Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:41:00
Deutsch
28.Mar 2024

Polynom = x^2-92x+13

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) = 13 = 13
f(1) = 39 = 3*13
f(2) = 167 = 167
f(3) = 127 = 127
f(4) = 339 = 3*113
f(5) = 211 = 211
f(6) = 503 = 503
f(7) = 291 = 3*97
f(8) = 659 = 659
f(9) = 367 = 367
f(10) = 807 = 3*269
f(11) = 439 = 439
f(12) = 947 = 947
f(13) = 507 = 3*13*13
f(14) = 1079 = 13*83
f(15) = 571 = 571
f(16) = 1203 = 3*401
f(17) = 631 = 631
f(18) = 1319 = 1319
f(19) = 687 = 3*229
f(20) = 1427 = 1427
f(21) = 739 = 739
f(22) = 1527 = 3*509
f(23) = 787 = 787
f(24) = 1619 = 1619
f(25) = 831 = 3*277
f(26) = 1703 = 13*131
f(27) = 871 = 13*67
f(28) = 1779 = 3*593
f(29) = 907 = 907
f(30) = 1847 = 1847
f(31) = 939 = 3*313
f(32) = 1907 = 1907
f(33) = 967 = 967
f(34) = 1959 = 3*653
f(35) = 991 = 991
f(36) = 2003 = 2003
f(37) = 1011 = 3*337
f(38) = 2039 = 2039
f(39) = 1027 = 13*79
f(40) = 2067 = 3*13*53
f(41) = 1039 = 1039
f(42) = 2087 = 2087
f(43) = 1047 = 3*349
f(44) = 2099 = 2099
f(45) = 1051 = 1051
f(46) = 2103 = 3*701
f(47) = 1051 = 1051
f(48) = 2099 = 2099
f(49) = 1047 = 3*349
f(50) = 2087 = 2087
f(51) = 1039 = 1039
f(52) = 2067 = 3*13*53
f(53) = 1027 = 13*79
f(54) = 2039 = 2039
f(55) = 1011 = 3*337
f(56) = 2003 = 2003
f(57) = 991 = 991
f(58) = 1959 = 3*653
f(59) = 967 = 967
f(60) = 1907 = 1907
f(61) = 939 = 3*313
f(62) = 1847 = 1847
f(63) = 907 = 907
f(64) = 1779 = 3*593
f(65) = 871 = 13*67
f(66) = 1703 = 13*131
f(67) = 831 = 3*277
f(68) = 1619 = 1619
f(69) = 787 = 787
f(70) = 1527 = 3*509
f(71) = 739 = 739
f(72) = 1427 = 1427
f(73) = 687 = 3*229
f(74) = 1319 = 1319
f(75) = 631 = 631
f(76) = 1203 = 3*401
f(77) = 571 = 571
f(78) = 1079 = 13*83
f(79) = 507 = 3*13*13
f(80) = 947 = 947
f(81) = 439 = 439
f(82) = 807 = 3*269
f(83) = 367 = 367
f(84) = 659 = 659
f(85) = 291 = 3*97
f(86) = 503 = 503
f(87) = 211 = 211
f(88) = 339 = 3*113
f(89) = 127 = 127
f(90) = 167 = 167
f(91) = 39 = 3*13
f(92) = 13 = 13
f(93) = 53 = 53
f(94) = 201 = 3*67
f(95) = 149 = 149
f(96) = 397 = 397
f(97) = 249 = 3*83
f(98) = 601 = 601
f(99) = 353 = 353
f(100) = 813 = 3*271

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^2-92x+13

f(0)=13
f(1)=3
f(2)=167
f(3)=127
f(4)=113
f(5)=211
f(6)=503
f(7)=97
f(8)=659
f(9)=367
f(10)=269
f(11)=439
f(12)=947
f(13)=1
f(14)=83
f(15)=571
f(16)=401
f(17)=631
f(18)=1319
f(19)=229
f(20)=1427
f(21)=739
f(22)=509
f(23)=787
f(24)=1619
f(25)=277
f(26)=131
f(27)=67
f(28)=593
f(29)=907
f(30)=1847
f(31)=313
f(32)=1907
f(33)=967
f(34)=653
f(35)=991
f(36)=2003
f(37)=337
f(38)=2039
f(39)=79
f(40)=53
f(41)=1039
f(42)=2087
f(43)=349
f(44)=2099
f(45)=1051
f(46)=701
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)=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)=149
f(96)=397
f(97)=1
f(98)=601
f(99)=353

b) Substitution of the polynom
The polynom f(x)=x^2-92x+13 could be written as f(y)= y^2-2103 with x=y+46

