Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:11:29
Deutsch
28.Mar 2024

Polynom = x^2-72x-23

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) = 23 = 23
f(1) = 47 = 47
f(2) = 163 = 163
f(3) = 115 = 5*23
f(4) = 295 = 5*59
f(5) = 179 = 179
f(6) = 419 = 419
f(7) = 239 = 239
f(8) = 535 = 5*107
f(9) = 295 = 5*59
f(10) = 643 = 643
f(11) = 347 = 347
f(12) = 743 = 743
f(13) = 395 = 5*79
f(14) = 835 = 5*167
f(15) = 439 = 439
f(16) = 919 = 919
f(17) = 479 = 479
f(18) = 995 = 5*199
f(19) = 515 = 5*103
f(20) = 1063 = 1063
f(21) = 547 = 547
f(22) = 1123 = 1123
f(23) = 575 = 5*5*23
f(24) = 1175 = 5*5*47
f(25) = 599 = 599
f(26) = 1219 = 23*53
f(27) = 619 = 619
f(28) = 1255 = 5*251
f(29) = 635 = 5*127
f(30) = 1283 = 1283
f(31) = 647 = 647
f(32) = 1303 = 1303
f(33) = 655 = 5*131
f(34) = 1315 = 5*263
f(35) = 659 = 659
f(36) = 1319 = 1319
f(37) = 659 = 659
f(38) = 1315 = 5*263
f(39) = 655 = 5*131
f(40) = 1303 = 1303
f(41) = 647 = 647
f(42) = 1283 = 1283
f(43) = 635 = 5*127
f(44) = 1255 = 5*251
f(45) = 619 = 619
f(46) = 1219 = 23*53
f(47) = 599 = 599
f(48) = 1175 = 5*5*47
f(49) = 575 = 5*5*23
f(50) = 1123 = 1123
f(51) = 547 = 547
f(52) = 1063 = 1063
f(53) = 515 = 5*103
f(54) = 995 = 5*199
f(55) = 479 = 479
f(56) = 919 = 919
f(57) = 439 = 439
f(58) = 835 = 5*167
f(59) = 395 = 5*79
f(60) = 743 = 743
f(61) = 347 = 347
f(62) = 643 = 643
f(63) = 295 = 5*59
f(64) = 535 = 5*107
f(65) = 239 = 239
f(66) = 419 = 419
f(67) = 179 = 179
f(68) = 295 = 5*59
f(69) = 115 = 5*23
f(70) = 163 = 163
f(71) = 47 = 47
f(72) = 23 = 23
f(73) = 25 = 5*5
f(74) = 125 = 5*5*5
f(75) = 101 = 101
f(76) = 281 = 281
f(77) = 181 = 181
f(78) = 445 = 5*89
f(79) = 265 = 5*53
f(80) = 617 = 617
f(81) = 353 = 353
f(82) = 797 = 797
f(83) = 445 = 5*89
f(84) = 985 = 5*197
f(85) = 541 = 541
f(86) = 1181 = 1181
f(87) = 641 = 641
f(88) = 1385 = 5*277
f(89) = 745 = 5*149
f(90) = 1597 = 1597
f(91) = 853 = 853
f(92) = 1817 = 23*79
f(93) = 965 = 5*193
f(94) = 2045 = 5*409
f(95) = 1081 = 23*47
f(96) = 2281 = 2281
f(97) = 1201 = 1201
f(98) = 2525 = 5*5*101
f(99) = 1325 = 5*5*53
f(100) = 2777 = 2777

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^2-72x-23

f(0)=23
f(1)=47
f(2)=163
f(3)=5
f(4)=59
f(5)=179
f(6)=419
f(7)=239
f(8)=107
f(9)=1
f(10)=643
f(11)=347
f(12)=743
f(13)=79
f(14)=167
f(15)=439
f(16)=919
f(17)=479
f(18)=199
f(19)=103
f(20)=1063
f(21)=547
f(22)=1123
f(23)=1
f(24)=1
f(25)=599
f(26)=53
f(27)=619
f(28)=251
f(29)=127
f(30)=1283
f(31)=647
f(32)=1303
f(33)=131
f(34)=263
f(35)=659
f(36)=1319
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
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)=101
f(76)=281
f(77)=181
f(78)=89
f(79)=1
f(80)=617
f(81)=353
f(82)=797
f(83)=1
f(84)=197
f(85)=541
f(86)=1181
f(87)=641
f(88)=277
f(89)=149
f(90)=1597
f(91)=853
f(92)=1
f(93)=193
f(94)=409
f(95)=1
f(96)=2281
f(97)=1201
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-72x-23 could be written as f(y)= y^2-1319 with x=y+36

