Inhaltsverzeichnis

Development of
Algorithmic Constructions

15:25:42
Deutsch
28.Mar 2024

Polynom = x^2+53x-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) = 19 = 19
f(2) = 37 = 37
f(3) = 95 = 5*19
f(4) = 155 = 5*31
f(5) = 217 = 7*31
f(6) = 281 = 281
f(7) = 347 = 347
f(8) = 415 = 5*83
f(9) = 485 = 5*97
f(10) = 557 = 557
f(11) = 631 = 631
f(12) = 707 = 7*101
f(13) = 785 = 5*157
f(14) = 865 = 5*173
f(15) = 947 = 947
f(16) = 1031 = 1031
f(17) = 1117 = 1117
f(18) = 1205 = 5*241
f(19) = 1295 = 5*7*37
f(20) = 1387 = 19*73
f(21) = 1481 = 1481
f(22) = 1577 = 19*83
f(23) = 1675 = 5*5*67
f(24) = 1775 = 5*5*71
f(25) = 1877 = 1877
f(26) = 1981 = 7*283
f(27) = 2087 = 2087
f(28) = 2195 = 5*439
f(29) = 2305 = 5*461
f(30) = 2417 = 2417
f(31) = 2531 = 2531
f(32) = 2647 = 2647
f(33) = 2765 = 5*7*79
f(34) = 2885 = 5*577
f(35) = 3007 = 31*97
f(36) = 3131 = 31*101
f(37) = 3257 = 3257
f(38) = 3385 = 5*677
f(39) = 3515 = 5*19*37
f(40) = 3647 = 7*521
f(41) = 3781 = 19*199
f(42) = 3917 = 3917
f(43) = 4055 = 5*811
f(44) = 4195 = 5*839
f(45) = 4337 = 4337
f(46) = 4481 = 4481
f(47) = 4627 = 7*661
f(48) = 4775 = 5*5*191
f(49) = 4925 = 5*5*197
f(50) = 5077 = 5077
f(51) = 5231 = 5231
f(52) = 5387 = 5387
f(53) = 5545 = 5*1109
f(54) = 5705 = 5*7*163
f(55) = 5867 = 5867
f(56) = 6031 = 37*163
f(57) = 6197 = 6197
f(58) = 6365 = 5*19*67
f(59) = 6535 = 5*1307
f(60) = 6707 = 19*353
f(61) = 6881 = 7*983
f(62) = 7057 = 7057
f(63) = 7235 = 5*1447
f(64) = 7415 = 5*1483
f(65) = 7597 = 71*107
f(66) = 7781 = 31*251
f(67) = 7967 = 31*257
f(68) = 8155 = 5*7*233
f(69) = 8345 = 5*1669
f(70) = 8537 = 8537
f(71) = 8731 = 8731
f(72) = 8927 = 79*113
f(73) = 9125 = 5*5*5*73
f(74) = 9325 = 5*5*373
f(75) = 9527 = 7*1361
f(76) = 9731 = 37*263
f(77) = 9937 = 19*523
f(78) = 10145 = 5*2029
f(79) = 10355 = 5*19*109
f(80) = 10567 = 10567
f(81) = 10781 = 10781
f(82) = 10997 = 7*1571
f(83) = 11215 = 5*2243
f(84) = 11435 = 5*2287
f(85) = 11657 = 11657
f(86) = 11881 = 109*109
f(87) = 12107 = 12107
f(88) = 12335 = 5*2467
f(89) = 12565 = 5*7*359
f(90) = 12797 = 67*191
f(91) = 13031 = 83*157
f(92) = 13267 = 13267
f(93) = 13505 = 5*37*73
f(94) = 13745 = 5*2749
f(95) = 13987 = 71*197
f(96) = 14231 = 7*19*107
f(97) = 14477 = 31*467
f(98) = 14725 = 5*5*19*31
f(99) = 14975 = 5*5*599
f(100) = 15227 = 15227

