Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:33:08
Deutsch
29.Mar 2024

Polynom = x^2+36x-179

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) = 179 = 179
f(1) = 71 = 71
f(2) = 103 = 103
f(3) = 31 = 31
f(4) = 19 = 19
f(5) = 13 = 13
f(6) = 73 = 73
f(7) = 61 = 61
f(8) = 173 = 173
f(9) = 113 = 113
f(10) = 281 = 281
f(11) = 169 = 13*13
f(12) = 397 = 397
f(13) = 229 = 229
f(14) = 521 = 521
f(15) = 293 = 293
f(16) = 653 = 653
f(17) = 361 = 19*19
f(18) = 793 = 13*61
f(19) = 433 = 433
f(20) = 941 = 941
f(21) = 509 = 509
f(22) = 1097 = 1097
f(23) = 589 = 19*31
f(24) = 1261 = 13*97
f(25) = 673 = 673
f(26) = 1433 = 1433
f(27) = 761 = 761
f(28) = 1613 = 1613
f(29) = 853 = 853
f(30) = 1801 = 1801
f(31) = 949 = 13*73
f(32) = 1997 = 1997
f(33) = 1049 = 1049
f(34) = 2201 = 31*71
f(35) = 1153 = 1153
f(36) = 2413 = 19*127
f(37) = 1261 = 13*97
f(38) = 2633 = 2633
f(39) = 1373 = 1373
f(40) = 2861 = 2861
f(41) = 1489 = 1489
f(42) = 3097 = 19*163
f(43) = 1609 = 1609
f(44) = 3341 = 13*257
f(45) = 1733 = 1733
f(46) = 3593 = 3593
f(47) = 1861 = 1861
f(48) = 3853 = 3853
f(49) = 1993 = 1993
f(50) = 4121 = 13*317
f(51) = 2129 = 2129
f(52) = 4397 = 4397
f(53) = 2269 = 2269
f(54) = 4681 = 31*151
f(55) = 2413 = 19*127
f(56) = 4973 = 4973
f(57) = 2561 = 13*197
f(58) = 5273 = 5273
f(59) = 2713 = 2713
f(60) = 5581 = 5581
f(61) = 2869 = 19*151
f(62) = 5897 = 5897
f(63) = 3029 = 13*233
f(64) = 6221 = 6221
f(65) = 3193 = 31*103
f(66) = 6553 = 6553
f(67) = 3361 = 3361
f(68) = 6893 = 61*113
f(69) = 3533 = 3533
f(70) = 7241 = 13*557
f(71) = 3709 = 3709
f(72) = 7597 = 71*107
f(73) = 3889 = 3889
f(74) = 7961 = 19*419
f(75) = 4073 = 4073
f(76) = 8333 = 13*641
f(77) = 4261 = 4261
f(78) = 8713 = 8713
f(79) = 4453 = 61*73
f(80) = 9101 = 19*479
f(81) = 4649 = 4649
f(82) = 9497 = 9497
f(83) = 4849 = 13*373
f(84) = 9901 = 9901
f(85) = 5053 = 31*163
f(86) = 10313 = 10313
f(87) = 5261 = 5261
f(88) = 10733 = 10733
f(89) = 5473 = 13*421
f(90) = 11161 = 11161
f(91) = 5689 = 5689
f(92) = 11597 = 11597
f(93) = 5909 = 19*311
f(94) = 12041 = 12041
f(95) = 6133 = 6133
f(96) = 12493 = 13*31*31
f(97) = 6361 = 6361
f(98) = 12953 = 12953
f(99) = 6593 = 19*347
f(100) = 13421 = 13421

