Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:07:07
Deutsch
29.Mar 2024

Polynom = x^2-68x-37

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) = 37 = 37
f(1) = 13 = 13
f(2) = 169 = 13*13
f(3) = 29 = 29
f(4) = 293 = 293
f(5) = 11 = 11
f(6) = 409 = 409
f(7) = 29 = 29
f(8) = 517 = 11*47
f(9) = 71 = 71
f(10) = 617 = 617
f(11) = 83 = 83
f(12) = 709 = 709
f(13) = 47 = 47
f(14) = 793 = 13*61
f(15) = 13 = 13
f(16) = 869 = 11*79
f(17) = 113 = 113
f(18) = 937 = 937
f(19) = 121 = 11*11
f(20) = 997 = 997
f(21) = 1 = 1
f(22) = 1049 = 1049
f(23) = 67 = 67
f(24) = 1093 = 1093
f(25) = 139 = 139
f(26) = 1129 = 1129
f(27) = 143 = 11*13
f(28) = 1157 = 13*89
f(29) = 73 = 73
f(30) = 1177 = 11*107
f(31) = 37 = 37
f(32) = 1189 = 29*41
f(33) = 149 = 149
f(34) = 1193 = 1193
f(35) = 149 = 149
f(36) = 1189 = 29*41
f(37) = 37 = 37
f(38) = 1177 = 11*107
f(39) = 73 = 73
f(40) = 1157 = 13*89
f(41) = 143 = 11*13
f(42) = 1129 = 1129
f(43) = 139 = 139
f(44) = 1093 = 1093
f(45) = 67 = 67
f(46) = 1049 = 1049
f(47) = 1 = 1
f(48) = 997 = 997
f(49) = 121 = 11*11
f(50) = 937 = 937
f(51) = 113 = 113
f(52) = 869 = 11*79
f(53) = 13 = 13
f(54) = 793 = 13*61
f(55) = 47 = 47
f(56) = 709 = 709
f(57) = 83 = 83
f(58) = 617 = 617
f(59) = 71 = 71
f(60) = 517 = 11*47
f(61) = 29 = 29
f(62) = 409 = 409
f(63) = 11 = 11
f(64) = 293 = 293
f(65) = 29 = 29
f(66) = 169 = 13*13
f(67) = 13 = 13
f(68) = 37 = 37
f(69) = 1 = 1
f(70) = 103 = 103
f(71) = 11 = 11
f(72) = 251 = 251
f(73) = 41 = 41
f(74) = 407 = 11*37
f(75) = 61 = 61
f(76) = 571 = 571
f(77) = 41 = 41
f(78) = 743 = 743
f(79) = 13 = 13
f(80) = 923 = 13*71
f(81) = 127 = 127
f(82) = 1111 = 11*101
f(83) = 151 = 151
f(84) = 1307 = 1307
f(85) = 11 = 11
f(86) = 1511 = 1511
f(87) = 101 = 101
f(88) = 1723 = 1723
f(89) = 229 = 229
f(90) = 1943 = 29*67
f(91) = 257 = 257
f(92) = 2171 = 13*167
f(93) = 143 = 11*13
f(94) = 2407 = 29*83
f(95) = 79 = 79
f(96) = 2651 = 11*241
f(97) = 347 = 347
f(98) = 2903 = 2903
f(99) = 379 = 379
f(100) = 3163 = 3163

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^2-68x-37

f(0)=37
f(1)=13
f(2)=1
f(3)=29
f(4)=293
f(5)=11
f(6)=409
f(7)=1
f(8)=47
f(9)=71
f(10)=617
f(11)=83
f(12)=709
f(13)=1
f(14)=61
f(15)=1
f(16)=79
f(17)=113
f(18)=937
f(19)=1
f(20)=997
f(21)=1
f(22)=1049
f(23)=67
f(24)=1093
f(25)=139
f(26)=1129
f(27)=1
f(28)=89
f(29)=73
f(30)=107
f(31)=1
f(32)=41
f(33)=149
f(34)=1193
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)=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)=103
f(71)=1
f(72)=251
f(73)=1
f(74)=1
f(75)=1
f(76)=571
f(77)=1
f(78)=743
f(79)=1
f(80)=1
f(81)=127
f(82)=101
f(83)=151
f(84)=1307
f(85)=1
f(86)=1511
f(87)=1
f(88)=1723
f(89)=229
f(90)=1
f(91)=257
f(92)=167
f(93)=1
f(94)=1
f(95)=1
f(96)=241
f(97)=347
f(98)=2903
f(99)=379

b) Substitution of the polynom
The polynom f(x)=x^2-68x-37 could be written as f(y)= y^2-1193 with x=y+34