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-36
f'(x)>2x-73 with x > 36

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

23, 47, 163, 5, 59, 179, 419, 239, 107, 1, 643, 347, 743, 79, 167, 439, 919, 479, 199, 103, 1063, 547, 1123, 1, 1, 599, 53, 619, 251, 127, 1283, 647, 1303, 131, 263, 659, 1319, 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, 101, 281, 181, 89, 1, 617, 353, 797, 1, 197, 541, 1181, 641, 277, 149, 1597, 853, 1, 193, 409, 1, 2281, 1201, 1, 1, 2777, 1453, 3037, 317, 661, 1721, 3581, 1861, 773, 401, 4157, 2153, 4457, 461, 953, 1, 5081, 2621, 1, 557, 5737, 2953, 1, 1, 257, 3301, 6781, 1, 1429, 733, 7517, 3853, 1, 809, 1657, 4241, 8681, 4441, 1, 929, 9497, 211, 1, 1013, 2069, 5281, 10781, 5501, 449, 229, 11677, 5953, 1, 1237, 2521, 6421, 1, 6661, 2713, 1381, 14057, 311, 14557, 1481, 1, 1, 15581, 1, 3221, 1637, 1, 1, 1, 349, 709, 9001, 1, 9281, 3769, 1913, 19417, 1, 19997, 2029, 1, 1, 359, 467, 4357, 1, 22397, 11353, 23017, 2333, 4729, 11981, 24281, 12301, 997, 1, 25577, 12953, 26237, 2657, 5381, 1, 27581, 607, 5653, 2861, 1259, 14653, 631, 3001, 6073, 15361, 31081, 1, 6361, 3217, 32537, 16453, 1, 673, 1361, 1, 34781, 17581, 7109, 3593, 1579, 18353, 37097, 1, 7577, 19141, 823, 19541, 1, 3989, 683, 20353, 41117, 4153, 8389, 1, 1, 21601, 1, 881, 563, 22453, 45337, 1, 9241, 23321, 1, 23761, 1, 1, 48857, 1, 49757, 5021, 10133, 25561, 51581, 26021, 10501, 5297, 53437, 26953, 54377, 1097, 2213, 27901, 2447, 1, 1, 1, 58217, 1, 59197, 1, 12037, 30341, 1, 30841, 12437, 6269, 63197, 601, 64217, 6473, 13049, 1, 839, 1, 2693, 1, 677, 1, 3019, 6997, 1, 35521, 1523, 36061, 14533, 7321, 73757, 701, 74857, 7541, 15193, 38261, 77081, 38821, 15641, 7877, 79337, 39953, 3499, 1621, 653, 1787, 82781, 41681, 1, 1, 1811, 42853, 86297, 8689, 17497, 44041, 88681, 44641, 17977, 9049, 91097, 45853, 92317, 9293, 1, 1, 94781, 47701, 1, 1933, 1093, 48953, 1, 1, 19961, 50221, 101081, 1, 1, 10301, 103657, 52153, 1019, 1, 1, 1, 107581, 54121, 947, 10957, 110237, 2411, 111577, 1, 4517, 719, 114281, 1223, 1, 11633, 117017, 1, 1, 11909, 23957, 1, 121181, 1, 24517, 12329, 123997, 2711, 751, 12613, 1103, 63781, 787, 1217, 5189, 2609, 2791, 1, 132637, 13337, 26821, 67421, 135581, 68161, 1, 13781, 1091, 69653, 140057, 14081, 1231, 71161, 1, 1, 28921, 14537, 1, 73453, 2503, 2969, 1, 1, 1151, 75781, 30469, 15313, 1033, 1, 1187, 15629, 1, 78941, 1483, 3467, 32057, 1, 7039, 81353, 163517, 16433, 33029, 82981, 166781, 1783, 6737, 1, 3209, 85453, 863, 17257, 1, 87121, 175081, 87961, 35353, 17761, 7759, 89653, 1, 1, 36373, 887, 183581, 92221, 37061, 18617, 3529, 1999, 1, 3793, 7621, 95701, 3259, 96581, 1, 1, 195817, 1667, 197597, 1, 39877, 1, 8747, 1279, 40597, 20389, 204797, 1, 1931, 20753, 1, 104681, 1, 105601, 1697, 4261, 213977, 107453, 2137, 1, 43541, 109321, 9547, 110261, 44293, 967, 1, 112153, 225257, 22621, 45433, 1, 229081, 115021, 983, 23197, 232937, 116953, 1, 1, 