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-46
f'(x)>2x-93 with x > 46

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

13, 3, 167, 127, 113, 211, 503, 97, 659, 367, 269, 439, 947, 1, 83, 571, 401, 631, 1319, 229, 1427, 739, 509, 787, 1619, 277, 131, 67, 593, 907, 1847, 313, 1907, 967, 653, 991, 2003, 337, 2039, 79, 53, 1039, 2087, 349, 2099, 1051, 701, 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, 149, 397, 1, 601, 353, 271, 461, 1033, 191, 1, 1, 499, 809, 1741, 311, 1993, 1061, 751, 1193, 2521, 443, 2797, 1, 1, 1613, 3373, 587, 3673, 1913, 1327, 2069, 4297, 743, 4621, 2393, 1, 197, 1, 911, 5641, 2909, 1999, 3089, 6361, 1091, 6733, 3461, 2371, 281, 577, 1283, 1, 4049, 2767, 4253, 8713, 1487, 9133, 4673, 3187, 4889, 769, 1, 1, 5333, 3631, 1, 11353, 1931, 11821, 6029, 4099, 6269, 12781, 1, 1021, 6761, 4591, 7013, 14281, 2423, 14797, 7529, 5107, 7793, 1, 2687, 1, 641, 5647, 8609, 17497, 2963, 18061, 9173, 6211, 9461, 19213, 3251, 19801, 773, 523, 1, 21001, 1, 21613, 1, 7411, 11273, 22861, 3863, 23497, 11909, 619, 941, 24793, 1, 25453, 12893, 8707, 13229, 1, 4523, 27481, 13913, 9391, 1097, 2221, 4871, 29581, 14969, 10099, 15329, 463, 5231, 1, 16061, 10831, 16433, 2557, 431, 33997, 17189, 11587, 17573, 35533, 5987, 36313, 18353, 1, 18749, 37897, 491, 1, 19553, 13171, 19961, 761, 6791, 41161, 20789, 13999, 1, 42841, 7211, 3361, 1697, 14851, 1, 857, 7643, 691, 23369, 15727, 23813, 48073, 8087, 48973, 1901, 1279, 25169, 643, 8543, 51721, 1, 17551, 26561, 53593, 9011, 54541, 27509, 1423, 2153, 1, 9491, 727, 28961, 19471, 29453, 1, 1, 60397, 30449, 1, 2381, 4801, 10487, 1, 31973, 21487, 613, 65497, 11003, 66541, 33533, 22531, 34061, 5281, 887, 617, 35129, 23599, 673, 71881, 12071, 72973, 36761, 24691, 37313, 75181, 971, 5869, 1, 1, 38993, 78553, 13187, 79693, 599, 26947, 40709, 1, 13763, 6397, 3221, 28111, 42461, 85513, 1, 1, 43649, 1, 44249, 89101, 14951, 90313, 1, 2347, 46073, 92761, 1, 93997, 47309, 1, 47933, 96493, 16187, 1459, 49193, 2539, 3833, 100297, 16823, 101581, 647, 1, 1, 104173, 17471, 1, 53069, 1, 4133, 8317, 18131, 109453, 55061, 36931, 55733, 883, 18803, 113497, 57089, 38287, 57773, 8941, 1499, 1, 1, 39667, 1, 120397, 20183, 1, 61253, 1, 61961, 124633, 1607, 9697, 63389, 42499, 64109, 128941, 21611, 1571, 1237, 43951, 66293, 133321, 22343, 10369, 1, 45427, 1, 1, 23087, 139273, 1321, 1, 70769, 142297, 1, 143821, 1, 3727, 73061, 146893, 24611, 1879, 74609, 49999, 75389, 151561, 25391, 153133, 76961, 3967, 5981, 1049, 26183, 157897, 79349, 1, 80153, 1, 26987, 1, 81773, 54787, 6353, 1, 27803, 167641, 1, 56431, 85061, 170953, 28631, 3257, 86729, 58099, 1307, 13537, 2267, 1399, 89261, 59791, 929, 181081, 30323, 3449, 1, 61507, 92693, 186253, 2399, 14461, 94433, 63247, 1, 191497, 32063, 193261, 97073, 65011, 97961, 2029, 1, 15277, 7673, 997, 100649, 202201, 33851, 204013, 102461, 68611, 1, 207661, 34763, 209497, 8093, 5419, 1, 213193, 1, 2591, 1367, 72307, 108929, 218797, 1, 220681, 110813, 1, 8597, 224473, 37571, 226381, 1, 76099, 1451, 230221, 1, 232153, 1, 78031, 9041, 1, 