Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:14:12
Deutsch
29.Mar 2024

Polynom = x^2-43x+3

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) = 3 = 3
f(1) = 39 = 3*13
f(2) = 79 = 79
f(3) = 117 = 3*3*13
f(4) = 153 = 3*3*17
f(5) = 187 = 11*17
f(6) = 219 = 3*73
f(7) = 249 = 3*83
f(8) = 277 = 277
f(9) = 303 = 3*101
f(10) = 327 = 3*109
f(11) = 349 = 349
f(12) = 369 = 3*3*41
f(13) = 387 = 3*3*43
f(14) = 403 = 13*31
f(15) = 417 = 3*139
f(16) = 429 = 3*11*13
f(17) = 439 = 439
f(18) = 447 = 3*149
f(19) = 453 = 3*151
f(20) = 457 = 457
f(21) = 459 = 3*3*3*17
f(22) = 459 = 3*3*3*17
f(23) = 457 = 457
f(24) = 453 = 3*151
f(25) = 447 = 3*149
f(26) = 439 = 439
f(27) = 429 = 3*11*13
f(28) = 417 = 3*139
f(29) = 403 = 13*31
f(30) = 387 = 3*3*43
f(31) = 369 = 3*3*41
f(32) = 349 = 349
f(33) = 327 = 3*109
f(34) = 303 = 3*101
f(35) = 277 = 277
f(36) = 249 = 3*83
f(37) = 219 = 3*73
f(38) = 187 = 11*17
f(39) = 153 = 3*3*17
f(40) = 117 = 3*3*13
f(41) = 79 = 79
f(42) = 39 = 3*13
f(43) = 3 = 3
f(44) = 47 = 47
f(45) = 93 = 3*31
f(46) = 141 = 3*47
f(47) = 191 = 191
f(48) = 243 = 3*3*3*3*3
f(49) = 297 = 3*3*3*11
f(50) = 353 = 353
f(51) = 411 = 3*137
f(52) = 471 = 3*157
f(53) = 533 = 13*41
f(54) = 597 = 3*199
f(55) = 663 = 3*13*17
f(56) = 731 = 17*43
f(57) = 801 = 3*3*89
f(58) = 873 = 3*3*97
f(59) = 947 = 947
f(60) = 1023 = 3*11*31
f(61) = 1101 = 3*367
f(62) = 1181 = 1181
f(63) = 1263 = 3*421
f(64) = 1347 = 3*449
f(65) = 1433 = 1433
f(66) = 1521 = 3*3*13*13
f(67) = 1611 = 3*3*179
f(68) = 1703 = 13*131
f(69) = 1797 = 3*599
f(70) = 1893 = 3*631
f(71) = 1991 = 11*181
f(72) = 2091 = 3*17*41
f(73) = 2193 = 3*17*43
f(74) = 2297 = 2297
f(75) = 2403 = 3*3*3*89
f(76) = 2511 = 3*3*3*3*31
f(77) = 2621 = 2621
f(78) = 2733 = 3*911
f(79) = 2847 = 3*13*73
f(80) = 2963 = 2963
f(81) = 3081 = 3*13*79
f(82) = 3201 = 3*11*97
f(83) = 3323 = 3323
f(84) = 3447 = 3*3*383
f(85) = 3573 = 3*3*397
f(86) = 3701 = 3701
f(87) = 3831 = 3*1277
f(88) = 3963 = 3*1321
f(89) = 4097 = 17*241
f(90) = 4233 = 3*17*83
f(91) = 4371 = 3*31*47
f(92) = 4511 = 13*347
f(93) = 4653 = 3*3*11*47
f(94) = 4797 = 3*3*13*41
f(95) = 4943 = 4943
f(96) = 5091 = 3*1697
f(97) = 5241 = 3*1747
f(98) = 5393 = 5393
f(99) = 5547 = 3*43*43
f(100) = 5703 = 3*1901