9473, 118901, 238781, 119881, 1, 1051, 1, 121853, 10639, 1, 1, 2099, 1669, 1, 1, 25169, 4283, 2699, 1423, 1, 51349, 1, 1031, 129901, 10433, 5237, 262877, 1039, 11519, 26597, 53401, 5827, 5077, 1, 1, 1, 1, 137153, 275357, 1, 1, 2963, 3539, 140321, 1, 28277, 283837, 142453, 285977, 5741, 1, 6287, 5477, 145681, 2543, 1, 2917, 147853, 296797, 29789, 59797, 150041, 301181, 151141, 1291, 30449, 305597, 153353, 307817, 30893, 1, 155581, 312281, 1, 1, 1, 316777, 6911, 319037, 1, 1, 161221, 323581, 1, 65173, 1, 1433, 164653, 1, 33161, 66553, 1, 335081, 1889, 67481, 33857, 1877, 7411, 342077, 1373, 1, 172801, 1, 173981, 69829, 1, 351517, 176353, 353897, 35509, 71257, 3803, 358681, 1747, 1, 36229, 363497, 182353, 1321, 36713, 3203, 2339, 1, 8087, 14929, 7489, 3511, 1, 378137, 1, 1, 190921, 383081, 192161, 1307, 1, 388057, 194653, 7369, 39181, 1, 197161, 395581, 8627, 79621, 39937, 17419, 1951, 2089, 8089, 16229, 1, 1, 204781, 1, 41213, 413417, 2053, 1, 41729, 83717, 1409, 421181, 211241, 1, 42509, 18539, 2707, 429017, 1871, 1, 216481, 1, 1, 17477, 1753, 439577, 2477, 442237, 44357, 1, 1, 1, 224461, 90053, 45161, 452957, 227153, 7723, 1987, 91673, 4337, 20047, 3919, 92761, 1, 466537, 1, 2879, 9413, 1, 236701, 2843, 238081, 1, 1, 480317, 240853, 483097, 48449, 97177, 4597, 21247, 1, 98297, 2143, 4799, 247853, 497117, 49853, 99989, 250681, 502781, 252101, 1, 10141, 508477, 1, 511337, 1, 102841, 1, 517081, 2423, 103993, 2267, 522857, 262153, 22859, 52721, 105733, 1, 531581, 266521, 2017, 53597, 6803, 4567, 540377, 10837, 1, 1, 1, 273881, 109849, 55073, 552217, 1, 1, 1, 111637, 1, 561181, 1, 2129, 56569, 6373, 2239, 570217, 57173, 114649, 3229, 1, 1, 23173, 11617, 12391, 1613, 585437, 1, 117701, 1, 591581, 296561, 5171, 59621, 597757, 299653, 600857, 1, 1, 1, 607081, 1867, 122041, 1, 613337, 5801, 1, 1, 4957, 2371, 622781, 1, 5443, 62753, 10663, 13711, 632297, 63389, 1, 5399, 2789, 320141, 128377, 1, 645097, 6101, 648317, 1, 1, 326581, 654781, 6983, 26321, 1, 3323, 14411, 664537, 66617, 5807, 334721, 671081, 336361, 134873, 67601, 677657, 339653, 1, 68261, 1, 1777, 687581, 344621, 138181, 69257, 14771, 1, 3541, 2797, 1, 351301, 704281, 1, 141529, 1, 711017, 1, 9043, 1, 1, 2207, 721181, 361441, 144917, 1, 727997, 364853, 1, 1, 1, 1, 1, 16087, 29669, 14869, 1, 373453, 748637, 75037, 1, 376921, 7481, 378661, 151813, 76081, 762557, 1, 1873, 76781, 153913, 385661, 773081, 8243, 1, 1, 1, 390953, 783677, 1, 31489, 394501, 790781, 7477, 2011, 79613, 797917, 399853, 6311, 80329, 161017, 1, 808681, 405241, 162457, 81409, 815897, 8699, 1, 3571, 1, 412481, 1, 7817, 1, 3329, 6367, 417953, 837737, 1, 168281, 421621, 6451, 423461, 169753, 85061, 852457, 5407, 4783, 1, 3659, 430861, 37547, 432721, 1949, 3779, 871037, 4079, 874777, 1, 35141, 1, 3433, 1, 177209, 88793, 1, 445853, 4937, 1, 1, 2131, 4271, 451541, 3851, 3943, 908797, 455353, 39679, 91453, 183289, 459181, 920281, 461101, 7393, 18521,