39503, 237997, 119489, 79987, 120473, 2141, 40487, 1637, 122453, 81967, 123449, 19069, 3191, 1, 125453, 1, 126461, 1289, 42491, 255961, 128489, 85999, 129509, 1, 3347, 20161, 131561, 1, 1979, 266221, 44543, 268297, 134669, 90127, 1201, 1, 45587, 21121, 10601, 92227, 138869, 3359, 46643, 280921, 1, 94351, 142061, 5381, 47711, 287341, 11093, 1, 145289, 291661, 1, 2243, 1, 1249, 1, 298201, 1, 300397, 1187, 7759, 11681, 304813, 1, 307033, 154073, 103087, 1, 3943, 52103, 313741, 157433, 105331, 12197, 24481, 53231, 320521, 160829, 107599, 161969, 325081, 54371, 1217, 1, 109891, 3121, 25537, 4271, 334297, 167729, 112207, 168893, 5059, 56687, 341293, 171233, 114547, 3253, 345997, 4451, 1, 174773, 116911, 175961, 1303, 59051, 355501, 178349, 119299, 179549, 1, 60251, 27901, 13997, 121711, 1621, 367561, 61463, 369997, 1, 124147, 2251, 374893, 62687, 377353, 14561, 9739, 190529, 1, 1, 5743, 193013, 1, 2459, 389773, 65171, 392281, 196769, 1, 15233, 4787, 66431, 399853, 1531, 1, 201833, 404941, 1, 4201, 1, 2579, 1, 31741, 1, 415213, 208253, 139267, 209549, 420397, 1, 423001, 1, 141871, 213461, 32941, 5507, 430861, 1, 1, 217409, 1, 72911, 438793, 1, 147151, 221393, 444121, 5711, 34369, 224069, 149827, 1, 2143, 1, 2309, 228113, 1, 229469, 460297, 76943, 35617, 1, 155251, 1, 468493, 78311, 471241, 3527, 157999, 237689, 1, 79691, 479533, 1, 1, 1, 485101, 81083, 487897, 1, 163567, 1, 6247, 82487, 496333, 1, 12799, 1481, 501997, 83903, 3023, 253133, 1, 3067, 510553, 85331, 1, 2027, 1523, 19913, 39937, 86771, 522073, 261761, 174991, 263213, 527881, 88223, 530797, 266129, 177907, 267593, 41281, 6899, 10181, 2789, 180847, 272009, 545497, 91163, 548461, 274973, 1, 276461, 1973, 7127, 1, 2473, 186799, 280949, 563401, 94151, 566413, 283961, 2833, 285473, 572461, 1, 44269, 22193, 192847, 290033, 1, 97187, 4463, 293093, 195907, 294629, 590797, 1, 1, 22901, 15307, 299261, 600073, 100271, 1, 3643, 202099, 1, 609421, 1, 612553, 4583, 15787, 23741, 618841, 1951, 621997, 1867, 208387, 313373, 4217, 104987, 631513, 4007, 2549, 24473, 49069, 2011, 641101, 1, 214771, 2543, 4943, 108191, 650761, 326189, 1, 327809, 3889, 8447, 660493, 3413, 221251, 1, 667021, 111443, 670297, 1759, 224527, 5039, 1, 8699, 52321, 340913, 227827, 342569, 686797, 114743, 1721, 3061, 2383, 347561, 696793, 116411, 53857, 26993, 234499, 352589, 13337, 1, 1789, 2389, 1873, 357653, 717001, 119783, 720397, 27773, 1, 362753, 13721, 121487, 730633, 366173, 244687, 367889, 737497, 123203, 1, 371333, 19087, 28697, 747853, 1, 751321, 376529, 1, 378269, 1, 1, 761773, 381761, 3229, 29501, 4549, 1, 772297, 4663, 258607, 388793, 779353, 1, 1, 392333, 262147, 1, 1, 10151, 793561, 397673, 265711, 7537, 1, 1, 4211, 403049, 1, 404849, 811501, 10427, 62701, 408461, 272911, 7741, 822361, 137363, 825997, 1, 2851, 1, 833293, 139187, 64381, 32261, 280207, 6287, 844297, 141023, 1, 424913, 1, 426761, 855373, 142871, 859081, 33113, 22123, 4457, 6823, 144731, 870253, 436061, 291331, 3343, 1, 146603, 3137, 5591, 22699, 1, 1, 1, 7901, 447353, 5639, 449249, 900397, 150383, 904201, 453053,