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+53x-73

f(0)=73
f(1)=19
f(2)=37
f(3)=5
f(4)=31
f(5)=7
f(6)=281
f(7)=347
f(8)=83
f(9)=97
f(10)=557
f(11)=631
f(12)=101
f(13)=157
f(14)=173
f(15)=947
f(16)=1031
f(17)=1117
f(18)=241
f(19)=1
f(20)=1
f(21)=1481
f(22)=1
f(23)=67
f(24)=71
f(25)=1877
f(26)=283
f(27)=2087
f(28)=439
f(29)=461
f(30)=2417
f(31)=2531
f(32)=2647
f(33)=79
f(34)=577
f(35)=1
f(36)=1
f(37)=3257
f(38)=677
f(39)=1
f(40)=521
f(41)=199
f(42)=3917
f(43)=811
f(44)=839
f(45)=4337
f(46)=4481
f(47)=661
f(48)=191
f(49)=197
f(50)=5077
f(51)=5231
f(52)=5387
f(53)=1109
f(54)=163
f(55)=5867
f(56)=1
f(57)=6197
f(58)=1
f(59)=1307
f(60)=353
f(61)=983
f(62)=7057
f(63)=1447
f(64)=1483
f(65)=107
f(66)=251
f(67)=257
f(68)=233
f(69)=1669
f(70)=8537
f(71)=8731
f(72)=113
f(73)=1
f(74)=373
f(75)=1361
f(76)=263
f(77)=523
f(78)=2029
f(79)=109
f(80)=10567
f(81)=10781
f(82)=1571
f(83)=2243
f(84)=2287
f(85)=11657
f(86)=1
f(87)=12107
f(88)=2467
f(89)=359
f(90)=1
f(91)=1
f(92)=13267
f(93)=1
f(94)=2749
f(95)=1
f(96)=1
f(97)=467
f(98)=1
f(99)=599

b) Substitution of the polynom
The polynom f(x)=x^2+53x-73 could be written as f(y)= y^2-775.25 with x=y-26.5