6. Sequence of the polynom (only primes)

23, 47, 163, 5, 59, 179, 419, 239, 107, 643, 347, 743, 79, 167, 439, 919, 479, 199, 103, 1063, 547, 1123, 599, 53, 619, 251, 127, 1283, 647, 1303, 131, 263, 659, 1319, 101, 281, 181, 89, 617, 353, 797, 197, 541, 1181, 641, 277, 149, 1597, 853, 193, 409, 2281, 1201, 2777, 1453, 3037, 317, 661, 1721, 3581, 1861, 773, 401, 4157, 2153, 4457, 461, 953, 5081, 2621, 557, 5737, 2953, 257, 3301, 6781, 1429, 733, 7517, 3853, 809, 1657, 4241, 8681, 4441, 929, 9497, 211, 1013, 2069, 5281, 10781, 5501, 449, 229, 11677, 5953, 1237, 2521, 6421, 6661, 2713, 1381, 14057, 311, 14557, 1481, 15581, 3221, 1637, 349, 709, 9001, 9281, 3769, 1913, 19417, 19997, 2029, 359, 467, 4357, 22397, 11353, 23017, 2333, 4729, 11981, 24281, 12301, 997, 25577, 12953, 26237, 2657, 5381, 27581, 607, 5653, 2861, 1259, 14653, 631, 3001, 6073, 15361, 31081, 6361, 3217, 32537, 16453, 673, 1361, 34781, 17581, 7109, 3593, 1579, 18353, 37097, 7577, 19141, 823, 19541, 3989, 683, 20353, 41117, 4153, 8389, 21601, 881, 563, 22453, 45337, 9241, 23321, 23761, 48857, 49757, 5021, 10133, 25561, 51581, 26021, 10501, 5297, 53437, 26953, 54377, 1097, 2213, 27901, 2447, 58217, 59197, 12037, 30341, 30841, 12437, 6269, 63197, 601, 64217, 6473, 13049, 839, 2693, 677, 3019, 6997, 35521, 1523, 36061, 14533, 7321, 73757, 701, 74857, 7541, 15193, 38261, 77081, 38821, 15641, 7877, 79337, 39953, 3499, 1621, 653, 1787, 82781, 41681, 1811, 42853, 86297, 8689, 17497, 44041, 88681, 44641, 17977, 9049, 91097, 45853, 92317, 9293, 94781, 47701, 1933, 1093, 48953, 19961, 50221, 101081, 10301, 103657, 52153, 1019, 107581, 54121, 947, 10957, 110237, 2411, 111577, 4517, 719, 114281, 1223, 11633, 117017, 11909, 23957, 121181, 24517, 12329, 123997, 2711, 751, 12613, 1103, 63781, 787, 1217, 5189, 2609, 2791, 132637, 13337, 26821, 67421, 135581, 68161, 13781, 1091, 69653, 140057, 14081, 1231, 71161, 28921, 14537, 73453, 2503, 2969, 1151, 75781, 30469, 15313, 1033, 1187, 15629, 78941, 1483, 3467, 32057, 7039, 81353, 163517, 16433, 33029, 82981, 166781, 1783, 6737, 3209, 85453, 863, 17257, 87121, 175081, 87961, 35353, 17761, 7759, 89653, 36373, 887, 183581, 92221, 37061, 18617, 3529, 1999, 3793, 7621, 95701, 3259, 96581, 195817, 1667, 197597, 39877, 8747, 1279, 40597, 20389, 204797, 1931, 20753, 104681, 105601, 1697, 4261, 213977, 107453, 2137, 43541, 109321, 9547, 110261, 44293, 967, 112153, 225257, 22621, 45433, 229081, 115021, 983, 23197, 232937, 116953, 9473, 118901, 238781, 119881, 1051, 121853, 10639, 2099, 1669, 25169, 4283, 2699, 1423, 51349, 1031, 129901, 10433, 5237, 262877, 