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^2-43x+3

f(0)=3
f(1)=13
f(2)=79
f(3)=1
f(4)=17
f(5)=11
f(6)=73
f(7)=83
f(8)=277
f(9)=101
f(10)=109
f(11)=349
f(12)=41
f(13)=43
f(14)=31
f(15)=139
f(16)=1
f(17)=439
f(18)=149
f(19)=151
f(20)=457
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=47
f(45)=1
f(46)=1
f(47)=191
f(48)=1
f(49)=1
f(50)=353
f(51)=137
f(52)=157
f(53)=1
f(54)=199
f(55)=1
f(56)=1
f(57)=89
f(58)=97
f(59)=947
f(60)=1
f(61)=367
f(62)=1181
f(63)=421
f(64)=449
f(65)=1433
f(66)=1
f(67)=179
f(68)=131
f(69)=599
f(70)=631
f(71)=181
f(72)=1
f(73)=1
f(74)=2297
f(75)=1
f(76)=1
f(77)=2621
f(78)=911
f(79)=1
f(80)=2963
f(81)=1
f(82)=1
f(83)=3323
f(84)=383
f(85)=397
f(86)=3701
f(87)=1277
f(88)=1321
f(89)=241
f(90)=1
f(91)=1
f(92)=347
f(93)=1
f(94)=1
f(95)=4943
f(96)=1697
f(97)=1747
f(98)=5393
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-43x+3 could be written as f(y)= y^2-459.25 with x=y+21.5

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-21.5
f'(x)>2x-44 with x > 21