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+26.5
f'(x)>2x+52

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, 19, 37, 5, 31, 7, 281, 347, 83, 97, 557, 631, 101, 157, 173, 947, 1031, 1117, 241, 1, 1, 1481, 1, 67, 71, 1877, 283, 2087, 439, 461, 2417, 2531, 2647, 79, 577, 1, 1, 3257, 677, 1, 521, 199, 3917, 811, 839, 4337, 4481, 661, 191, 197, 5077, 5231, 5387, 1109, 163, 5867, 1, 6197, 1, 1307, 353, 983, 7057, 1447, 1483, 107, 251, 257, 233, 1669, 8537, 8731, 113, 1, 373, 1361, 263, 523, 2029, 109, 10567, 10781, 1571, 2243, 2287, 11657, 1, 12107, 2467, 359, 1, 1, 13267, 1, 2749, 1, 1, 467, 1, 599, 15227, 137, 15737, 457, 3251, 1, 1, 17047, 3463, 3517, 2551, 18131, 1, 1, 3793, 1013, 19531, 149, 4021, 4079, 151, 20981, 21277, 863, 1, 331, 22481, 22787, 1, 1, 641, 3433, 1, 4933, 1, 25307, 1, 1, 751, 5323, 26947, 27281, 27617, 5591, 5659, 4091, 397, 29327, 1187, 1201, 821, 389, 4441, 1, 6361, 1693, 32531, 491, 6653, 1, 1097, 34381, 34757, 7027, 7103, 35897, 1, 991, 7411, 7489, 1, 38231, 1, 223, 1, 39827, 40231, 40637, 8209, 8291, 5981, 42281, 42697, 8623, 8707, 1, 44381, 1, 1, 9133, 1487, 1, 701, 499, 1367, 443, 48731, 49177, 1, 2003, 50527, 7283, 51437, 1, 1, 52817, 53281, 757, 1549, 10937, 2903, 55631, 2953, 11317, 1, 8221, 58031, 1, 11801, 1, 269, 1951, 1, 2459, 1, 62477, 62981, 63487, 12799, 1, 823, 3449, 66047, 13313, 13417, 67607, 9733, 967, 1, 1, 1, 1913, 71317, 2053, 14479, 72937, 1, 74027, 1, 601, 569, 76231, 2477, 1, 15581, 78467, 79031, 1, 16033, 1, 81307, 2213, 82457, 16607, 2389, 313, 1, 4493, 17191, 911, 1103, 1, 88327, 3557, 3581, 90127, 90731, 613, 1, 1, 617, 1, 94397, 1, 1, 13751, 5099, 1, 1033, 19753, 99397, 1493, 1, 20261, 20389, 102587, 103231, 953, 1, 1, 1091, 1, 107137, 21559, 1, 5743, 15683, 5813, 1, 1, 1051, 113131, 1559, 3271, 743, 1, 116531, 1, 23581, 23719, 17041, 119981, 120677, 971, 1, 122777, 1, 1, 24979, 25121, 126317, 127031, 127747, 25693, 3691, 3511, 130631, 131357, 26417, 1, 1609, 19183, 135017, 1429, 27299, 1, 4451, 138727, 797, 1, 140977, 141731, 3851, 28649, 1, 20681, 145531, 146297, 1, 29567, 859, 149381, 1129, 30187, 1597, 2089, 153281, 154067, 30971, 4447, 156437, 157231, 4271, 6353, 1277, 1171, 1, 5227, 32569, 1, 1009, 8699, 1019, 1, 33547, 2309, 1583, 170207, 433, 929, 24671, 173531, 174367, 1, 1, 176887, 1, 1, 7177, 7211, 9533, 181981, 9623, 36739, 5273, 1181, 1709, 6037, 1213, 1021, 189757, 1, 191507, 1, 38653, 1303, 1931, 1831, 5623, 2081, 2797, 10499, 1327, 1, 8087, 1, 1, 204887, 1, 41341, 1087, 1381, 29921, 42073, 42257, 212207, 213131, 2579, 1, 1, 1, 217781, 218717, 1, 44119, 221537, 1, 1, 1, 9013, 3187, 227231, 1319, 6547, 46021, 231067, 1, 12263, 1, 2473, 503, 236881, 237857, 1291, 47963, 240797, 241781, 1, 48751, 1579, 7927, 246731, 1297, 9949, 1427, 250727, 13249, 252737, 2671, 50951, 255767, 36683, 257797, 1399, 1, 260857, 261881, 1, 7541, 1, 1, 3761, 4001, 1, 1, 2039, 272231, 1, 10973, 2203, 1, 8951, 39791, 1, 56131, 281717, 3407, 