1039, 11519, 26597, 53401, 5827, 5077, 137153, 275357, 2963, 3539, 140321, 28277, 283837, 142453, 285977, 5741, 6287, 5477, 145681, 2543, 2917, 147853, 296797, 29789, 59797, 150041, 301181, 151141, 1291, 30449, 305597, 153353, 307817, 30893, 155581, 312281, 316777, 6911, 319037, 161221, 323581, 65173, 1433, 164653, 33161, 66553, 335081, 1889, 67481, 33857, 1877, 7411, 342077, 1373, 172801, 173981, 69829, 351517, 176353, 353897, 35509, 71257, 3803, 358681, 1747, 36229, 363497, 182353, 1321, 36713, 3203, 2339, 8087, 14929, 7489, 3511, 378137, 190921, 383081, 192161, 1307, 388057, 194653, 7369, 39181, 197161, 395581, 8627, 79621, 39937, 17419, 1951, 2089, 8089, 16229, 204781, 41213, 413417, 2053, 41729, 83717, 1409, 421181, 211241, 42509, 18539, 2707, 429017, 1871, 216481, 17477, 1753, 439577, 2477, 442237, 44357, 224461, 90053, 45161, 452957, 227153, 7723, 1987, 91673, 4337, 20047, 3919, 92761, 466537, 2879, 9413, 236701, 2843, 238081, 480317, 240853, 483097, 48449, 97177, 4597, 21247, 98297, 2143, 4799, 247853, 497117, 49853, 99989, 250681, 502781, 252101, 10141, 508477, 511337, 102841, 517081, 2423, 103993, 2267, 522857, 262153, 22859, 52721, 105733, 531581, 266521, 2017, 53597, 6803, 4567, 540377, 10837, 273881, 109849, 55073, 552217, 111637, 561181, 2129, 56569, 6373, 2239, 570217, 57173, 114649, 3229, 23173, 11617, 12391, 1613, 585437, 117701, 591581, 296561, 5171, 59621, 597757, 299653, 600857, 607081, 1867, 122041, 613337, 5801, 4957, 2371, 622781, 5443, 62753, 10663, 13711, 632297, 63389, 5399, 2789, 320141, 128377, 645097, 6101, 648317, 326581, 654781, 6983, 26321, 3323, 14411, 664537, 66617, 5807, 334721, 671081, 336361, 134873, 67601, 677657, 339653, 68261, 1777, 687581, 344621, 138181, 69257, 14771, 3541, 2797, 351301, 704281, 141529, 711017, 9043, 2207, 721181, 361441, 144917, 727997, 364853, 16087, 29669, 14869, 373453, 748637, 75037, 376921, 7481, 378661, 151813, 76081, 762557, 1873, 76781, 153913, 385661, 773081, 8243, 390953, 783677, 31489, 394501, 790781, 7477, 2011, 79613, 797917, 399853, 6311, 80329, 161017, 808681, 405241, 162457, 81409, 815897, 8699, 3571, 412481, 7817, 3329, 6367, 417953, 837737, 168281, 421621, 6451, 423461, 169753, 85061, 852457, 5407, 4783, 3659, 430861, 37547, 432721, 1949, 3779, 871037, 4079, 874777, 35141, 3433, 177209, 88793, 445853, 4937, 2131, 4271, 451541, 3851, 3943, 908797, 455353, 39679, 91453, 183289, 459181, 920281, 461101, 7393, 18521,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-72x-23 and
the reducible primes which appear as divisor for the first time
p | x^2-72x-23 and p < x^2-72x-23