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

3, 13, 79, 1, 17, 11, 73, 83, 277, 101, 109, 349, 41, 43, 31, 139, 1, 439, 149, 151, 457, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 191, 1, 1, 353, 137, 157, 1, 199, 1, 1, 89, 97, 947, 1, 367, 1181, 421, 449, 1433, 1, 179, 131, 599, 631, 181, 1, 1, 2297, 1, 1, 2621, 911, 1, 2963, 1, 1, 3323, 383, 397, 3701, 1277, 1321, 241, 1, 1, 347, 1, 1, 4943, 1697, 1747, 5393, 1, 1901, 5861, 223, 229, 577, 167, 1, 1, 2341, 2399, 1, 839, 859, 193, 2699, 251, 197, 2887, 227, 1, 1, 1049, 311, 1, 1, 10253, 317, 3557, 10883, 1, 419, 887, 3917, 307, 12197, 1, 4217, 1171, 1, 1483, 1, 271, 1, 14303, 373, 4931, 1, 1699, 1, 15797, 5351, 5437, 1, 1, 1, 1, 653, 1, 1069, 1, 1, 19001, 6427, 6521, 1, 2237, 2269, 1, 7001, 1, 1, 7297, 569, 1, 1, 1, 571, 7907, 8011, 1, 8221, 757, 25301, 1, 1, 1, 8867, 1, 1, 9199, 9311, 1663, 1, 3217, 29297, 1, 769, 30341, 787, 1, 1013, 3529, 1, 2953, 10949, 11071, 33581, 11317, 673, 1, 433, 1, 35831, 12071, 1109, 1193, 12457, 1, 38153, 4283, 4327, 39341, 1019, 13381, 3119, 1, 811, 41771, 1, 4733, 43013, 467, 1, 44273, 1, 1, 1, 1, 1, 593, 15761, 15907, 2833, 953, 16349, 1151, 1, 509, 3911, 17099, 1327, 479, 1, 1, 53597, 6007, 1, 55001, 1, 1097, 3319, 1459, 617, 4451, 2161, 2179, 1, 19937, 20101, 5527, 20431, 1, 1, 6977, 541, 63803, 1, 1, 827, 1, 2011, 1423, 7489, 7547, 68447, 1, 23167, 5387, 547, 1823, 71633, 1, 2693, 1, 1447, 24781, 74891, 25147, 1, 1867, 659, 8629, 1, 26261, 26449, 1, 1, 27017, 2633, 9133, 1, 4903, 1, 1, 1979, 1, 1, 86861, 1, 1087, 88643, 1, 29947, 607, 1, 1, 1, 1, 1, 1289, 1, 739, 1, 32191, 1, 8893, 10939, 1, 99713, 1, 821, 7817, 1, 34301, 6091, 1, 1, 105491, 35381, 1, 107453, 36037, 2789, 109433, 941, 1, 111431, 1, 37591, 2767, 1, 2251, 6793, 12907, 12983, 9041, 1, 3049, 1, 40099, 1301, 1667, 1511, 1, 2879, 883, 41737, 125921, 1, 1, 9851, 1, 14389, 130223, 43649, 43891, 4271, 1, 44621, 134597, 1367, 1163, 136811, 3527, 46099, 8179, 2741, 1, 1, 5261, 1, 1, 1, 1, 863, 48871, 3779, 148151, 16547, 16633, 150473, 50417, 1, 1, 51199, 51461, 155171, 1, 17417, 12119, 1123, 53047, 1, 4871, 53849, 907, 6043, 6073, 1, 1, 1, 1879, 1, 1, 15427, 18947, 1, 172181, 1, 1, 1, 1427, 58787, 1, 1, 1, 1, 3541, 1951, 182297, 61051, 1, 929, 2293, 6911, 187463, 1, 63067, 14621, 63649, 1487, 1471, 1, 1, 11491, 1, 65707, 1, 66301, 1, 2069, 1723, 1, 203393, 68099, 937, 2609, 1, 6301, 6737, 1, 1, 1, 70841, 1, 214373, 1669, 72077, 217163, 2203, 24337, 219971, 73637, 73951, 7187, 5737, 74897, 1021, 1481, 1, 20773, 76487, 1, 5381, 1889, 1, 234281, 8713, 1, 1229, 1, 7247, 240131, 1, 1, 1, 1, 1, 1249, 82349, 1, 19157, 7577, 1, 1, 1, 28229, 255071, 85361, 1993, 258113, 5081, 5101, 