 = 41*97
f(76) = 8137 = 79*103
f(77) = 4161 = 3*19*73
f(78) = 8509 = 67*127
f(79) = 4349 = 4349
f(80) = 8889 = 3*2963
f(81) = 4541 = 19*239
f(82) = 9277 = 9277
f(83) = 4737 = 3*1579
f(84) = 9673 = 17*569
f(85) = 4937 = 4937
f(86) = 10077 = 3*3359
f(87) = 5141 = 53*97
f(88) = 10489 = 17*617
f(89) = 5349 = 3*1783
f(90) = 10909 = 10909
f(91) = 5561 = 67*83
f(92) = 11337 = 3*3779
f(93) = 5777 = 53*109
f(94) = 11773 = 61*193
f(95) = 5997 = 3*1999
f(96) = 12217 = 19*643
f(97) = 6221 = 6221
f(98) = 12669 = 3*41*103
f(99) = 6449 = 6449
f(100) = 13129 = 19*691

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+32x-71

f(0)=71
f(1)=19
f(2)=3
f(3)=17
f(4)=73
f(5)=1
f(6)=157
f(7)=101
f(8)=83
f(9)=149
f(10)=349
f(11)=67
f(12)=457
f(13)=257
f(14)=191
f(15)=317
f(16)=41
f(17)=127
f(18)=829
f(19)=449
f(20)=1
f(21)=521
f(22)=1117
f(23)=199
f(24)=1
f(25)=677
f(26)=479
f(27)=761
f(28)=1609
f(29)=283
f(30)=1789
f(31)=941
f(32)=659
f(33)=61
f(34)=53
f(35)=379
f(36)=2377
f(37)=1
f(38)=863
f(39)=1
f(40)=1
f(41)=487
f(42)=3037
f(43)=1
f(44)=1091
f(45)=1697
f(46)=3517
f(47)=607
f(48)=3769
f(49)=1949
f(50)=79
f(51)=2081
f(52)=4297
f(53)=739
f(54)=269
f(55)=2357
f(56)=1619
f(57)=1
f(58)=271
f(59)=883
f(60)=5449
f(61)=2801
f(62)=1
f(63)=2957
f(64)=6073
f(65)=1039
f(66)=6397
f(67)=193
f(68)=2243
f(69)=3449
f(70)=7069
f(71)=1
f(72)=7417
f(73)=3797
f(74)=2591
f(75)=97
f(76)=103
f(77)=1
f(78)=1
f(79)=4349
f(80)=2963
f(81)=239
f(82)=9277
f(83)=1579
f(84)=569
f(85)=4937
f(86)=3359
f(87)=1
f(88)=617
f(89)=1783
f(90)=10909
f(91)=1
f(92)=3779
f(93)=109
f(94)=1
f(95)=1999
f(96)=643
f(97)=6221
f(98)=1
f(99)=6449

b) Substitution of the polynom
The polynom f(x)=x^2+32x-71 could be written as f(y)= y^2-327 with x=y-16