3593, 56983, 8171, 287057, 3947, 2111, 58057, 3067, 2683, 2207, 2917, 59141, 59359, 1, 4211, 2753, 1721, 1, 303377, 304481, 4561, 61339, 61561, 44131, 1, 10037, 1, 1, 16553, 315631, 1223, 63577, 63803, 1, 3181, 322417, 1, 9277, 1999, 4139, 328127, 13171, 13217, 331577, 47533, 17573, 593, 3539, 337367, 338531, 9181, 9739, 1, 1, 1, 1, 2237, 1, 49871, 350281, 1, 1, 70769, 355037, 18749, 51061, 1, 1, 2423, 1, 363437, 1, 10453, 619, 1433, 2447, 74143, 1, 373157, 1, 375607, 75367, 1, 19963, 3923, 1, 1, 1, 1, 386731, 387977, 15569, 15619, 1, 392981, 1499, 1, 1, 643, 399281, 57221, 80363, 4243, 404357, 1, 406907, 81637, 11699, 410747, 412031, 4261, 1, 1, 417187, 1, 419777, 16843, 1, 1, 6343, 426287, 1, 85781, 22643, 1, 432847, 1, 1, 62401, 438131, 3889, 1, 88423, 2251, 2833, 1, 89491, 89759, 450137, 451481, 23833, 1, 1, 456877, 1783, 1907, 92189, 92461, 14957, 2143, 6389, 93553, 93827, 1789, 7043, 473257, 1, 95203, 6043, 1, 480167, 1, 96589, 69191, 485731, 5869, 19541, 19597, 491327, 492731, 1, 99109, 99391, 498367, 499781, 501197, 100523, 14401, 719, 1, 1, 1, 102233, 512597, 73433, 1, 1543, 103669, 519787, 521231, 522677, 1, 21023, 733, 1523, 2663, 106279, 1, 2063, 1, 5021, 1, 108037, 541657, 543131, 77801, 1, 1, 1, 550531, 17807, 3571, 1, 556487, 557981, 15121, 1, 1, 29683, 80783, 566987, 1697, 114001, 7829, 573031, 574547, 1, 1627, 579107, 580631, 582157, 1, 117043, 769, 2441, 1, 1, 1, 773, 595981, 85361, 1, 1, 1, 603731, 1, 121369, 17383, 7349, 611531, 613097, 1, 3331, 9221, 4657, 620957, 6553, 124823, 625697, 1777, 2699, 18013, 1, 2239, 1, 1, 25537, 25601, 1, 643231, 644837, 129289, 1, 1, 651281, 4909, 1657, 1, 6781, 4813, 1997, 132527, 18979, 665897, 667531, 669167, 134161, 134489, 10061, 2609, 1, 1, 1, 682327, 35999, 685637, 1, 137791, 1, 692281, 1933, 139123, 139457, 3221, 1, 3677, 140797, 1, 2137, 19163, 101531, 7499, 1, 1, 1, 719177, 1, 4129, 724277, 725981, 10861, 145879, 146221, 732817, 104933, 736247, 3989, 147937, 1, 743131, 1, 1, 7877, 1, 24251, 1, 151051, 1, 1, 10711, 762227, 30559, 1, 1, 769231, 1, 4177, 1, 776267, 40949, 779797, 1, 22381, 785107, 786881, 1, 1, 158443, 3019, 113683, 797567, 159871, 1, 21701, 4937, 26017, 1, 6481, 1, 1, 42923, 163469, 1, 117281, 822781, 8501, 1, 1, 3307, 831881, 119101, 167107, 1, 1, 2687, 7459, 1, 1, 848387, 1, 1, 34157, 34231, 1, 1, 10903, 5569, 5581, 866917, 1979, 23531, 1, 174877, 876257, 878131, 880007, 9283, 176753, 6659, 1, 12527, 178261, 178639, 895087, 1, 128411, 1, 36107, 904577, 906481, 24551, 182059, 1, 914117, 916031, 48313, 183973, 1, 1, 132233, 927557, 185897, 186283, 2351, 11839, 6841, 26833, 5087, 9337, 2677, 13337, 7591, 1, 136111, 1, 8941, 10091, 192121, 6131, 964531, 138071, 1, 194087, 972407, 14543, 1, 1, 1, 31687, 31751, 986267, 197651, 2713, 52223, 1, 52433, 39929, 40009, 1002227, 8887, 1006237, 28807, 2083, 1012267, 1, 4217, 203663, 204067, 146051, 1, 1026407, 2897, 10847, 1032497, 54449, 1, 207721, 208129, 1042687, 1, 33767, 41953, 1,