23743, 1, 9749, 20327, 88427, 88771, 2647, 1, 2897, 270461, 1, 2753, 6673, 2129, 7069, 1, 1, 92951, 1, 31219, 31337, 283097, 1, 3067, 286301, 95791, 96149, 22271, 1, 1, 6229, 2389, 5783, 1583, 1, 99401, 1567, 33377, 1, 1, 1, 101599, 23531, 102337, 9337, 2837, 34483, 34607, 312581, 6151, 6173, 2273, 105691, 1, 10301, 1, 1, 4421, 1213, 1, 326153, 1381, 109481, 329591, 36749, 36877, 1, 6553, 8599, 336521, 3631, 1, 8293, 1, 38039, 7309, 1291, 1, 347051, 8929, 116471, 1, 1, 1, 1, 3821, 1, 1873, 1, 120047, 2393, 40283, 3109, 364961, 1, 1, 368597, 1, 123677, 1, 2441, 1, 375923, 125717, 126127, 29201, 126949, 1, 383321, 14243, 1, 387047, 1, 3167, 390791, 130681, 4229, 23209, 1, 44119, 2357, 12109, 2843, 402131, 2861, 3137, 405947, 45247, 45389, 409781, 1877, 1, 1, 10639, 8161, 1, 15511, 15559, 5077, 140897, 1789, 425297, 3307, 12967, 1, 47837, 3691, 1, 1, 145267, 1613, 8597, 8623, 441101, 1, 49307, 445097, 11447, 149257, 1, 150151, 2063, 453143, 1, 16883, 1, 1, 1847, 1, 1, 11897, 9901, 3989, 52009, 2383, 156941, 1, 3407, 5107, 158777, 6047, 53233, 1, 1, 161087, 1, 28591, 14771, 162947, 490241, 1, 2029, 1, 165287, 5347, 12163, 12823, 1, 3517, 56039, 56197, 507197, 9973, 1, 5273, 170971, 171449, 515783, 1, 1, 520103, 1, 174331, 1, 1, 175781, 11251, 1, 1, 1, 1, 2153, 41351, 1, 13859, 17483, 60383, 1, 1, 182627, 1, 50077, 184111, 184607, 7607, 4759, 1, 1, 1, 187597, 1, 1, 17191, 3623, 2347, 21179, 3797, 191621, 14779, 577901, 1, 193649, 2927, 1, 1, 587063, 6329, 1, 14431, 2503, 198251, 45869, 1, 1, 54631, 4273, 1, 605603, 202387, 202907, 35899, 1, 7573, 614981, 15809, 18731, 1, 5051, 1, 2591, 69557, 1, 1, 210257, 2087, 633953, 19259, 1, 37573, 1, 71327, 643547, 215051, 215587, 648371, 216661, 217199, 1, 1, 1, 1, 2267, 1, 662951, 1, 13063, 2411, 1, 6779, 6661, 4783, 225349, 14419, 17419, 5279, 52511, 1, 1, 1, 20887, 230311, 1, 13613, 1, 697601, 1, 1997, 702623, 18059, 1693, 64333, 236449, 237011, 1, 79379, 1, 717797, 239831, 1, 3271, 1, 1, 728003, 1, 1, 733133, 1, 245521, 1, 1, 247241, 743447, 1, 9221, 57587, 14713, 14747, 1, 251857, 1, 759053, 84533, 1, 69481, 8237, 19687, 769547, 19777, 257687, 774821, 86287, 1, 1, 15331, 23747, 785423, 262399, 3329, 4679, 1, 1, 19417, 265961, 6199, 801461, 1, 1801, 9721, 89849, 5297, 47779, 20873, 1, 62897, 273157, 8831, 74827, 1, 1, 3301, 1, 1, 834023, 278617, 1, 1, 1, 1, 845021, 282287, 2801, 27437, 284131, 284747, 9619, 95327, 1, 1, 1, 22189, 867233, 1, 1, 51343, 5717, 97397, 1, 293441, 294067, 1, 22717, 295949, 1, 33023, 3677, 2137, 299099, 299731, 1, 300997, 1, 1, 3257, 1, 912521, 23447, 305449, 918263, 306727, 307367, 924023, 1, 1, 1, 310577, 2239, 1, 1, 1, 2713, 34939, 1, 86113, 7717, 317047, 953093, 318349,