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+16
f'(x)>2x+31

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

71, 19, 3, 17, 73, 1, 157, 101, 83, 149, 349, 67, 457, 257, 191, 317, 41, 127, 829, 449, 1, 521, 1117, 199, 1, 677, 479, 761, 1609, 283, 1789, 941, 659, 61, 53, 379, 2377, 1, 863, 1, 1, 487, 3037, 1, 1091, 1697, 3517, 607, 3769, 1949, 79, 2081, 4297, 739, 269, 2357, 1619, 1, 271, 883, 5449, 2801, 1, 2957, 6073, 1039, 6397, 193, 2243, 3449, 7069, 1, 7417, 3797, 2591, 97, 103, 1, 1, 4349, 2963, 239, 9277, 1579, 569, 4937, 3359, 1, 617, 1783, 10909, 1, 3779, 109, 1, 1999, 643, 6221, 1, 6449, 691, 131, 13597, 6917, 4691, 421, 14557, 2467, 1, 7649, 1, 7901, 16057, 2719, 16573, 443, 139, 8681, 1, 1, 18169, 9221, 367, 9497, 19273, 3259, 1, 10061, 6803, 1, 151, 3547, 21577, 10937, 389, 661, 22777, 3847, 1231, 1, 1, 12161, 347, 4159, 1, 1, 163, 13121, 1, 4483, 373, 13781, 547, 743, 28573, 1, 1721, 1, 1, 15149, 30649, 5167, 31357, 1, 10691, 16217, 32797, 5527, 33529, 997, 11423, 17321, 1, 1, 431, 18077, 641, 18461, 37309, 1, 929, 1, 12959, 1, 409, 6679, 2381, 20441, 13763, 20849, 2477, 1, 42937, 1, 14591, 1163, 44617, 7507, 1109, 433, 15443, 227, 1, 467, 48073, 1, 16319, 1453, 631, 1, 2671, 25601, 1, 1, 2767, 8839, 877, 26981, 18143, 27449, 3257, 1, 1, 1, 1123, 1, 58237, 9787, 59209, 1571, 20063, 30341, 1, 541, 787, 31337, 21059, 1873, 64189, 263, 1069, 1933, 22079, 33377, 67273, 11299, 1289, 34421, 1217, 34949, 70429, 11827, 1, 36017, 1423, 1, 1009, 1, 4397, 37649, 1, 38201, 1877, 12919, 773, 39317, 26399, 2099, 80329, 1, 1, 1, 27539, 41597, 83773, 827, 1, 701, 28703, 647, 2129, 1, 859, 44537, 1, 45137, 4783, 1, 5417, 46349, 1637, 311, 1, 15859, 95773, 48197, 1, 48821, 98269, 1, 99529, 821, 33599, 1237, 991, 1, 103357, 1, 34883, 1, 1049, 1, 1, 683, 36191, 54617, 109897, 18427, 111229, 55949, 1, 1381, 6701, 1, 6067, 57977, 2287, 58661, 6211, 1, 119389, 60041, 1, 60737, 122173, 20479, 1, 62141, 1, 3697, 1523, 21187, 127837, 1, 1051, 64997, 1951, 1153, 132169, 66449, 44543, 67181, 2549, 22639, 136573, 967, 2707, 69401, 2633, 1, 8297, 70901, 1, 1, 144073, 1, 1, 73181, 49043, 1013, 148669, 24907, 150217, 4441, 50591, 919, 153337, 1511, 1597, 77849, 52163, 4139, 158077, 26479, 159673, 1, 53759, 81041, 162889, 27283, 9677, 1, 1, 83477, 1, 28099, 887, 85121, 1, 1409, 9091, 28927, 2389, 2137, 3089, 1669, 1193, 1, 1291, 90149, 60383, 1, 182857, 1, 1, 92717, 62099, 93581, 188029, 1657, 1741, 1, 63839, 1, 11369, 32359, 1, 97961, 1, 98849, 198589, 33247, 1327, 1, 67391, 101537, 203977, 34147, 10831, 103349, 69203, 6133, 1, 35059, 5153, 1, 71039, 107021, 947, 35983, 216829, 108881, 1, 1, 220573, 36919, 1493, 5879, 1, 1, 3709, 1993, 13421, 1181, 1447, 1627, 2297, 1, 1, 1, 78623, 1499, 1459, 39799, 1471, 1, 4241, 1, 243709, 2399, 1, 123341, 82559, 124337, 4093, 1019, 251677, 126341, 1, 1, 255709, 42787, 15161, 1931, 1, 6863, 15401, 1, 263869, 6971, 88643, 133481, 1, 44839, 270073, 1, 1093, 136601, 