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+36x-179

f(0)=179
f(1)=71
f(2)=103
f(3)=31
f(4)=19
f(5)=13
f(6)=73
f(7)=61
f(8)=173
f(9)=113
f(10)=281
f(11)=1
f(12)=397
f(13)=229
f(14)=521
f(15)=293
f(16)=653
f(17)=1
f(18)=1
f(19)=433
f(20)=941
f(21)=509
f(22)=1097
f(23)=1
f(24)=97
f(25)=673
f(26)=1433
f(27)=761
f(28)=1613
f(29)=853
f(30)=1801
f(31)=1
f(32)=1997
f(33)=1049
f(34)=1
f(35)=1153
f(36)=127
f(37)=1
f(38)=2633
f(39)=1373
f(40)=2861
f(41)=1489
f(42)=163
f(43)=1609
f(44)=257
f(45)=1733
f(46)=3593
f(47)=1861
f(48)=3853
f(49)=1993
f(50)=317
f(51)=2129
f(52)=4397
f(53)=2269
f(54)=151
f(55)=1
f(56)=4973
f(57)=197
f(58)=5273
f(59)=2713
f(60)=5581
f(61)=1
f(62)=5897
f(63)=233
f(64)=6221
f(65)=1
f(66)=6553
f(67)=3361
f(68)=1
f(69)=3533
f(70)=557
f(71)=3709
f(72)=107
f(73)=3889
f(74)=419
f(75)=4073
f(76)=641
f(77)=4261
f(78)=8713
f(79)=1
f(80)=479
f(81)=4649
f(82)=9497
f(83)=373
f(84)=9901
f(85)=1
f(86)=10313
f(87)=5261
f(88)=10733
f(89)=421
f(90)=11161
f(91)=5689
f(92)=11597
f(93)=311
f(94)=12041
f(95)=6133
f(96)=1
f(97)=6361
f(98)=12953
f(99)=347

b) Substitution of the polynom
The polynom f(x)=x^2+36x-179 could be written as f(y)= y^2-503 with x=y-18