6. Sequence of the polynom (only primes)

13, 3, 167, 127, 113, 211, 503, 97, 659, 367, 269, 439, 947, 83, 571, 401, 631, 1319, 229, 1427, 739, 509, 787, 1619, 277, 131, 67, 593, 907, 1847, 313, 1907, 967, 653, 991, 2003, 337, 2039, 79, 53, 1039, 2087, 349, 2099, 1051, 701, 149, 397, 601, 353, 271, 461, 1033, 191, 499, 809, 1741, 311, 1993, 1061, 751, 1193, 2521, 443, 2797, 1613, 3373, 587, 3673, 1913, 1327, 2069, 4297, 743, 4621, 2393, 197, 911, 5641, 2909, 1999, 3089, 6361, 1091, 6733, 3461, 2371, 281, 577, 1283, 4049, 2767, 4253, 8713, 1487, 9133, 4673, 3187, 4889, 769, 5333, 3631, 11353, 1931, 11821, 6029, 4099, 6269, 12781, 1021, 6761, 4591, 7013, 14281, 2423, 14797, 7529, 5107, 7793, 2687, 641, 5647, 8609, 17497, 2963, 18061, 9173, 6211, 9461, 19213, 3251, 19801, 773, 523, 21001, 21613, 7411, 11273, 22861, 3863, 23497, 11909, 619, 941, 24793, 25453, 12893, 8707, 13229, 4523, 27481, 13913, 9391, 1097, 2221, 4871, 29581, 14969, 10099, 15329, 463, 5231, 16061, 10831, 16433, 2557, 431, 33997, 17189, 11587, 17573, 35533, 5987, 36313, 18353, 18749, 37897, 491, 19553, 13171, 19961, 761, 6791, 41161, 20789, 13999, 42841, 7211, 3361, 1697, 14851, 857, 7643, 691, 23369, 15727, 23813, 48073, 8087, 48973, 1901, 1279, 25169, 643, 8543, 51721, 17551, 26561, 53593, 9011, 54541, 27509, 1423, 2153, 9491, 727, 28961, 19471, 29453, 60397, 30449, 2381, 4801, 10487, 31973, 21487, 613, 65497, 11003, 66541, 33533, 22531, 34061, 5281, 887, 617, 35129, 23599, 673, 71881, 12071, 72973, 36761, 24691, 37313, 75181, 971, 5869, 38993, 78553, 13187, 79693, 599, 26947, 40709, 13763, 6397, 3221, 28111, 42461, 85513, 43649, 44249, 89101, 14951, 90313, 2347, 46073, 92761, 93997, 47309, 47933, 96493, 16187, 1459, 49193, 2539, 3833, 100297, 16823, 101581, 647, 104173, 17471, 53069, 4133, 8317, 18131, 109453, 55061, 36931, 55733, 883, 18803, 113497, 57089, 38287, 57773, 8941, 1499, 39667, 120397, 20183, 61253, 61961, 124633, 1607, 9697, 63389, 42499, 64109, 128941, 21611, 1571, 1237, 43951, 66293, 133321, 22343, 10369, 45427, 23087, 139273, 1321, 70769, 142297, 143821, 3727, 73061, 146893, 24611, 1879, 74609, 49999, 75389, 151561, 25391, 153133, 76961, 3967, 5981, 1049, 26183, 157897, 79349, 80153, 26987, 81773, 54787, 6353, 27803, 167641, 56431, 85061, 170953, 28631, 3257, 86729, 58099, 1307, 13537, 2267, 1399, 89261, 59791, 929, 181081, 30323, 3449, 61507, 92693, 186253, 2399, 14461, 94433, 63247, 191497, 32063, 193261, 97073, 65011, 97961, 2029, 15277, 7673, 997, 100649, 202201, 33851, 204013, 102461, 68611, 207661, 34763, 209497, 8093, 5419, 213193, 2591, 1367, 72307, 108929, 218797, 220681, 110813, 8597, 224473, 37571, 226381, 76099, 1451, 230221, 232153, 78031, 9041, 39503, 237997, 119489, 79987, 120473, 2141, 40487, 1637, 122453, 81967, 123449, 19069, 3191, 125453, 126461, 1289, 42491, 255961, 128489, 85999, 