6. Sequence of the polynom (only primes)

73, 19, 37, 5, 31, 7, 281, 347, 83, 97, 557, 631, 101, 157, 173, 947, 1031, 1117, 241, 1481, 67, 71, 1877, 283, 2087, 439, 461, 2417, 2531, 2647, 79, 577, 3257, 677, 521, 199, 3917, 811, 839, 4337, 4481, 661, 191, 197, 5077, 5231, 5387, 1109, 163, 5867, 6197, 1307, 353, 983, 7057, 1447, 1483, 107, 251, 257, 233, 1669, 8537, 8731, 113, 373, 1361, 263, 523, 2029, 109, 10567, 10781, 1571, 2243, 2287, 11657, 12107, 2467, 359, 13267, 2749, 467, 599, 15227, 137, 15737, 457, 3251, 17047, 3463, 3517, 2551, 18131, 3793, 1013, 19531, 149, 4021, 4079, 151, 20981, 21277, 863, 331, 22481, 22787, 641, 3433, 4933, 25307, 751, 5323, 26947, 27281, 27617, 5591, 5659, 4091, 397, 29327, 1187, 1201, 821, 389, 4441, 6361, 1693, 32531, 491, 6653, 1097, 34381, 34757, 7027, 7103, 35897, 991, 7411, 7489, 38231, 223, 39827, 40231, 40637, 8209, 8291, 5981, 42281, 42697, 8623, 8707, 44381, 9133, 1487, 701, 499, 1367, 443, 48731, 49177, 2003, 50527, 7283, 51437, 52817, 53281, 757, 1549, 10937, 2903, 55631, 2953, 11317, 8221, 58031, 11801, 269, 1951, 2459, 62477, 62981, 63487, 12799, 823, 3449, 66047, 13313, 13417, 67607, 9733, 967, 1913, 71317, 2053, 14479, 72937, 74027, 601, 569, 76231, 2477, 15581, 78467, 79031, 16033, 81307, 2213, 82457, 16607, 2389, 313, 4493, 17191, 911, 1103, 88327, 3557, 3581, 90127, 90731, 613, 617, 94397, 13751, 5099, 1033, 19753, 99397, 1493, 20261, 20389, 102587, 103231, 953, 1091, 107137, 21559, 5743, 15683, 5813, 1051, 113131, 1559, 3271, 743, 116531, 23581, 23719, 17041, 119981, 120677, 971, 122777, 24979, 25121, 126317, 127031, 127747, 25693, 3691, 3511, 130631, 131357, 26417, 1609, 19183, 135017, 1429, 27299, 4451, 138727, 797, 140977, 141731, 3851, 28649, 20681, 145531, 146297, 29567, 859, 149381, 1129, 30187, 1597, 2089, 153281, 154067, 30971, 4447, 156437, 157231, 4271, 6353, 1277, 1171, 5227, 32569, 1009, 8699, 1019, 33547, 2309, 1583, 170207, 433, 929, 24671, 173531, 174367, 176887, 7177, 7211, 9533, 181981, 9623, 36739, 5273, 1181, 1709, 6037, 1213, 1021, 189757, 191507, 38653, 1303, 1931, 1831, 5623, 2081, 2797, 10499, 1327, 8087, 204887, 41341, 1087, 1381, 29921, 42073, 42257, 212207, 213131, 2579, 217781, 218717, 44119, 221537, 9013, 3187, 227231, 1319, 6547, 46021, 231067, 12263, 2473, 503, 236881, 237857, 1291, 47963, 240797, 241781, 48751, 1579, 7927, 246731, 1297, 9949, 1427, 250727, 13249, 252737, 2671, 50951, 255767, 36683, 257797, 1399, 260857, 261881, 7541, 3761, 4001, 2039, 272231, 10973, 2203, 8951, 39791, 56131, 281717, 3407, 3593, 56983, 8171, 287057, 3947, 2111, 58057, 3067, 2683, 2207, 2917, 59141, 59359, 4211, 2753, 1721, 303377, 304481, 4561, 61339, 61561, 44131, 10037, 16553, 315631, 1223, 63577, 63803, 3181, 322417, 9277, 1999, 4139, 328127, 13171, 13217, 331577, 47533, 17573, 593, 3539, 337367, 338531, 9181, 9739, 2237, 49871, 350281, 70769, 355037, 18749, 51061, 2423, 363437, 10453, 619, 1433, 2447, 74143, 373157, 375607, 75367, 19963, 3923, 386731, 387977, 15569, 15619, 392981, 1499, 643, 399281, 57221, 80363, 4243, 404357, 406907, 81637, 11699, 410747, 412031, 4261, 417187, 419777, 16843, 6343, 426287, 85781, 22643, 432847, 62401, 438131, 3889, 88423, 2251, 2833, 89491, 89759, 450137, 451481, 23833, 456877, 1783, 1907, 92189, 92461, 14957, 2143, 6389, 93553, 93827, 1789, 7043, 473257, 95203, 6043, 480167, 96589, 69191, 485731, 5869, 19541, 19597, 491327, 492731, 99109, 99391, 498367, 499781, 501197, 100523, 14401, 719, 102233, 512597, 73433, 1543, 103669, 519787, 521231, 522677, 21023, 733, 1523, 2663, 106279, 2063, 5021, 108037, 541657, 543131, 77801, 550531, 17807, 3571, 556487, 557981, 15121, 29683, 80783, 566987, 1697, 114001, 7829, 573031, 574547, 1627, 579107, 580631, 582157, 117043, 769, 2441, 773, 595981, 85361, 603731, 121369, 17383, 7349, 611531, 613097, 3331, 9221, 4657, 620957, 6553, 124823, 625697, 1777, 2699, 18013, 2239, 25537, 25601, 643231, 644837, 129289, 651281, 4909, 1657, 6781, 4813, 1997, 132527, 18979, 665897, 667531, 669167, 134161, 134489, 10061, 2609, 682327, 35999, 685637, 137791, 692281, 1933, 139123, 139457, 3221, 3677, 140797, 2137, 19163, 101531, 7499, 719177, 4129, 724277, 725981, 10861, 145879, 146221, 732817, 104933, 736247, 3989, 147937, 743131, 7877, 24251, 151051, 10711, 762227, 30559, 769231, 4177, 776267, 40949, 779797, 22381, 785107, 786881, 158443, 3019, 113683, 797567, 159871, 21701, 4937, 26017, 6481, 42923, 163469, 117281, 822781, 8501, 3307, 831881, 119101, 167107, 2687, 7459, 848387, 34157, 34231, 10903, 5569, 5581, 866917, 1979, 23531, 174877, 876257, 878131, 880007, 9283, 176753, 6659, 12527, 178261, 178639, 895087, 128411, 36107, 904577, 906481, 24551, 182059, 914117, 916031, 48313, 183973, 132233, 927557, 185897, 186283, 2351, 11839, 6841, 26833, 5087, 9337, 2677, 13337, 7591, 136111, 8941, 10091, 192121, 6131, 964531, 138071, 194087, 972407, 14543, 31687, 31751, 986267, 197651, 2713, 52223, 52433, 39929, 40009, 1002227, 8887, 1006237, 28807, 2083, 1012267, 4217, 203663, 204067, 146051, 1026407, 2897, 10847, 1032497, 54449, 207721, 208129, 1042687, 33767, 41953,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+53x-73 and
the reducible primes which appear as divisor for the first time
p | x^2+53x-73 and p < x^2+53x-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)
11010731.0000000.7000001.0000000.0000000.0000000.000000
21008135460.8100000.3500000.8100008.1000005.00000015.333333
31.0007352414940.7350000.2410000.7350009.0740746.88571410.739130
410.0007.2571.6955.5620.7257000.1695000.7257009.8734697.03319511.259109
5100.00071.86113.19458.6670.7186100.1319400.7186109.9023017.78407110.547825
61.000.000714.351108.003606.3480.7143510.1080030.7143519.9407338.18576610.335419
710.000.0007.112.340913.6306.198.7100.7112340.0913630.7112349.9563668.45930210.223023
8100.000.00070.874.6207.914.57362.960.0470.7087460.0791460.7087469.9650228.66277710.156960
91.000.000.000706.871.17069.813.684637.057.4860.7068710.0698140.7068719.9735448.82090310.118440
1010.000.000.0007.053.976.927624.828.9186.429.148.0090.7053980.0624830.7053989.9791568.94995010.091944