6. Sequence of the polynom (only primes)

3, 13, 79, 17, 11, 73, 83, 277, 101, 109, 349, 41, 43, 31, 139, 439, 149, 151, 457, 47, 191, 353, 137, 157, 199, 89, 97, 947, 367, 1181, 421, 449, 1433, 179, 131, 599, 631, 181, 2297, 2621, 911, 2963, 3323, 383, 397, 3701, 1277, 1321, 241, 347, 4943, 1697, 1747, 5393, 1901, 5861, 223, 229, 577, 167, 2341, 2399, 839, 859, 193, 2699, 251, 197, 2887, 227, 1049, 311, 10253, 317, 3557, 10883, 419, 887, 3917, 307, 12197, 4217, 1171, 1483, 271, 14303, 373, 4931, 1699, 15797, 5351, 5437, 653, 1069, 19001, 6427, 6521, 2237, 2269, 7001, 7297, 569, 571, 7907, 8011, 8221, 757, 25301, 8867, 9199, 9311, 1663, 3217, 29297, 769, 30341, 787, 1013, 3529, 2953, 10949, 11071, 33581, 11317, 673, 433, 35831, 12071, 1109, 1193, 12457, 38153, 4283, 4327, 39341, 1019, 13381, 3119, 811, 41771, 4733, 43013, 467, 44273, 593, 15761, 15907, 2833, 953, 16349, 1151, 509, 3911, 17099, 1327, 479, 53597, 6007, 55001, 1097, 3319, 1459, 617, 4451, 2161, 2179, 19937, 20101, 5527, 20431, 6977, 541, 63803, 827, 2011, 1423, 7489, 7547, 68447, 23167, 5387, 547, 1823, 71633, 2693, 1447, 24781, 74891, 25147, 1867, 659, 8629, 26261, 26449, 27017, 2633, 9133, 4903, 1979, 86861, 1087, 88643, 29947, 607, 1289, 739, 32191, 8893, 10939, 99713, 821, 7817, 34301, 6091, 105491, 35381, 107453, 36037, 2789, 109433, 941, 111431, 37591, 2767, 2251, 6793, 12907, 12983, 9041, 3049, 40099, 1301, 1667, 1511, 2879, 883, 41737, 125921, 9851, 14389, 130223, 43649, 43891, 4271, 44621, 134597, 1367, 1163, 136811, 3527, 46099, 8179, 2741, 5261, 863, 48871, 3779, 148151, 16547, 16633, 150473, 50417, 51199, 51461, 155171, 17417, 12119, 1123, 53047, 4871, 53849, 907, 6043, 6073, 1879, 15427, 18947, 172181, 1427, 58787, 3541, 1951, 182297, 61051, 929, 2293, 6911, 187463, 63067, 14621, 63649, 1487, 1471, 11491, 65707, 66301, 2069, 1723, 203393, 68099, 937, 2609, 6301, 6737, 70841, 214373, 1669, 72077, 217163, 2203, 24337, 219971, 73637, 73951, 7187, 5737, 74897, 1021, 1481, 20773, 76487, 5381, 1889, 234281, 8713, 1229, 7247, 240131, 1249, 82349, 19157, 7577, 28229, 255071, 85361, 1993, 258113, 5081, 5101, 23743, 9749, 20327, 88427, 88771, 2647, 2897, 270461, 2753, 6673, 2129, 7069, 92951, 31219, 31337, 283097, 3067, 286301, 95791, 96149, 22271, 6229, 2389, 5783, 1583, 99401, 1567, 33377, 101599, 23531, 102337, 9337, 2837, 34483, 34607, 312581, 6151, 6173, 2273, 105691, 10301, 4421, 1213, 326153, 1381, 109481, 329591, 36749, 36877, 6553, 8599, 336521, 3631, 8293, 38039, 7309, 1291, 347051, 8929, 116471, 3821, 1873, 120047, 2393, 40283, 3109, 364961, 368597, 123677, 2441, 375923, 125717, 126127, 29201, 126949, 383321, 14243, 387047, 3167, 390791, 130681, 4229, 23209, 44119, 2357, 12109, 2843, 402131, 2861, 3137, 405947, 45247, 45389, 409781, 1877, 10639, 8161, 15511, 15559, 5077, 140897, 1789, 425297, 3307, 12967, 47837, 3691, 145267, 1613, 8597, 8623, 441101, 49307, 445097, 11447, 149257, 150151, 2063, 453143, 16883, 1847, 11897, 9901, 3989, 52009, 2383, 156941, 3407, 5107, 158777, 6047, 53233, 161087, 28591, 14771, 162947, 490241, 2029, 165287, 5347, 12163, 12823, 3517, 56039, 56197, 507197, 9973, 5273, 170971, 171449, 515783, 520103, 174331, 175781, 11251, 2153, 41351, 13859, 17483, 60383, 182627, 50077, 184111, 184607, 7607, 4759, 187597, 17191, 3623, 2347, 21179, 3797, 191621, 14779, 577901, 193649, 2927, 587063, 6329, 14431, 2503, 198251, 45869, 54631, 4273, 605603, 202387, 202907, 35899, 7573, 614981, 15809, 18731, 5051, 2591, 69557, 210257, 2087, 633953, 19259, 37573, 71327, 643547, 215051, 215587, 648371, 216661, 217199, 2267, 662951, 13063, 2411, 6779, 6661, 4783, 225349, 14419, 17419, 5279, 52511, 20887, 230311, 13613, 697601, 1997, 702623, 18059, 1693, 64333, 236449, 237011, 79379, 717797, 239831, 3271, 728003, 733133, 245521, 247241, 743447, 9221, 57587, 14713, 14747, 251857, 759053, 84533, 69481, 8237, 19687, 769547, 19777, 257687, 774821, 86287, 15331, 23747, 785423, 262399, 3329, 4679, 19417, 265961, 6199, 801461, 1801, 9721, 89849, 5297, 47779, 20873, 62897, 273157, 8831, 74827, 3301, 834023, 278617, 845021, 282287, 2801, 27437, 284131, 284747, 9619, 95327, 22189, 867233, 51343, 5717, 97397, 293441, 294067, 22717, 295949, 33023, 3677, 2137, 299099, 299731, 300997, 3257, 912521, 23447, 305449, 918263, 306727, 307367, 924023, 310577, 2239, 2713, 34939, 86113, 7717, 317047, 953093, 318349,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-43x+3 and
the reducible primes which appear as divisor for the first time
p | x^2-43x+3 and p < x^2-43x+3