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)
1109720.9000000.7000000.2000000.0000000.0000000.000000
21005035150.5000000.3500000.1500005.5555555.0000007.500000
31.0007253283970.7250000.3280000.39700014.5000009.37142826.466667
410.0007.4832.3445.1390.7483000.2344000.51390010.3213807.14634112.944585
5100.00074.00718.03455.9730.7400700.1803400.5597309.8900187.69368610.891808
61.000.000730.794147.652583.1420.7307940.1476520.5831429.8746608.18742410.418273
710.000.0007.253.0981.245.3666.007.7320.7253100.1245370.6007739.9249558.43446710.302348
8100.000.00072.103.75610.770.13961.333.6170.7210380.1077010.6133369.9410988.64817110.209113
91.000.000.000717.763.22794.898.127622.865.1000.7177630.0948980.6228659.9545888.81122610.155362
1010.000.000.0007.151.575.047848.413.8196.303.161.2280.7151580.0848410.6303169.9636978.94025910.119625


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
416151140.9375000.6875000.2500001.8750001.8333332.000000
532292090.9062500.6250000.2812501.9333331.8181822.250000
6643322110.5156250.3437500.1718751.1379311.1000001.222222
71287349240.5703120.3828120.1875002.2121212.2272732.181818
825616996730.6601560.3750000.2851562.3150681.9591843.041667
95123631811820.7089840.3535160.3554692.1479291.8854172.493151
101.0247453364090.7275390.3281250.3994142.0523421.8563542.247253
112.0481.5205979230.7421880.2915040.4506842.0402681.7767862.256724
124.0963.0571.0911.9660.7463380.2663570.4799802.0111841.8274712.130011
138.1926.1311.9594.1720.7484130.2391360.5092772.0055611.7956002.122075
1416.38412.2283.6178.6110.7463380.2207640.5255741.9944541.8463502.063998
1532.76824.3696.67217.6970.7436830.2036130.5400701.9928851.8446232.055162
1665.53648.58712.36336.2240.7413790.1886440.5527341.9938041.8529682.046901
17131.07296.75123.03573.7160.7381520.1757430.5624081.9912941.8632212.035004
18262.144192.87543.309149.5660.7357600.1652110.5705491.9935191.8801392.028949
19524.288384.25081.587302.6630.7328990.1556150.5772841.9922231.8838352.023608
201.048.576766.211154.208612.0030.7307160.1470640.5836521.9940431.8901052.022061
212.097.1521.528.279292.5411.235.7380.7287400.1394940.5892461.9945931.8970552.019170
224.194.3043.049.536555.6892.493.8470.7270660.1324870.5945791.9954051.8995252.018103
238.388.6086.086.5901.057.5885.029.0020.7255780.1260740.5995041.9959071.9032012.016564
2416.777.21612.151.2192.017.95410.133.2650.7242690.1202790.6039901.9963921.9080722.014965
2533.554.43224.257.2143.860.77820.396.4360.7229210.1150600.6078611.9962781.9132142.012820
2667.108.86448.431.8897.401.37341.030.5160.7216910.1102890.6114021.9965971.9170682.011651
27134.217.72896.713.95314.208.98582.504.9680.7205750.1058650.6147101.9969071.9197772.010820
28268.435.456193.150.47327.327.777165.822.6960.7195420.1018040.6177381.9971311.9232742.009851
29536.870.912385.782.39952.634.527333.147.8720.7185760.0980390.6205361.9973151.9260452.009061
301.073.741.824770.599.735101.521.121669.078.6140.7176770.0945490.6231281.9974981.9287932.008353
312.147.483.6481.539.389.658196.077.5261.343.312.1320.7168340.0913060.6255281.9976511.9313962.007704
324.294.967.2963.075.413.295379.121.2842.696.292.0110.7160500.0882710.6277791.9978131.9335272.007197
338.589.934.5926.144.503.384733.901.9275.410.601.4570.7153140.0854370.6298771.9979441.9357972.006682
3417.179.869.18412.277.110.8331.422.197.79110.854.913.0420.7146220.0827830.6318391.9980641.9378582.006230