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
244311.0000000.7500000.2500001.3333331.000000inf
388621.0000000.7500000.2500002.0000002.0000002.000000
416161061.0000000.6250000.3750002.0000001.6666673.000000
5322917120.9062500.5312500.3750001.8125001.7000002.000000
6645627290.8750000.4218750.4531251.9310341.5882352.416667
712810043570.7812500.3359380.4453121.7857141.5925931.965517
8256196791170.7656250.3085940.4570311.9600001.8372092.052632
95123901392510.7617190.2714840.4902341.9897961.7594942.145299
101.0247542475070.7363280.2412110.4951171.9333331.7769782.019920
112.0481.5044331.0710.7343750.2114260.5229491.9946951.7530362.112426
124.0962.9987732.2250.7319340.1887210.5432131.9933511.7852192.077498
138.1925.9561.4264.5300.7270510.1740720.5529791.9866581.8447612.035955
1416.38411.8522.6049.2480.7233890.1589360.5644531.9899261.8260872.041501
1532.76823.5684.86318.7050.7192380.1484070.5708311.9885251.8675122.022599
1665.53647.1218.97838.1430.7190090.1369930.5820161.9993641.8461852.039187
17131.07294.12416.82177.3030.7181090.1283340.5897751.9974961.8735802.026663
18262.144187.85031.645156.2050.7165910.1207160.5958751.9957721.8812792.020685
19524.288374.89659.722315.1740.7150570.1139110.6011471.9957201.8872492.017695
201.048.576748.962112.903636.0590.7142660.1076730.6065931.9977861.8904762.018120
212.097.1521.495.793214.0111.281.7820.7132500.1020480.6112011.9971551.8955302.015193
224.194.3042.987.497407.2312.580.2660.7122750.0970910.6151831.9972661.9028512.013030
238.388.6085.968.021775.5525.192.4690.7114440.0924530.6189911.9976661.9044522.012378
2416.777.21611.922.0661.480.43810.441.6280.7106110.0882410.6223701.9976581.9088832.010918
2533.554.43223.818.1152.835.25620.982.8590.7098350.0844970.6253381.9978181.9151472.009539
2667.108.86447.588.1045.438.79442.149.3100.7091180.0810440.6280741.9979791.9182732.008750
27134.217.72895.089.83410.443.19784.646.6370.7084740.0778080.6306671.9981851.9201312.008257
28268.435.456190.018.77320.093.973169.924.8000.7078750.0748560.6330191.9983081.9241212.007461
29536.870.912379.743.09938.713.778341.029.3210.7073270.0721100.6352171.9984511.9266362.006943
301.073.741.824758.940.49774.690.717684.249.7800.7068180.0695610.6372571.9985631.9293062.006425
312.147.483.6481.516.864.900144.310.5091.372.554.3910.7063450.0672000.6391451.9986611.9321082.005926
324.294.967.2963.031.818.500279.132.2902.752.686.2100.7059000.0649910.6409101.9987401.9342482.005521
338.589.934.5926.060.054.801540.481.0845.519.573.7170.7054830.0629200.6425631.9988191.9362902.005159
3417.179.869.18412.113.461.9821.047.582.82811.065.879.1540.7050960.0609770.6441191.9989031.9382412.004843