6689, 2699, 2683, 2617, 1, 8221, 14767, 1, 282697, 2677, 1, 142949, 1, 1, 2207, 1, 1229, 1151, 1, 1, 1, 1361, 5839, 149441, 299977, 1, 302173, 151637, 1429, 8039, 306589, 51283, 308809, 1867, 1, 9181, 313273, 1, 315517, 1, 1451, 3889, 4051, 53527, 16963, 161717, 108191, 1, 17203, 54667, 1, 1, 1, 166301, 333757, 55819, 1, 168617, 112799, 1, 4799, 56983, 6473, 9059, 1187, 173297, 347773, 3061, 350137, 10333, 117503, 176849, 1, 3491, 5857, 1, 119891, 180437, 2851, 1, 4993, 2203, 1, 1787, 1373, 1, 1, 1, 124739, 187721, 22157, 62983, 1399, 190181, 1, 2423, 1607, 1, 6337, 2731, 3163, 10271, 1259, 3851, 1, 1, 1973, 11701, 399097, 1, 401629, 201449, 1, 2777, 406717, 1283, 4931, 1567, 137279, 3083, 1, 1, 416989, 5101, 1, 210437, 2689, 1, 424777, 1307, 1, 3019, 430009, 71887, 432637, 3557, 145091, 12841, 2269, 3853, 440569, 1, 147743, 11699, 3511, 1, 1, 1, 1, 2333, 453949, 75883, 456649, 3137, 9007, 230357, 462073, 77239, 1439, 3821, 1543, 2417, 1, 78607, 1, 1, 158591, 1, 9029, 79987, 6779, 14197, 161363, 3623, 6163, 4787, 489673, 12923, 1, 246941, 495289, 4357, 12149, 1, 166979, 251177, 1, 84199, 1753, 254021, 169823, 1, 30137, 85627, 1, 258317, 1, 1, 520957, 1, 1, 262649, 2473, 264101, 529657, 1, 5273, 267017, 1733, 1, 2819, 89983, 541369, 1, 181439, 1, 547273, 91459, 2141, 14519, 3023, 5233, 32717, 2267, 559177, 280337, 1, 281837, 565177, 94447, 568189, 1, 190403, 1, 30223, 95959, 3823, 289397, 10181, 1, 8707, 1, 586429, 17293, 196499, 295517, 1, 1, 4547, 298601, 199583, 300149, 601849, 1, 604957, 303257, 11923, 1, 6301, 1, 36137, 3049, 205823, 309521, 15137, 103699, 1, 312677, 3943, 314261, 3299, 1, 33331, 1, 4003, 319037, 1, 6287, 642877, 4079, 1, 5309, 7823, 1, 1, 1, 218591, 8017, 1, 1, 1, 17471, 1, 333581, 39341, 5881, 672073, 336857, 225119, 4637, 678649, 113383, 681949, 2459, 228419, 8377, 688573, 1, 691897, 1, 12197, 1, 9839, 116707, 36943, 351797, 235091, 2543, 11617, 118387, 10627, 6733, 238463, 358541, 42281, 120079, 722173, 6829, 1, 19139, 8783, 1, 1, 19319, 245279, 1, 739273, 123499, 742717, 1, 248723, 21997, 12289, 1, 10607, 1, 6151, 379157, 1, 126967, 763549, 382649, 13457, 1, 770557, 1, 774073, 387917, 1, 1, 781129, 130483, 1, 1, 262739, 4759, 791773, 6961, 795337, 9721, 1, 1, 10993, 134047, 1, 23761, 269891, 2687, 813277, 1, 15413, 409349, 273503, 5791, 824137, 137659, 2293, 1, 1, 416621, 43951, 139483, 1, 1, 1, 2833, 1, 141319, 6691, 425801, 284483, 2239, 1, 2347, 2213, 1, 288191, 3307, 8597, 1, 21269, 5531, 291923, 1, 8539, 2069, 883273, 442577, 1, 444461, 890809, 148783, 894589, 448241, 15761, 450137, 53069, 2843, 1, 1, 17839, 455849, 4591, 2879, 1, 459677, 307091, 5843, 925117, 1, 1, 6947, 2237, 1, 936697, 2143, 940573, 1, 3793, 473201, 948349, 3863, 9817, 2927, 1, 2939, 6907, 1, 3041, 3803, 18979, 2539, 51151, 1, 1, 488897, 17189, 490877, 983737, 4007, 1, 494849, 330563, 496841, 995677, 1, 2087, 1, 4583, 502841, 19013, 1, 3761, 506861, 338579, 26783, 1, 170299, 1, 512921, 4337, 514949,