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+18
f'(x)>2x+35

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

179, 71, 103, 31, 19, 13, 73, 61, 173, 113, 281, 1, 397, 229, 521, 293, 653, 1, 1, 433, 941, 509, 1097, 1, 97, 673, 1433, 761, 1613, 853, 1801, 1, 1997, 1049, 1, 1153, 127, 1, 2633, 1373, 2861, 1489, 163, 1609, 257, 1733, 3593, 1861, 3853, 1993, 317, 2129, 4397, 2269, 151, 1, 4973, 197, 5273, 2713, 5581, 1, 5897, 233, 6221, 1, 6553, 3361, 1, 3533, 557, 3709, 107, 3889, 419, 4073, 641, 4261, 8713, 1, 479, 4649, 9497, 373, 9901, 1, 10313, 5261, 10733, 421, 11161, 5689, 11597, 311, 12041, 6133, 1, 6361, 12953, 347, 13421, 6829, 1069, 7069, 1, 1, 139, 7561, 15373, 601, 15881, 8069, 863, 8329, 16921, 661, 563, 8861, 947, 9133, 18541, 1, 1, 9689, 19661, 9973, 20233, 331, 1601, 1, 21401, 571, 21997, 11149, 1, 881, 167, 619, 23833, 12073, 401, 953, 25097, 1, 25741, 13033, 26393, 431, 2081, 13693, 1459, 14029, 389, 14369, 2237, 14713, 1567, 15061, 983, 15413, 31181, 1213, 191, 1, 32621, 16493, 33353, 1297, 1, 907, 34841, 17609, 35597, 17989, 2797, 967, 523, 1, 1223, 1, 1, 1, 1, 19949, 211, 20353, 41113, 1597, 2207, 683, 701, 21589, 43597, 1693, 2339, 22433, 45293, 22861, 46153, 23293, 3617, 1, 227, 24169, 48781, 1, 3821, 1319, 491, 823, 51481, 25969, 1, 1, 751, 26893, 239, 27361, 569, 2141, 1811, 28309, 57097, 28789, 58061, 1, 1, 29761, 60013, 30253, 61001, 1, 251, 31249, 1, 1, 64013, 32261, 65033, 2521, 2131, 33289, 1, 33809, 68141, 1, 69193, 1, 1, 35393, 977, 1, 5569, 36469, 1, 37013, 74573, 37561, 5821, 38113, 76781, 38669, 1277, 39229, 4159, 3061, 80153, 40361, 81293, 40933, 4339, 1, 83597, 42089, 84761, 307, 85933, 43261, 6701, 43853, 88301, 44449, 2887, 2371, 6977, 643, 859, 46261, 93133, 2467, 743, 1, 95597, 48109, 857, 48733, 1, 3797, 1361, 49993, 100621, 1, 1, 1, 7937, 51913, 104473, 52561, 1, 1, 8237, 1, 1777, 1759, 109721, 1, 111053, 4297, 1583, 56533, 1063, 3011, 1, 1, 116461, 58573, 117833, 3119, 1229, 359, 9277, 60649, 121997, 1979, 123401, 62053, 9601, 62761, 126233, 63473, 6719, 64189, 129097, 4993, 4211, 65633, 6947, 66361, 1181, 1, 134921, 67829, 136397, 1, 797, 69313, 1, 70061, 787, 3727, 142381, 71569, 11069, 1, 4691, 3847, 146953, 1, 911, 5741, 150041, 1033, 151597, 1249, 1487, 1, 154733, 77761, 1, 78553, 157901, 79349, 12269, 80149, 1, 80953, 162713, 81761, 12641, 1163, 165961, 83389, 167597, 1, 169241, 1, 2341, 4519, 172553, 86693, 174221, 6733, 175897, 4651, 1, 89213, 5783, 1, 13921, 1, 182681, 1, 1901, 439, 1, 93493, 9887, 1, 189593, 95233, 191341, 7393, 10163, 96989, 194861, 1009, 6343, 1, 198413, 1, 200201, 100549, 201997, 101449, 1, 5387, 1619, 3331, 207433, 104173, 16097, 5531, 211097, 467, 2917, 1753, 1, 8297, 216653, 108793, 218521, 1, 1, 8513, 11699, 1, 1, 3631, 1, 113513, 1, 1013, 229897, 115429, 231821, 487, 17981, 117361, 7603, 1621, 1423, 1, 239597, 1, 241561, 1, 243533, 1187, 245513, 499, 247501, 124249, 249497, 1, 251501, 503, 19501, 127261, 8243, 128273, 257561, 129289, 1051, 1, 261641, 2153, 1579, 132361, 1, 1, 2503, 1, 269897, 135469, 271981, 10501, 4493, 1, 276173, 