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)
11010461.0000000.4000001.0000000.0000000.0000000.000000
21005119320.5100000.1900000.5100005.1000004.7500005.333333
31.0006431195240.6430000.1190000.64300012.6078436.26315816.375000
410.0006.8158275.9880.6815000.0827000.68150010.5987566.94958011.427481
5100.00068.7826.23162.5510.6878200.0623100.68782010.0927367.53446210.446059
61.000.000688.33551.592636.7430.6883350.0515920.68833510.0074878.27989110.179582
710.000.0006.887.856438.3666.449.4900.6887860.0438370.68878610.0065468.49678210.128875
8100.000.00068.925.2103.797.45265.127.7580.6892520.0379750.68925210.0067728.66274310.098125
91.000.000.000689.605.29633.513.591656.091.7050.6896050.0335140.68960510.0051258.82528410.073918
1010.000.000.0006.898.915.768299.878.0256.599.037.7430.6898920.0299880.68989210.0041518.94795210.058103


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
244221.0000000.5000000.5000001.3333331.0000002.000000
388441.0000000.5000000.5000002.0000002.0000002.000000
416155100.9375000.3125000.6250001.8750001.2500002.500000
532197120.5937500.2187500.3750001.2666671.4000001.200000
6642911180.4531250.1718750.2812501.5263161.5714291.500000
71286923460.5390620.1796880.3593752.3793102.0909092.555556
8256149381110.5820310.1484380.4335942.1594201.6521742.413043
9512315662490.6152340.1289060.4863282.1140941.7368422.243243
101.0246581225360.6425780.1191410.5234382.0888891.8484852.152611
112.0481.3602121.1480.6640620.1035160.5605472.0668691.7377052.141791
124.0962.7733852.3880.6770020.0939940.5830082.0389701.8160382.080139
138.1925.5716844.8870.6800540.0834960.5965582.0090161.7766232.046482
1416.38411.1841.2689.9160.6826170.0773930.6052252.0075391.8538012.029057
1532.76822.4862.32220.1640.6862180.0708620.6153562.0105511.8312302.033481
1665.53645.0854.28940.7960.6879430.0654450.6224982.0050251.8471152.023210
17131.07290.1657.97382.1920.6879040.0608290.6270751.9998891.8589412.014707
18262.144180.29415.157165.1370.6877670.0578190.6299481.9996011.9010412.009161
19524.288360.76628.585332.1810.6881070.0545220.6335852.0009871.8859272.011548
201.048.576721.86453.981667.8830.6884230.0514800.6369432.0009201.8884382.010600
212.097.1521.443.978102.4111.341.5670.6885420.0488330.6397092.0003461.8971682.008686
224.194.3042.887.901195.1102.692.7910.6885290.0465180.6420111.9999621.9051662.007198
238.388.6085.777.628372.1005.405.5280.6887470.0443580.6443892.0006321.9071292.007407
2416.777.21611.557.994710.83610.847.1580.6889100.0423690.6465412.0004741.9103362.006679
2533.554.43223.120.5281.360.52821.760.0000.6890450.0405470.6484992.0003931.9139832.006055
2667.108.86446.249.6942.608.63443.641.0600.6891740.0388720.6503022.0003741.9173692.005563
27134.217.72892.516.4495.011.20687.505.2430.6893010.0373360.6519652.0003691.9210082.005113
28268.435.456185.064.2509.643.081175.421.1690.6894180.0359230.6534952.0003391.9243032.004693
29536.870.912370.183.91818.581.737351.602.1810.6895210.0346110.6549102.0002991.9269502.004332
301.073.741.824740.466.67335.855.737704.610.9360.6896130.0333930.6562202.0002671.9296232.004000
312.147.483.6481.481.126.93969.262.7121.411.864.2270.6897030.0322530.6574502.0002611.9317052.003750
324.294.967.2962.962.626.655133.971.5042.828.655.1510.6897900.0311930.6585982.0002521.9342522.003489
338.589.934.5925.925.978.612259.395.4665.666.583.1460.6898750.0301980.6596772.0002451.9361992.003278
3417.179.869.18411.853.321.454502.768.88611.350.552.5680.6899540.0292650.6606892.0002301.9382332.003068