6. Sequence of the polynom (only primes)

71, 19, 3, 17, 73, 157, 101, 83, 149, 349, 67, 457, 257, 191, 317, 41, 127, 829, 449, 521, 1117, 199, 677, 479, 761, 1609, 283, 1789, 941, 659, 61, 53, 379, 2377, 863, 487, 3037, 1091, 1697, 3517, 607, 3769, 1949, 79, 2081, 4297, 739, 269, 2357, 1619, 271, 883, 5449, 2801, 2957, 6073, 1039, 6397, 193, 2243, 3449, 7069, 7417, 3797, 2591, 97, 103, 4349, 2963, 239, 9277, 1579, 569, 4937, 3359, 617, 1783, 10909, 3779, 109, 1999, 643, 6221, 6449, 691, 131, 13597, 6917, 4691, 421, 14557, 2467, 7649, 7901, 16057, 2719, 16573, 443, 139, 8681, 18169, 9221, 367, 9497, 19273, 3259, 10061, 6803, 151, 3547, 21577, 10937, 389, 661, 22777, 3847, 1231, 12161, 347, 4159, 163, 13121, 4483, 373, 13781, 547, 743, 28573, 1721, 15149, 30649, 5167, 31357, 10691, 16217, 32797, 5527, 33529, 997, 11423, 17321, 431, 18077, 641, 18461, 37309, 929, 12959, 409, 6679, 2381, 20441, 13763, 20849, 2477, 42937, 14591, 1163, 44617, 7507, 1109, 433, 15443, 227, 467, 48073, 16319, 1453, 631, 2671, 25601, 2767, 8839, 877, 26981, 18143, 27449, 3257, 1123, 58237, 9787, 59209, 1571, 20063, 30341, 541, 787, 31337, 21059, 1873, 64189, 263, 1069, 1933, 22079, 33377, 67273, 11299, 1289, 34421, 1217, 34949, 70429, 11827, 36017, 1423, 1009, 4397, 37649, 38201, 1877, 12919, 773, 39317, 26399, 2099, 80329, 27539, 41597, 83773, 827, 701, 28703, 647, 2129, 859, 44537, 45137, 4783, 5417, 46349, 1637, 311, 15859, 95773, 48197, 48821, 98269, 99529, 821, 33599, 1237, 991, 103357, 34883, 1049, 683, 36191, 54617, 109897, 18427, 111229, 55949, 1381, 6701, 6067, 57977, 2287, 58661, 6211, 119389, 60041, 60737, 122173, 20479, 62141, 3697, 1523, 21187, 127837, 1051, 64997, 1951, 1153, 132169, 66449, 44543, 67181, 2549, 22639, 136573, 967, 2707, 69401, 2633, 8297, 70901, 144073, 73181, 49043, 1013, 148669, 24907, 150217, 4441, 50591, 919, 153337, 1511, 1597, 77849, 52163, 4139, 158077, 26479, 159673, 53759, 81041, 162889, 27283, 9677, 83477, 28099, 887, 85121, 1409, 9091, 28927, 2389, 2137, 3089, 1669, 1193, 1291, 90149, 60383, 182857, 92717, 62099, 93581, 188029, 1657, 1741, 63839, 11369, 32359, 97961, 98849, 198589, 33247, 1327, 67391, 101537, 203977, 34147, 10831, 103349, 69203, 6133, 35059, 5153, 71039, 107021, 947, 35983, 216829, 108881, 220573, 36919, 1493, 5879, 3709, 1993, 13421, 1181, 1447, 1627, 2297, 78623, 1499, 1459, 39799, 1471, 4241, 243709, 2399, 123341, 82559, 124337, 4093, 1019, 251677, 126341, 255709, 42787, 15161, 1931, 6863, 15401, 263869, 6971, 88643, 133481, 44839, 270073, 1093, 136601, 6689, 2699, 2683, 2617, 8221, 14767, 282697, 2677, 142949, 2207, 1229, 1151, 1361, 5839, 149441, 