1429, 1, 7351, 21569, 140729, 1871, 141793, 1, 1, 1697, 4643, 2557, 145009, 3001, 1, 293261, 11321, 295433, 1, 297613, 1, 1, 1, 301997, 151549, 2843, 1, 16127, 1, 23741, 154873, 310861, 1, 4289, 157109, 1, 158233, 317593, 159361, 1, 8447, 10391, 12433, 324397, 1171, 326681, 8627, 5393, 12697, 1289, 1, 333581, 5399, 335897, 1327, 26017, 169693, 1783, 2801, 18047, 2423, 26557, 173209, 1237, 174389, 1, 175573, 352333, 13597, 1, 177953, 3467, 5779, 1, 13873, 361901, 181553, 364313, 9619, 366733, 183973, 1, 185189, 11987, 9811, 374041, 187633, 28961, 188861, 378953, 190093, 1637, 1, 5407, 14813, 386381, 193813, 1, 1823, 391373, 15101, 1283, 197569, 1, 198829, 398921, 2741, 30881, 3301, 403993, 2089, 3947, 203909, 31469, 6619, 1951, 10867, 1, 1, 6833, 1, 419401, 11071, 421997, 1873, 424601, 16381, 427213, 1, 1877, 215573, 6091, 2027, 33469, 7039, 23039, 219533, 440393, 220861, 1, 222193, 23459, 223529, 448397, 224869, 14551, 17401, 1999, 227561, 3023, 228913, 459181, 17713, 3323, 1, 464621, 1543, 467353, 234361, 36161, 1, 2083, 1867, 475597, 3359, 1, 239873, 481133, 241261, 7933, 1, 2719, 18773, 25763, 2383, 1553, 7963, 495113, 1, 1, 4093, 3943, 1, 3623, 252509, 1, 1, 509293, 255361, 1, 1, 1, 13591, 1, 2677, 2179, 8423, 523673, 1, 526573, 264013, 1, 3739, 1483, 20533, 5003, 1, 1, 1, 541193, 271333, 2203, 272809, 547097, 2663, 550061, 275773, 2239, 277261, 1, 278753, 559001, 280249, 561997, 21673, 2251, 1, 1, 1747, 1, 1, 574061, 287789, 577097, 289309, 8171, 15307, 1, 9431, 5479, 293893, 9661, 295429, 45569, 296969, 595481, 298513, 1, 1571, 31667, 23201, 604781, 4153, 2609, 304729, 32159, 23561, 614153, 9931, 617293, 309433, 620441, 311009, 47969, 312589, 626761, 314173, 1, 16619, 1, 317353, 636301, 318949, 9007, 16871, 642701, 24781, 6271, 1, 1, 1, 652361, 25153, 655597, 1, 658841, 330233, 34847, 1, 1, 333493, 1, 2411, 35363, 336769, 1, 338413, 1709, 340061, 681773, 1, 685081, 1, 688397, 3557, 691721, 1, 4603, 1, 698393, 350033, 701741, 1, 705097, 353389, 54497, 355073, 6911, 356761, 2441, 1, 1, 2591, 721997, 1, 1, 1723, 728813, 28097, 732233, 366973, 1, 368689, 1, 28493, 742541, 1889, 745993, 373861, 749453, 375593, 57917, 377329, 1, 1, 759881, 3559, 4517, 2347, 766873, 1, 24851, 6329, 773897, 29833, 777421, 1, 780953, 1, 1787, 30241, 788041, 12739, 1, 2293, 795161, 398473, 61441, 400261, 42227, 402053, 805901, 403849, 1, 1, 1, 1, 11503, 409261, 820333, 1, 823961, 1, 4201, 414709, 2837, 1, 834893, 1, 838553, 420193, 5167, 422029, 2099, 423869, 849581, 425713, 2459, 5857, 65921, 429413, 1, 431269, 2063, 3833, 868121, 33461, 45887, 1, 875593, 438733, 879341, 33893, 1, 442489, 12491, 3499, 4663, 446261, 1, 1, 2309, 1, 901997, 1, 69677, 23887, 909613, 455761, 913433, 457673, 6599, 35353, 2297, 1931, 7283, 463433, 928793, 35797, 1, 467293, 936521, 469229, 1, 15199, 3823, 6481, 948173, 6691, 952073, 477013, 1, 1, 8971, 480929, 31091, 1, 967753, 1, 971693, 486833, 7019, 2927, 8669, 1987, 1, 1, 987533, 494761, 1, 1, 1, 498749, 2003, 8209, 32371, 1, 4079, 2011, 1, 506773, 1015561, 508789, 1, 39293, 1, 1, 1027693, 3083, 1031753, 39761,