129509, 3347, 20161, 131561, 1979, 266221, 44543, 268297, 134669, 90127, 1201, 45587, 21121, 10601, 92227, 138869, 3359, 46643, 280921, 94351, 142061, 5381, 47711, 287341, 11093, 145289, 291661, 2243, 1249, 298201, 300397, 1187, 7759, 11681, 304813, 307033, 154073, 103087, 3943, 52103, 313741, 157433, 105331, 12197, 24481, 53231, 320521, 160829, 107599, 161969, 325081, 54371, 1217, 109891, 3121, 25537, 4271, 334297, 167729, 112207, 168893, 5059, 56687, 341293, 171233, 114547, 3253, 345997, 4451, 174773, 116911, 175961, 1303, 59051, 355501, 178349, 119299, 179549, 60251, 27901, 13997, 121711, 1621, 367561, 61463, 369997, 124147, 2251, 374893, 62687, 377353, 14561, 9739, 190529, 5743, 193013, 2459, 389773, 65171, 392281, 196769, 15233, 4787, 66431, 399853, 1531, 201833, 404941, 4201, 2579, 31741, 415213, 208253, 139267, 209549, 420397, 423001, 141871, 213461, 32941, 5507, 430861, 217409, 72911, 438793, 147151, 221393, 444121, 5711, 34369, 224069, 149827, 2143, 2309, 228113, 229469, 460297, 76943, 35617, 155251, 468493, 78311, 471241, 3527, 157999, 237689, 79691, 479533, 485101, 81083, 487897, 163567, 6247, 82487, 496333, 12799, 1481, 501997, 83903, 3023, 253133, 3067, 510553, 85331, 2027, 1523, 19913, 39937, 86771, 522073, 261761, 174991, 263213, 527881, 88223, 530797, 266129, 177907, 267593, 41281, 6899, 10181, 2789, 180847, 272009, 545497, 91163, 548461, 274973, 276461, 1973, 7127, 2473, 186799, 280949, 563401, 94151, 566413, 283961, 2833, 285473, 572461, 44269, 22193, 192847, 290033, 97187, 4463, 293093, 195907, 294629, 590797, 22901, 15307, 299261, 600073, 100271, 3643, 202099, 609421, 612553, 4583, 15787, 23741, 618841, 1951, 621997, 1867, 208387, 313373, 4217, 104987, 631513, 4007, 2549, 24473, 49069, 2011, 641101, 214771, 2543, 4943, 108191, 650761, 326189, 327809, 3889, 8447, 660493, 3413, 221251, 667021, 111443, 670297, 1759, 224527, 5039, 8699, 52321, 340913, 227827, 342569, 686797, 114743, 1721, 3061, 2383, 347561, 696793, 116411, 53857, 26993, 234499, 352589, 13337, 1789, 2389, 1873, 357653, 717001, 119783, 720397, 27773, 362753, 13721, 121487, 730633, 366173, 244687, 367889, 737497, 123203, 371333, 19087, 28697, 747853, 751321, 376529, 378269, 761773, 381761, 3229, 29501, 4549, 772297, 4663, 258607, 388793, 779353, 392333, 262147, 10151, 793561, 397673, 265711, 7537, 4211, 403049, 404849, 811501, 10427, 62701, 408461, 272911, 7741, 822361, 137363, 825997, 2851, 833293, 139187, 64381, 32261, 280207, 6287, 844297, 141023, 424913, 426761, 855373, 142871, 859081, 33113, 22123, 4457, 6823, 144731, 870253, 436061, 291331, 3343, 146603, 3137, 5591, 22699, 7901, 447353, 5639, 449249, 900397, 150383, 904201, 453053,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-92x+13 and
the reducible primes which appear as divisor for the first time
p | x^2-92x+13 and p < x^2-92x+13