299977, 302173, 151637, 1429, 8039, 306589, 51283, 308809, 1867, 9181, 313273, 315517, 1451, 3889, 4051, 53527, 16963, 161717, 108191, 17203, 54667, 166301, 333757, 55819, 168617, 112799, 4799, 56983, 6473, 9059, 1187, 173297, 347773, 3061, 350137, 10333, 117503, 176849, 3491, 5857, 119891, 180437, 2851, 4993, 2203, 1787, 1373, 124739, 187721, 22157, 62983, 1399, 190181, 2423, 1607, 6337, 2731, 3163, 10271, 1259, 3851, 1973, 11701, 399097, 401629, 201449, 2777, 406717, 1283, 4931, 1567, 137279, 3083, 416989, 5101, 210437, 2689, 424777, 1307, 3019, 430009, 71887, 432637, 3557, 145091, 12841, 2269, 3853, 440569, 147743, 11699, 3511, 2333, 453949, 75883, 456649, 3137, 9007, 230357, 462073, 77239, 1439, 3821, 1543, 2417, 78607, 158591, 9029, 79987, 6779, 14197, 161363, 3623, 6163, 4787, 489673, 12923, 246941, 495289, 4357, 12149, 166979, 251177, 84199, 1753, 254021, 169823, 30137, 85627, 258317, 520957, 262649, 2473, 264101, 529657, 5273, 267017, 1733, 2819, 89983, 541369, 181439, 547273, 91459, 2141, 14519, 3023, 5233, 32717, 2267, 559177, 280337, 281837, 565177, 94447, 568189, 190403, 30223, 95959, 3823, 289397, 10181, 8707, 586429, 17293, 196499, 295517, 4547, 298601, 199583, 300149, 601849, 604957, 303257, 11923, 6301, 36137, 3049, 205823, 309521, 15137, 103699, 312677, 3943, 314261, 3299, 33331, 4003, 319037, 6287, 642877, 4079, 5309, 7823, 218591, 8017, 17471, 333581, 39341, 5881, 672073, 336857, 225119, 4637, 678649, 113383, 681949, 2459, 228419, 8377, 688573, 691897, 12197, 9839, 116707, 36943, 351797, 235091, 2543, 11617, 118387, 10627, 6733, 238463, 358541, 42281, 120079, 722173, 6829, 19139, 8783, 19319, 245279, 739273, 123499, 742717, 248723, 21997, 12289, 10607, 6151, 379157, 126967, 763549, 382649, 13457, 770557, 774073, 387917, 781129, 130483, 262739, 4759, 791773, 6961, 795337, 9721, 10993, 134047, 23761, 269891, 2687, 813277, 15413, 409349, 273503, 5791, 824137, 137659, 2293, 416621, 43951, 139483, 2833, 141319, 6691, 425801, 284483, 2239, 2347, 2213, 288191, 3307, 8597, 21269, 5531, 291923, 8539, 2069, 883273, 442577, 444461, 890809, 148783, 894589, 448241, 15761, 450137, 53069, 2843, 17839, 455849, 4591, 2879, 459677, 307091, 5843, 925117, 6947, 2237, 936697, 2143, 940573, 3793, 473201, 948349, 3863, 9817, 2927, 2939, 6907, 3041, 3803, 18979, 2539, 51151, 488897, 17189, 490877, 983737, 4007, 494849, 330563, 496841, 995677, 2087, 4583, 502841, 19013, 3761, 506861, 338579, 26783, 170299, 512921, 4337, 514949,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+32x-71 and
the reducible primes which appear as divisor for the first time
p | x^2+32x-71 and p < x^2+32x-71