6. Sequence of the polynom (only primes)

179, 71, 103, 31, 19, 13, 73, 61, 173, 113, 281, 397, 229, 521, 293, 653, 433, 941, 509, 1097, 97, 673, 1433, 761, 1613, 853, 1801, 1997, 1049, 1153, 127, 2633, 1373, 2861, 1489, 163, 1609, 257, 1733, 3593, 1861, 3853, 1993, 317, 2129, 4397, 2269, 151, 4973, 197, 5273, 2713, 5581, 5897, 233, 6221, 6553, 3361, 3533, 557, 3709, 107, 3889, 419, 4073, 641, 4261, 8713, 479, 4649, 9497, 373, 9901, 10313, 5261, 10733, 421, 11161, 5689, 11597, 311, 12041, 6133, 6361, 12953, 347, 13421, 6829, 1069, 7069, 139, 7561, 15373, 601, 15881, 8069, 863, 8329, 16921, 661, 563, 8861, 947, 9133, 18541, 9689, 19661, 9973, 20233, 331, 1601, 21401, 571, 21997, 11149, 881, 167, 619, 23833, 12073, 401, 953, 25097, 25741, 13033, 26393, 431, 2081, 13693, 1459, 14029, 389, 14369, 2237, 14713, 1567, 15061, 983, 15413, 31181, 1213, 191, 32621, 16493, 33353, 1297, 907, 34841, 17609, 35597, 17989, 2797, 967, 523, 1223, 19949, 211, 20353, 41113, 1597, 2207, 683, 701, 21589, 43597, 1693, 2339, 22433, 45293, 22861, 46153, 23293, 3617, 227, 24169, 48781, 3821, 1319, 491, 823, 51481, 25969, 751, 26893, 239, 27361, 569, 2141, 1811, 28309, 57097, 28789, 58061, 29761, 60013, 30253, 61001, 251, 31249, 64013, 32261, 65033, 2521, 2131, 33289, 33809, 68141, 69193, 35393, 977, 5569, 36469, 37013, 74573, 37561, 5821, 38113, 76781, 38669, 1277, 39229, 4159, 3061, 80153, 40361, 81293, 40933, 4339, 83597, 42089, 84761, 307, 85933, 43261, 6701, 43853, 88301, 44449, 2887, 2371, 6977, 643, 859, 46261, 93133, 2467, 743, 95597, 48109, 857, 48733, 3797, 1361, 49993, 100621, 7937, 51913, 104473, 52561, 8237, 1777, 1759, 109721, 111053, 4297, 1583, 56533, 1063, 3011, 116461, 58573, 117833, 3119, 1229, 359, 9277, 60649, 121997, 1979, 123401, 62053, 9601, 62761, 126233, 63473, 6719, 64189, 129097, 4993, 4211, 65633, 6947, 66361, 1181, 134921, 67829, 136397, 797, 69313, 70061, 787, 3727, 142381, 71569, 11069, 4691, 3847, 146953, 911, 5741, 150041, 1033, 151597, 1249, 1487, 154733, 77761, 78553, 157901, 79349, 12269, 80149, 80953, 162713, 81761, 12641, 1163, 165961, 83389, 167597, 169241, 2341, 4519, 172553, 86693, 174221, 6733, 175897, 4651, 89213, 5783, 13921, 182681, 1901, 439, 93493, 9887, 189593, 95233, 191341, 7393, 10163, 96989, 194861, 1009, 6343, 198413, 200201, 100549, 201997, 101449, 5387, 1619, 3331, 207433, 104173, 16097, 5531, 211097, 467, 2917, 1753, 8297, 216653, 108793, 218521, 8513, 11699, 3631, 113513, 1013, 229897, 115429, 231821, 487, 17981, 117361, 7603, 1621, 1423, 239597, 241561, 243533, 1187, 245513, 499, 247501, 124249, 249497, 251501, 503, 19501, 127261, 8243, 128273, 257561, 129289, 1051, 261641, 2153, 1579, 132361, 2503, 269897, 135469, 271981, 10501, 4493, 276173, 1429, 7351, 21569, 140729, 1871, 141793, 1697, 4643, 2557, 145009, 3001, 293261, 11321, 295433, 