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)
11011471.1000000.4000001.1000000.0000000.0000000.000000
21004916330.4900000.1600000.4900004.4545454.0000004.714286
31.0007821885940.7820000.1880000.78200015.95918411.75000018.000000
410.0008.0631.3646.6990.8063000.1364000.80630010.3107417.25531911.277778
5100.00078.91110.68568.2260.7891100.1068500.7891109.7868047.83357810.184505
61.000.000773.51487.399686.1150.7735140.0873990.7735149.8023608.17959810.056503
710.000.0007.615.460738.5456.876.9150.7615460.0738540.7615469.8452788.45026910.022977
8100.000.00075.256.4686.397.39668.859.0720.7525650.0639740.7525659.8820648.66216210.013076
91.000.000.000745.600.74356.474.361689.126.3820.7456010.0564740.7456019.9074658.82771010.007779
1010.000.000.0007.401.106.382505.396.2756.895.710.1070.7401110.0505400.7401119.9263678.94912810.006452


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
389451.1250000.5000000.6250001.8000002.0000001.666667
416165111.0000000.3125000.6875001.7777781.2500002.200000
5323210221.0000000.3125000.6875002.0000002.0000002.000000
6644514310.7031250.2187500.4843751.4062501.4000001.409091
71287225470.5625000.1953120.3671881.6000001.7857141.516129
8256179521270.6992190.2031250.4960942.4861112.0800002.702128
9512388992890.7578120.1933590.5644532.1675981.9038462.275591
101.0247991916080.7802730.1865230.5937502.0592781.9292932.103806
112.0481.6273491.2780.7944340.1704100.6240232.0362951.8272252.101974
124.0963.2926482.6440.8037110.1582030.6455082.0233561.8567342.068858
138.1926.6091.1505.4590.8067630.1403810.6663822.0075941.7746912.064675
1416.38413.1462.11011.0360.8023680.1287840.6735841.9891061.8347832.021616
1532.76826.1273.92822.1990.7973330.1198730.6774601.9874491.8616112.011508
1665.53651.9417.26744.6740.7925570.1108860.6816711.9880201.8500512.012433
17131.072103.20113.63389.5680.7873610.1040120.6833501.9868891.8760152.004925
18262.144205.06325.663179.4000.7822530.0978970.6843571.9870251.8824182.002948
19524.288407.70548.276359.4290.7776360.0920790.6855561.9881941.8811522.003506
201.048.576810.83791.260719.5770.7732740.0870320.6862421.9887841.8903802.002000
212.097.1521.613.192173.1921.440.0000.7692300.0825840.6866461.9895391.8977862.001176
224.194.3043.211.105329.4092.881.6960.7655870.0785370.6870501.9905291.9019872.001178
238.388.6086.394.723627.4355.767.2880.7623100.0747960.6875141.9914401.9047292.001352
2416.777.21612.739.3031.196.91711.542.3860.7593220.0713420.6879801.9921591.9076352.001354
2533.554.43225.384.8632.292.14923.092.7140.7565280.0683110.6882171.9926421.9150442.000688
2667.108.86450.595.2664.395.71846.199.5480.7539280.0655010.6884271.9931271.9177282.000612
27134.217.728100.873.9458.442.31892.431.6270.7515690.0629000.6886691.9937431.9205782.000704
28268.435.456201.153.12516.250.404184.902.7210.7493540.0605370.6888161.9941041.9248752.000427
29536.870.912401.207.42231.316.749369.890.6730.7473070.0583320.6889751.9945371.9271372.000461
301.073.741.824800.380.88460.420.618739.960.2660.7454130.0562710.6891421.9949301.9293392.000484
312.147.483.6481.596.959.086116.724.6711.480.234.4150.7436420.0543540.6892881.9952491.9318682.000424
324.294.967.2963.186.814.525225.775.9592.961.038.5660.7419880.0525680.6894211.9955521.9342612.000385
338.589.934.5926.360.307.732437.176.5945.923.131.1380.7404370.0508940.6895431.9958201.9363292.000356
3417.179.869.18412.695.643.850847.337.68411.848.306.1660.7389840.0493220.6896621.9960741.9382052.000345