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
123301110
243301110
386423210
41610553322
532177105444
664279189675
71284315281313107
825679265322201819
9512139469336343435
101.0242478316462645962
112.048433146287108112106107
124.096773266507188197201187
138.1921.426483943344367354361
1416.3842.6048791.725643663659639
1532.7684.8631.6423.2211.2201.2191.1991.225
1665.5368.9783.0035.9752.2722.2032.2252.278
17131.07216.8215.58011.2414.2314.1474.2084.235
18262.14431.64510.57621.0697.9617.9237.8247.937
19524.28859.72219.95739.76515.04014.88414.87414.924
201.048.576112.90337.58575.31828.33028.14128.22428.208
212.097.152214.01171.362142.64953.45053.31253.65853.591
224.194.304407.231135.731271.500101.767101.690102.045101.729
238.388.608775.552258.556516.996193.646193.740194.304193.862
2416.777.2161.480.438493.824986.614369.972369.758370.670370.038
2533.554.4322.835.256945.5621.889.694708.950708.328709.407708.571
2667.108.8645.438.7941.813.0273.625.7671.360.2981.358.6041.360.4581.359.434
27134.217.72810.443.1973.479.6756.963.5222.610.4282.611.0492.611.3142.610.406
28268.435.45620.093.9736.697.41413.396.5595.024.3315.025.0255.021.7525.022.865
29536.870.91238.713.77812.905.92725.807.8519.680.1079.680.2969.675.6119.677.764
301.073.741.82474.690.71724.899.08649.791.63118.672.32518.677.22918.669.26718.671.896
312.147.483.648144.310.50948.106.61596.203.89436.078.39736.083.73136.074.26036.074.121
324.294.967.296279.132.29093.048.728186.083.56269.784.69369.788.85569.783.02869.775.714
338.589.934.592540.481.084180.167.870360.313.214135.121.294135.124.732135.122.612135.112.446
3417.179.869.1841.047.582.828349.201.334698.381.494261.912.072261.893.194261.893.101261.884.461