297613, 301997, 151549, 2843, 16127, 23741, 154873, 310861, 4289, 157109, 158233, 317593, 159361, 8447, 10391, 12433, 324397, 1171, 326681, 8627, 5393, 12697, 1289, 333581, 5399, 335897, 1327, 26017, 169693, 1783, 2801, 18047, 2423, 26557, 173209, 1237, 174389, 175573, 352333, 13597, 177953, 3467, 5779, 13873, 361901, 181553, 364313, 9619, 366733, 183973, 185189, 11987, 9811, 374041, 187633, 28961, 188861, 378953, 190093, 1637, 5407, 14813, 386381, 193813, 1823, 391373, 15101, 1283, 197569, 198829, 398921, 2741, 30881, 3301, 403993, 2089, 3947, 203909, 31469, 6619, 1951, 10867, 6833, 419401, 11071, 421997, 1873, 424601, 16381, 427213, 1877, 215573, 6091, 2027, 33469, 7039, 23039, 219533, 440393, 220861, 222193, 23459, 223529, 448397, 224869, 14551, 17401, 1999, 227561, 3023, 228913, 459181, 17713, 3323, 464621, 1543, 467353, 234361, 36161, 2083, 1867, 475597, 3359, 239873, 481133, 241261, 7933, 2719, 18773, 25763, 2383, 1553, 7963, 495113, 4093, 3943, 3623, 252509, 509293, 255361, 13591, 2677, 2179, 8423, 523673, 526573, 264013, 3739, 1483, 20533, 5003, 541193, 271333, 2203, 272809, 547097, 2663, 550061, 275773, 2239, 277261, 278753, 559001, 280249, 561997, 21673, 2251, 1747, 574061, 287789, 577097, 289309, 8171, 15307, 9431, 5479, 293893, 9661, 295429, 45569, 296969, 595481, 298513, 1571, 31667, 23201, 604781, 4153, 2609, 304729, 32159, 23561, 614153, 9931, 617293, 309433, 620441, 311009, 47969, 312589, 626761, 314173, 16619, 317353, 636301, 318949, 9007, 16871, 642701, 24781, 6271, 652361, 25153, 655597, 658841, 330233, 34847, 333493, 2411, 35363, 336769, 338413, 1709, 340061, 681773, 685081, 688397, 3557, 691721, 4603, 698393, 350033, 701741, 705097, 353389, 54497, 355073, 6911, 356761, 2441, 2591, 721997, 1723, 728813, 28097, 732233, 366973, 368689, 28493, 742541, 1889, 745993, 373861, 749453, 375593, 57917, 377329, 759881, 3559, 4517, 2347, 766873, 24851, 6329, 773897, 29833, 777421, 780953, 1787, 30241, 788041, 12739, 2293, 795161, 398473, 61441, 400261, 42227, 402053, 805901, 403849, 11503, 409261, 820333, 823961, 4201, 414709, 2837, 834893, 838553, 420193, 5167, 422029, 2099, 423869, 849581, 425713, 2459, 5857, 65921, 429413, 431269, 2063, 3833, 868121, 33461, 45887, 875593, 438733, 879341, 33893, 442489, 12491, 3499, 4663, 446261, 2309, 901997, 69677, 23887, 909613, 455761, 913433, 457673, 6599, 35353, 2297, 1931, 7283, 463433, 928793, 35797, 467293, 936521, 469229, 15199, 3823, 6481, 948173, 6691, 952073, 477013, 8971, 480929, 31091, 967753, 971693, 486833, 7019, 2927, 8669, 1987, 987533, 494761, 498749, 2003, 8209, 32371, 4079, 2011, 506773, 1015561, 508789, 39293, 1027693, 3083, 1031753, 39761,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+36x-179 and
the reducible primes which appear as divisor for the first time
p | x^2+36x-179 and p < x^2+36x-179