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)
11010551.0000000.5000001.0000000.0000000.0000000.000000
21008022580.8000000.2200000.8000008.0000004.40000011.600000
31.0007481346140.7480000.1340000.7480009.3500006.09090910.586206
410.0007.2761.0006.2760.7276000.1000000.7276009.7272737.46268710.221498
5100.00072.1437.69664.4470.7214300.0769600.7214309.9152007.69600010.268802
61.000.000717.14963.086654.0630.7171490.0630860.7171499.9406608.19724610.148851
710.000.0007.133.961533.1266.600.8350.7133960.0533130.7133969.9476698.45078210.092048
8100.000.00071.066.2464.612.83166.453.4150.7106620.0461280.7106629.9616818.65242210.067426
91.000.000.000708.565.22640.698.034667.867.1920.7085650.0406980.7085659.9704908.82278810.050156
1010.000.000.0007.069.132.085364.189.1866.704.942.8990.7069130.0364190.7069139.9766858.94857010.039336


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
245321.2500000.7500000.5000001.6666671.5000002.000000
388441.0000000.5000000.5000001.6000001.3333332.000000
416166101.0000000.3750000.6250002.0000001.5000002.500000
5323010200.9375000.3125000.6250001.8750001.6666672.000000
6645317360.8281250.2656250.5625001.7666671.7000001.800000
712810128730.7890620.2187500.5703121.9056601.6470592.027778
8256196441520.7656250.1718750.5937501.9405941.5714292.082192
9512389753140.7597660.1464840.6132811.9846941.7045452.065789
101.0247641366280.7460940.1328120.6132811.9640101.8133332.000000
112.0481.5062591.2470.7353520.1264650.6088871.9712041.9044121.985669
124.0962.9974682.5290.7316890.1142580.6174321.9900401.8069502.028067
138.1925.9768585.1180.7294920.1047360.6247561.9939941.8333332.023725
1416.38411.9091.55110.3580.7268680.0946660.6322021.9928051.8076922.023837
1532.76823.7362.83020.9060.7243650.0863650.6380001.9931141.8246292.018343
1665.53647.3335.25942.0740.7222440.0802460.6419981.9941441.8583042.012532
17131.07294.5069.84684.6600.7210240.0751190.6459051.9966201.8722192.012169
18262.144188.65018.466170.1840.7196430.0704420.6492001.9961701.8754822.010206
19524.288376.46634.942341.5240.7180520.0666470.6514051.9955791.8922342.006793
201.048.576751.97365.951686.0220.7171370.0628960.6542421.9974531.8874422.008708
212.097.1521.501.158125.2531.375.9050.7158080.0597250.6560831.9962921.8991832.005628
224.194.3042.997.398237.6162.759.7820.7146350.0566520.6579831.9967241.8970882.005794
238.388.6085.986.503452.4585.534.0450.7136470.0539370.6597101.9972331.9041562.005247
2416.777.21611.957.172863.18111.093.9910.7127030.0514500.6612531.9973551.9077592.004680
2533.554.43223.886.4411.652.44822.233.9930.7118710.0492470.6626251.9976661.9143702.004147
2667.108.86447.720.2633.169.57544.550.6880.7110870.0472300.6638571.9977971.9181092.003720
27134.217.72895.344.0416.087.28189.256.7600.7103680.0453540.6650151.9979781.9205352.003488
28268.435.456190.507.99111.711.438178.796.5530.7096980.0436290.6660691.9981111.9239192.003171
29536.870.912380.688.29222.563.157358.125.1350.7090870.0420270.6670601.9982801.9265912.002975
301.073.741.824760.756.76943.542.944717.213.8250.7085100.0405530.6679571.9983721.9298252.002691
312.147.483.6481.520.365.43184.105.6231.436.259.8080.7079750.0391650.6688111.9984911.9315562.002555
324.294.967.2963.038.595.410162.682.7422.875.912.6680.7074780.0378780.6696011.9985951.9342672.002362
338.589.934.5926.073.175.183315.023.8785.758.151.3050.7070110.0366740.6703371.9986781.9364312.002200
3417.179.869.18412.138.861.839610.599.37011.528.262.4690.7065750.0355420.6710331.9987671.9382642.002077