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
243120102
386150303
41611470506
5322091109011
66422913010012
712849222713101412
825696455137103712
9512181889378108112
101.0243361631731541016012
112.0485972953022941028112
124.0961.0915525395351053412
138.1921.9599999609501098712
1416.3843.6171.8291.7881.808101.78712
1532.7686.6723.4003.2723.323103.32712
1665.53612.3636.3006.0636.127106.21412
17131.07223.03511.68211.35311.4771011.53612
18262.14443.30921.83421.47521.6641021.62312
19524.28881.58741.02940.55840.7871040.77812
201.048.576154.20877.35076.85877.1221077.06412
212.097.152292.541146.893145.648146.19810146.32112
224.194.304555.689279.115276.574277.70710277.96012
238.388.6081.057.588531.408526.180528.51710529.04912
2416.777.2162.017.9541.013.7841.004.1701.009.231101.008.70112
2533.554.4323.860.7781.938.4611.922.3171.930.454101.930.30212
2667.108.8647.401.3733.713.9193.687.4543.700.555103.700.79612
27134.217.72814.208.9857.128.8487.080.1377.103.852107.105.11112
28268.435.45627.327.77713.707.44813.620.32913.664.1651013.663.59012
29536.870.91252.634.52726.400.19626.234.33126.318.1821026.316.32312
301.073.741.824101.521.12150.909.27550.611.84650.759.6301050.761.46912
312.147.483.648196.077.52698.317.59197.759.93598.039.8861098.037.61812
324.294.967.296379.121.284190.082.662189.038.622189.554.94810189.566.31412
338.589.934.592733.901.927367.928.863365.973.064366.961.45410366.940.45112
3417.179.869.1841.422.197.791712.942.540709.255.251711.118.14410711.079.62512


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
241010100
382020200
4164130202
5329450315
66411470416
7128249155496
8256733538279289
9512182839964266923
101.0244091912181426714060
112.048923448475300167293163
124.0961.9669511.015601385603377
138.1924.1722.0252.1471.2208621.238852
1416.3848.6114.2364.3752.4891.8392.4671.816
1532.76817.6978.7418.9565.0073.8715.0383.781
1665.53636.22418.00418.22010.1577.99210.1837.892
17131.07273.71636.77936.93720.57216.41220.47316.259
18262.144149.56674.68574.88141.18233.46941.20633.709
19524.288302.663150.939151.72482.94068.31782.96968.437
201.048.576612.003305.471306.532166.949139.132166.936138.986
212.097.1521.235.738616.537619.201335.780282.479335.563281.916
224.194.3042.493.8471.245.6691.248.178674.024573.018673.663573.142
238.388.6085.029.0022.513.0162.515.9861.352.9991.160.7551.354.1361.161.112
2416.777.21610.133.2655.064.6875.068.5782.716.4592.348.5502.718.2772.349.979
2533.554.43220.396.43610.193.00610.203.4305.450.5364.746.8915.452.2524.746.757
2667.108.86441.030.51620.505.99620.524.52010.933.3699.580.30710.934.7159.582.125
27134.217.72882.504.96841.240.49441.264.47421.922.67519.327.92121.929.01119.325.361
28268.435.456165.822.69682.886.80482.935.89243.954.78338.952.44043.962.93838.952.535
29536.870.912333.147.872166.527.377166.620.49588.106.79778.455.01588.124.24178.461.819
301.073.741.824669.078.614334.450.608334.628.006176.590.245157.936.188176.589.857157.962.324
312.147.483.6481.343.312.132671.479.751671.832.381353.836.328317.803.238353.836.713317.835.853
324.294.967.2962.696.292.0111.347.807.5751.348.484.436708.901.490639.212.283708.919.300639.258.938
338.589.934.5925.410.601.4572.704.655.5552.705.945.9021.420.081.6531.285.162.9031.420.122.4371.285.234.464
3417.179.869.18410.854.913.0425.426.234.2505.428.678.7922.844.466.4872.582.942.8792.844.457.4322.583.046.244


8. Check for existing Integer Sequences by OEIS

Found in Database : 23, 47, 163, 5, 59, 179, 419, 239, 107, 1, 643, 347, 743, 79, 167, 439, 919, 479, 199, 103,
Found in Database : 23, 47, 163, 5, 59, 179, 419, 239, 107, 643, 347, 743, 79, 167, 439, 919, 479, 199, 103, 1063, 547, 1123, 599, 53, 619, 251, 127, 1283, 647, 1303, 131, 263, 659, 1319,
Found in Database : 5, 23, 47, 53, 59, 79, 89, 101, 103, 107, 127, 131, 149,