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
384130112
4165140212
53210190514
664141130716
71282512136766
8256523913207196
9512998613457416
101.02419117813897896
112.0483493361316971676
124.0966486351332273136
138.1921.1501.1371356675716
1416.3842.1102.097131.05971.0386
1532.7683.9283.915131.94671.9696
1665.5367.2677.254133.61473.6406
17131.07213.63313.620136.80976.8116
18262.14425.66325.6501312.808712.8426
19524.28848.27648.2631324.029724.2346
201.048.57691.26091.2471345.434745.8136
212.097.152173.192173.1791386.380786.7996
224.194.304329.409329.39613164.4207164.9766
238.388.608627.435627.42213313.2127314.2106
2416.777.2161.196.9171.196.90413597.6657599.2396
2533.554.4322.292.1492.292.136131.145.07471.147.0626
2667.108.8644.395.7184.395.705132.196.67172.199.0346
27134.217.7288.442.3188.442.305134.221.08274.221.2236
28268.435.45616.250.40416.250.391138.125.62378.124.7686
29536.870.91231.316.74931.316.7361315.657.199715.659.5376
301.073.741.82460.420.61860.420.6051330.207.809730.212.7966
312.147.483.648116.724.671116.724.6581358.363.004758.361.6546
324.294.967.296225.775.959225.775.94613112.895.0577112.880.8896
338.589.934.592437.176.594437.176.58113218.591.5807218.585.0016
3417.179.869.184847.337.684847.337.67113423.680.0047423.657.6676


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
121000001
243111002
385312102
41611643314
532221475845
664312196979
712847252110121114
8256127517532313331
951228910818073737568
101.024608220387154149152153
112.0481.278479798321315331311
124.0962.6441.0141.629660637702645
138.1925.4592.1413.3171.3961.3601.3961.307
1416.38411.0364.3916.6442.7862.7312.8282.691
1532.76822.1998.99213.2065.5705.4855.6775.467
1665.53644.67418.44226.23111.24611.03511.36311.030
17131.07289.56837.43652.13122.65222.14422.72922.043
18262.144179.40076.102103.29745.49144.27445.35244.283
19524.288359.429154.319205.10991.28088.51490.87188.764
201.048.576719.577311.902407.674182.031178.007181.882177.657
212.097.1521.440.000628.593811.406363.852356.014364.440355.694
224.194.3042.881.6961.268.4291.613.266728.147712.675728.568712.306
238.388.6085.767.2882.556.8113.210.4761.456.8921.426.5951.457.3641.426.437
2416.777.21611.542.3865.148.0046.394.3812.916.1552.855.3952.915.3342.855.502
2533.554.43223.092.71410.354.49312.738.2205.832.0925.717.6945.830.2145.712.714
2667.108.86446.199.54820.819.89725.379.65011.662.44111.440.27211.661.03811.435.797
27134.217.72892.431.62741.844.38950.587.23723.327.87422.890.70723.326.67322.886.373
28268.435.456184.902.72184.055.913100.846.80746.654.72745.797.95646.648.12045.801.918
29536.870.912369.890.673168.793.654201.097.01893.303.42691.650.77493.295.01391.641.460
301.073.741.824739.960.266338.847.599401.112.666186.603.930183.378.265186.608.118183.369.953
312.147.483.6481.480.234.415680.009.545800.224.869373.174.294366.936.615373.191.332366.932.174
324.294.967.2962.961.038.5661.364.362.4421.596.676.123746.357.688734.141.978746.367.120734.171.780
338.589.934.5925.923.131.1382.736.846.5673.186.284.5701.492.649.1361.468.885.1571.492.636.0781.468.960.767
3417.179.869.18411.848.306.1665.489.060.6646.359.245.5012.985.258.2972.938.857.4082.985.242.1762.938.948.285


8. Check for existing Integer Sequences by OEIS

Found in Database : 13, 3, 167, 127, 113, 211, 503, 97, 659, 367, 269, 439, 947, 1, 83, 571, 401, 631, 1319, 229,
Found in Database : 13, 3, 167, 127, 113, 211, 503, 97, 659, 367, 269, 439, 947, 83, 571, 401, 631, 1319, 229, 1427, 739, 509, 787, 1619, 277, 131, 67, 593, 907, 1847, 313, 1907, 967, 653, 991, 2003, 337, 2039, 79,
Found in Database : 3, 13, 53, 67, 79, 83, 97, 113, 127, 131, 149,