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
122100101
242100101
384300211
4165400221
5327601222
66411642333
7128236156674
825638630107156
95126665819162011
101.024122611432263826
112.048212620451536246
124.0963856377939310198
138.1926846676170170178166
1416.3841.26861.260310323307328
1532.7682.32262.314567590573592
1665.5364.28964.2811.0631.0791.0651.082
17131.0727.97367.9651.9851.9951.9922.001
18262.14415.157615.1493.7923.7943.7853.786
19524.28828.585628.5777.2097.1107.1507.116
201.048.57653.981653.97313.55813.49413.51313.416
212.097.152102.4116102.40325.69225.58825.67925.452
224.194.304195.1106195.10248.86448.71848.76948.759
238.388.608372.1006372.09293.12592.97292.89493.109
2416.777.216710.8366710.828177.532177.799177.661177.844
2533.554.4321.360.52861.360.520339.670340.212340.158340.488
2667.108.8642.608.63462.608.626652.199652.388651.940652.107
27134.217.7285.011.20665.011.1981.252.6191.253.3011.252.5021.252.784
28268.435.4569.643.08169.643.0732.409.3892.411.2122.411.5202.410.960
29536.870.91218.581.737618.581.7294.644.5094.643.7624.646.8754.646.591
301.073.741.82435.855.737635.855.7298.963.6898.962.1898.964.7068.965.153
312.147.483.64869.262.712669.262.70417.315.40317.313.93017.317.37617.316.003
324.294.967.296133.971.5046133.971.49633.489.12033.485.90633.501.96733.494.511
338.589.934.592259.395.4666259.395.45864.845.69164.844.45664.863.84764.841.472
3417.179.869.184502.768.8866502.768.878125.686.674125.689.320125.700.664125.692.228


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
121100010
242111010
384222110
41610643331
53212753342
664181175364
71284623231091413
8256111565527273126
951224912612359696358
101.024536260276127133136140
112.0481.148578570280295277296
124.0962.3881.1911.197574595591628
138.1924.8872.4402.4471.2161.2121.1931.266
1416.3849.9164.9564.9602.4702.4802.4452.521
1532.76820.16410.08610.0784.9785.0885.0255.073
1665.53640.79620.43420.36210.18010.16610.19910.251
17131.07282.19241.19440.99820.58020.56820.55720.487
18262.144165.13782.99682.14141.28741.42241.18341.245
19524.288332.181166.770165.41182.67183.13283.22683.152
201.048.576667.883334.790333.093166.739166.883167.258167.003
212.097.1521.341.567672.393669.174334.779335.785335.604335.399
224.194.3042.692.7911.349.1671.343.624673.494672.914673.608672.775
238.388.6085.405.5282.707.7602.697.7681.350.5171.351.5741.352.3291.351.108
2416.777.21610.847.1585.431.0475.416.1112.711.3892.712.6272.712.3382.710.804
2533.554.43221.760.00010.897.02910.862.9715.439.9915.440.9915.439.1195.439.899
2667.108.86443.641.06021.856.19021.784.87010.907.89110.910.19510.910.85610.912.118
27134.217.72887.505.24343.824.48843.680.75521.872.61021.877.13121.876.04821.879.454
28268.435.456175.421.16987.844.05187.577.11843.847.59043.858.58443.855.29943.859.696
29536.870.912351.602.181176.055.241175.546.94087.895.33187.897.93787.905.04787.903.866
301.073.741.824704.610.936352.802.510351.808.426176.137.100176.147.887176.158.324176.167.625
312.147.483.6481.411.864.227706.889.529704.974.698352.939.055352.971.428352.962.191352.991.553
324.294.967.2962.828.655.1511.416.168.6901.412.486.461707.163.498707.160.081707.160.259707.171.313
338.589.934.5925.666.583.1462.836.829.8492.829.753.2971.416.657.5001.416.638.7301.416.637.8121.416.649.104
3417.179.869.18411.350.552.5685.682.095.0515.668.457.5172.837.650.8172.837.600.8252.837.653.8972.837.647.029


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 13, 79, 1, 17, 11, 73, 83, 277, 101, 109, 349, 41, 43, 31, 139, 1, 439, 149, 151,
Found in Database : 3, 13, 79, 17, 11, 73, 83, 277, 101, 109, 349, 41, 43, 31, 139, 439, 149, 151, 457,
Found in Database : 3, 11, 13, 17, 31, 41, 43, 47, 73, 79, 83, 89, 97, 101, 109, 131, 137, 139, 149,