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
241100001
382110101
4166331131
53212752343
6642915145789
712857292810151517
8256117625527283230
951225112712456626469
101.024507265242124128127128
112.0481.071537534259271270271
124.0962.2251.1291.096553572539561
138.1924.5302.3062.2241.1081.1751.1061.141
1416.3849.2484.7174.5312.2902.3752.3022.281
1532.76818.7059.5339.1724.6114.7324.7104.652
1665.53638.14319.41118.7329.5009.5899.5379.517
17131.07277.30339.33637.96719.20319.40219.38219.316
18262.144156.20579.35776.84838.96638.95939.03739.243
19524.288315.174160.248154.92678.53978.76878.89078.977
201.048.576636.059322.608313.451158.662159.046158.798159.553
212.097.1521.281.782649.260632.522319.965320.779320.050320.988
224.194.3042.580.2661.305.3831.274.883644.735645.860644.368645.303
238.388.6085.192.4692.624.3042.568.1651.296.6081.298.6081.298.6321.298.621
2416.777.21610.441.6285.276.7985.164.8302.609.9222.610.2352.610.5992.610.872
2533.554.43220.982.85910.595.88810.386.9715.245.4275.247.1525.246.5765.243.704
2667.108.86442.149.31021.274.50020.874.81010.536.00710.534.77410.540.69310.537.836
27134.217.72884.646.63742.711.93641.934.70121.157.57321.159.57921.163.09721.166.388
28268.435.456169.924.80085.704.65384.220.14742.474.67742.479.12742.483.61542.487.381
29536.870.912341.029.321171.945.569169.083.75285.244.61485.247.85585.265.21785.271.635
301.073.741.824684.249.780344.876.745339.373.035171.054.906171.050.212171.076.618171.068.044
312.147.483.6481.372.554.391691.584.652680.969.739343.135.754343.134.262343.140.187343.144.188
324.294.967.2962.752.686.2101.386.596.7401.366.089.470688.156.269688.164.772688.175.398688.189.771
338.589.934.5925.519.573.7172.779.611.9002.739.961.8171.379.879.9801.379.897.3871.379.876.2051.379.920.145
3417.179.869.18411.065.879.1545.571.328.9015.494.550.2532.766.505.9822.766.510.2442.766.416.2202.766.446.708


8. Check for existing Integer Sequences by OEIS

Found in Database : 73, 19, 37, 5, 31, 7, 281, 347, 83, 97, 557, 631, 101, 157, 173, 947, 1031, 1117, 241, 1,
Found in Database : 73, 19, 37, 5, 31, 7, 281, 347, 83, 97, 557, 631, 101, 157, 173, 947, 1031, 1117, 241, 1481, 67, 71, 1877, 283, 2087, 439, 461, 2417, 2531, 2647, 79, 577, 3257, 677,
Found in Database : 5, 7, 19, 31, 37, 67, 71, 73, 79, 83, 97, 101, 107, 109, 113, 137, 149,