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)
110111101.1000001.1000000.0000000.0000000.0000000.000000
21008670160.8600000.7000000.1600007.8181826.363636inf
31.0007874493380.7870000.4490000.3380009.1511636.41428621.125000
410.0007.6783.1434.5350.7678000.3143000.4535009.7560367.00000013.417160
5100.00075.01424.44550.5690.7501400.2444500.5056909.7699927.77760111.150826
61.000.000739.727199.025540.7020.7397270.1990250.5407029.8611868.14174710.692361
710.000.0007.327.8171.681.4335.646.3840.7327820.1681430.5646389.9061108.44835110.442691
8100.000.00072.764.08914.542.99758.221.0920.7276410.1454300.5822119.9298458.64916810.311217
91.000.000.000723.636.077128.188.037595.448.0400.7236360.1281880.5954489.9449638.81441710.227360
1010.000.000.0007.204.570.7411.146.097.8816.058.472.8600.7204570.1146100.6058479.9560698.94075610.174645


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)
123301.5000001.5000000.0000000.0000000.0000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
389901.1250001.1250000.0000001.8000001.800000-nan
416161601.0000001.0000000.0000001.7777781.777778-nan
532282710.8750000.8437500.0312501.7500001.687500inf
664564880.8750000.7500000.1250002.0000001.7777788.000000
712810985240.8515620.6640620.1875001.9464291.7708333.000000
8256206144620.8046880.5625000.2421881.8899081.6941182.583333
95124072571500.7949220.5019530.2929691.9757281.7847222.419355
101.0248074603470.7880860.4492190.3388671.9828011.7898832.313333
112.0481.6067968100.7841800.3886720.3955081.9900871.7304352.334294
124.0963.1871.4451.7420.7780760.3527830.4252931.9844331.8153272.150617
138.1926.3062.6523.6540.7697750.3237300.4460451.9786631.8352942.097589
1416.38412.4774.8707.6070.7615360.2972410.4642941.9785921.8363502.081828
1532.76824.7779.04615.7310.7561340.2760620.4800721.9858141.8574952.067964
1665.53649.27216.71132.5610.7518310.2549900.4968411.9886181.8473362.069862
17131.07298.09431.20566.8890.7483980.2380750.5103231.9908671.8673332.054267
18262.144195.35158.586136.7650.7452050.2234880.5217171.9914671.8774562.044656
19524.288389.121110.090279.0310.7421890.2099800.5322091.9919071.8791182.040222
201.048.576775.425208.000567.4250.7395030.1983640.5411391.9927611.8893632.033556
212.097.1521.545.961394.5551.151.4060.7371720.1881380.5490331.9936951.8968992.029177
224.194.3043.083.271749.2172.334.0540.7351090.1786270.5564821.9944041.8988912.027134
238.388.6086.151.2971.427.2154.724.0820.7332920.1701370.5631541.9950561.9049422.023981
2416.777.21612.273.4962.724.1559.549.3410.7315570.1623720.5691851.9952701.9087212.021417
2533.554.43224.491.9235.212.83619.279.0870.7299160.1553550.5745621.9955131.9135612.018892
2667.108.86448.886.2519.991.24838.895.0030.7284620.1488810.5795811.9960151.9166632.017471
27134.217.72897.588.09119.187.87778.400.2140.7270880.1429610.5841271.9962281.9204682.015689
28268.435.456194.832.78936.908.152157.924.6370.7258090.1374940.5883151.9964811.9235142.014339
29536.870.912389.032.42771.100.029317.932.3980.7246290.1324340.5921951.9967501.9264052.013191
301.073.741.824776.885.338137.137.967639.747.3710.7235310.1277200.5958111.9969681.9288032.012212
312.147.483.6481.551.567.508264.858.3001.286.709.2080.7225050.1233340.5991711.9971641.9313272.011277
324.294.967.2963.099.010.090512.157.3342.586.852.7560.7215450.1192460.6022991.9973411.9337032.010441
338.589.934.5926.190.299.444991.431.7225.198.867.7220.7206460.1154180.6052281.9975091.9357952.009727
3417.179.869.18412.366.110.9831.921.158.77610.444.952.2070.7198020.1118260.6079761.9976591.9377622.009082