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
122010101
243111101
384211111
4166412121
53210813151
664171518171
712828261121141
825644421211211
951275731331401
101.0241361341651691
112.048259257112311341
124.096468466122412421
138.192858856142714291
1416.3841.5511.549178117681
1532.7682.8302.82811.42111.4071
1665.5365.2595.25712.64012.6171
17131.0729.8469.84414.95414.8901
18262.14418.46618.46419.30519.1591
19524.28834.94234.940117.527117.4131
201.048.57665.95165.949132.983132.9661
212.097.152125.253125.251162.591162.6601
224.194.304237.616237.6141118.7991118.8151
238.388.608452.458452.4561226.3151226.1411
2416.777.216863.181863.1791431.2321431.9471
2533.554.4321.652.4481.652.4461825.4801826.9661
2667.108.8643.169.5753.169.57311.584.12411.585.4491
27134.217.7286.087.2816.087.27913.043.23313.044.0461
28268.435.45611.711.43811.711.43615.855.82915.855.6071
29536.870.91222.563.15722.563.155111.280.963111.282.1921
301.073.741.82443.542.94443.542.942121.768.251121.774.6911
312.147.483.64884.105.62384.105.621142.048.564142.057.0571
324.294.967.296162.682.742162.682.740181.333.355181.349.3851
338.589.934.592315.023.878315.023.8761157.500.0361157.523.8401
3417.179.869.184610.599.370610.599.3681305.291.1571305.308.2111


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
242111100
384131210
41610283331
532205156554
66436112591098
712873235019211716
8256152559741383736
951231411220283758076
101.024628234394151160157160
112.0481.247481766300321313313
124.0962.5291.0261.503603644634648
138.1925.1182.1242.9941.2321.2991.2671.320
1416.38410.3584.4255.9332.5362.6492.5382.635
1532.76820.9068.96211.9445.1305.3075.1975.272
1665.53642.07418.28123.79310.36310.72410.43310.554
17131.07284.66037.20047.46020.89721.39421.12021.249
18262.144170.18475.36594.81942.13143.02342.32142.709
19524.288341.524153.086188.43884.65286.00985.04185.822
201.048.576686.022309.766376.256170.758172.685170.397172.182
212.097.1521.375.905625.195750.710342.946345.280342.272345.407
224.194.3042.759.7821.260.1121.499.670688.653692.148687.240691.741
238.388.6085.534.0452.538.6442.995.4011.380.6011.387.7891.378.6781.386.977
2416.777.21611.093.9915.110.6895.983.3022.767.2702.781.0542.765.9382.779.729
2533.554.43222.233.99310.283.03311.950.9605.544.4705.573.8025.543.7305.571.991
2667.108.86444.550.68820.678.83623.871.85211.111.91411.167.66611.109.51011.161.598
27134.217.72889.256.76041.560.43947.696.32122.267.49322.362.24822.263.63622.363.383
28268.435.456178.796.55383.494.14295.302.41144.604.03144.793.85444.613.26844.785.400
29536.870.912358.125.135167.687.719190.437.41689.364.59589.695.39089.366.56789.698.583
301.073.741.824717.213.825336.661.564380.552.261178.995.346179.625.067178.986.642179.606.770
312.147.483.6481.436.259.808675.754.636760.505.172358.487.671359.649.745358.477.706359.644.686
324.294.967.2962.875.912.6681.356.002.3881.519.910.280717.903.606720.060.643717.861.709720.086.710
338.589.934.5925.758.151.3052.720.423.0323.037.728.2731.437.506.7231.441.561.9071.437.477.0221.441.605.653
3417.179.869.18411.528.262.4695.456.644.3356.071.618.1342.878.260.6522.885.879.3592.878.185.5652.885.936.893


8. Check for existing Integer Sequences by OEIS

Found in Database : 71, 19, 3, 17, 73, 1, 157, 101, 83, 149, 349, 67, 457, 257, 191, 317, 41, 127, 829, 449,
Found in Database : 71, 19, 3, 17, 73, 157, 101, 83, 149, 349, 67, 457, 257, 191, 317, 41, 127, 829, 449, 521, 1117, 199, 677, 479, 761, 1609, 283, 1789, 941, 659, 61, 53, 379, 2377, 863,
Found in Database : 3, 17, 19, 41, 53, 61, 67, 71, 73, 79, 83, 97, 101, 103, 109, 127, 131, 139, 149,