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
123120102
245320203
389631233
41616884273
532271215102123
664482127212223
7128854243402403
82561447668682713
951225713412312521273
101.02446023522522622293
112.04879639440239223993
124.0961.44572971671327273
138.1922.6521.3271.3251.32521.3223
1416.3844.8702.4352.4352.43222.4333
1532.7689.0464.5194.5274.51524.5263
1665.53616.7118.4108.3018.32428.3823
17131.07231.20515.71015.49515.618215.5823
18262.14458.58629.44429.14229.330229.2513
19524.288110.09055.29954.79154.956255.1293
201.048.576208.000104.374103.626103.8392104.1563
212.097.152394.555197.820196.735197.1272197.4233
224.194.304749.217375.773373.444374.3332374.8793
238.388.6081.427.215716.127711.088712.8522714.3583
2416.777.2162.724.1551.366.5141.357.6411.360.96321.363.1873
2533.554.4325.212.8362.614.9082.597.9282.605.50122.607.3303
2667.108.8649.991.2485.011.4494.979.7994.995.51324.995.7303
27134.217.72819.187.8779.624.5899.563.2889.594.13629.593.7363
28268.435.45636.908.15218.510.33818.397.81418.455.114218.453.0333
29536.870.91271.100.02935.656.78035.443.24935.549.667235.550.3573
301.073.741.824137.137.96768.772.69568.365.27268.577.481268.560.4813
312.147.483.648264.858.300132.801.782132.056.518132.441.0132132.417.2823
324.294.967.296512.157.334256.778.513255.378.821256.092.3942256.064.9353
338.589.934.592991.431.722497.032.254494.399.468495.743.6592495.688.0583
3417.179.869.1841.921.158.776963.067.821958.090.955960.592.4732960.566.2983


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
120000000
240000000
380000000
4160000000
5321101000
6648443122
71282410146675
825662273516161614
9512150708036404034
101.024347161186831007886
112.048810398412177231187215
124.0961.742889853391486413452
138.1923.6541.8561.798851987849967
1416.3847.6073.7873.8201.8222.0121.7712.002
1532.76815.7317.8407.8913.7374.1133.7104.171
1665.53632.56116.28716.2747.7698.4497.7628.581
17131.07266.88933.37433.51515.97817.36716.08017.464
18262.144136.76568.43668.32932.96335.33132.95435.517
19524.288279.031139.741139.29067.31971.69267.74672.274
201.048.576567.425284.288283.137137.251145.824137.946146.404
212.097.1521.151.406576.498574.908279.713295.309280.362296.022
224.194.3042.334.0541.167.9651.166.089568.643597.653569.458598.300
238.388.6084.724.0822.362.7972.361.2851.153.2191.208.3751.154.0181.208.470
2416.777.2169.549.3414.775.2364.774.1052.335.9022.438.3812.335.9432.439.115
2533.554.43219.279.0879.641.8169.637.2714.720.8694.917.8214.723.4854.916.912
2667.108.86438.895.00319.452.58519.442.4189.535.6719.907.9319.541.9539.909.448
27134.217.72878.400.21439.211.42139.188.79319.240.26319.956.36919.252.17419.951.408
28268.435.456157.924.63778.981.71878.942.91938.799.83940.159.40038.810.42840.154.970
29536.870.912317.932.398159.007.716158.924.68278.181.22880.778.19078.200.95280.772.028
301.073.741.824639.747.371319.952.686319.794.685157.453.559162.403.579157.478.752162.411.481
312.147.483.6481.286.709.208643.503.217643.205.991316.934.009326.409.903316.945.021326.420.275
324.294.967.2962.586.852.7561.293.692.5941.293.160.162637.598.548655.817.815637.617.147655.819.246
338.589.934.5925.198.867.7222.599.946.7692.598.920.9531.282.142.9351.317.234.6821.282.229.9041.317.260.201
3417.179.869.18410.444.952.2075.223.470.2765.221.481.9312.577.456.7632.645.021.0932.577.470.0362.645.004.315


8. Check for existing Integer Sequences by OEIS

Found in Database : 179, 71, 103, 31, 19, 13, 73, 61, 173, 113, 281, 1, 397, 229, 521, 293, 653, 1, 1, 433,
Found in Database : 179, 71, 103, 31, 19, 13, 73, 61, 173, 113, 281, 397, 229, 521, 293, 653, 433, 941, 509, 1097, 97, 673, 1433, 761, 1613, 853, 1801, 1997, 1049, 1153, 127, 2633, 1373,
Found in Database : 13, 19, 31, 61, 71, 73, 97, 103, 107, 113, 127, 139,