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-34
f'(x)>2x-69 with x > 35

4. Infinity of the sequence

The mathematical proof is analogue to the proof for the polynom f(x)=x^2+1

5. Sequence of the polynom with 1

37, 13, 1, 29, 293, 11, 409, 1, 47, 71, 617, 83, 709, 1, 61, 1, 79, 113, 937, 1, 997, 1, 1049, 67, 1093, 139, 1129, 1, 89, 73, 107, 1, 41, 149, 1193, 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, 103, 1, 251, 1, 1, 1, 571, 1, 743, 1, 1, 127, 101, 151, 1307, 1, 1511, 1, 1723, 229, 1, 257, 167, 1, 1, 1, 241, 347, 2903, 379, 3163, 1, 1, 223, 337, 1, 307, 1, 4283, 277, 4583, 1, 1, 631, 1, 1, 5531, 1, 1, 1, 6203, 797, 6551, 1, 6907, 443, 661, 233, 7643, 1, 1, 1, 647, 269, 8807, 563, 1, 1, 9623, 1229, 1, 641, 283, 1, 839, 1, 11351, 1447, 1, 1, 12263, 1, 439, 1621, 281, 1, 13691, 1, 1091, 1, 14683, 1867, 1381, 1931, 1, 499, 16231, 1031, 16763, 2129, 1, 1, 17851, 1, 1, 1, 311, 1, 19543, 1, 20123, 1, 1, 1, 1, 1, 21911, 2777, 1, 1427, 23143, 733, 2161, 3011, 24407, 1, 1, 1, 25703, 1627, 643, 1, 27031, 1, 1, 1753, 1, 449, 1, 1, 1, 3767, 30491, 1, 2837, 1973, 863, 367, 1, 4129, 33403, 2111, 1, 1, 521, 401, 35671, 4507, 3313, 1151, 37223, 2351, 38011, 4801, 1, 1, 1, 1, 40423, 1, 41243, 1, 42071, 1, 1, 677, 653, 1, 1, 433, 4133, 5737, 1, 1, 47207, 1489, 48091, 6067, 1, 1, 601, 1, 3907, 3203, 1783, 6521, 52631, 6637, 1847, 1, 54503, 859, 1, 6991, 4339, 547, 557, 1, 1423, 3677, 5393, 7477, 1471, 691, 607, 3863, 62311, 1, 4871, 1, 1, 1, 65371, 1, 6037, 1, 1823, 1, 1123, 8629, 5351, 1, 6421, 1, 71707, 821, 877, 1, 73883, 1163, 1, 4721, 76091, 1, 5939, 9721, 7121, 4931, 1, 1, 80603, 1, 2819, 1, 7537, 2609, 1, 1, 85243, 10729, 86423, 1, 1109, 1, 88807, 1, 90011, 1, 8293, 883, 1, 727, 1993, 1, 94907, 11941, 8741, 12097, 1, 1, 98663, 1, 7687, 967, 101207, 1, 907, 1, 1549, 1, 1, 13217, 106391, 13381, 1, 1, 8387, 857, 1, 13879, 3019, 1277, 113051, 1777, 114407, 7193, 115771, 14557, 9011, 1, 1, 7451, 991, 3769, 1, 1, 1, 15427, 124123, 1, 1, 1, 9767, 1451, 1759, 16141, 1459, 8161, 131303, 2063, 132763, 1, 1, 16871, 1, 1, 137191, 1, 138683, 1, 1361, 1, 1171, 1, 2017, 1, 144731, 1399, 11251, 18379, 2423, 4643, 149351, 853, 150907, 1, 1, 19157, 154043, 9677, 11971, 1, 157211, 19751, 14437, 1, 3413, 1, 1, 10177, 163643, 1, 1, 1597, 1, 953, 4111, 1, 15473, 21379, 171863, 21587, 173531, 5449, 1, 11003, 1237, 1709, 1669, 2039, 1297, 11321, 1433, 2857, 183707, 23071, 1733, 1, 187163, 1, 1321, 1, 2297, 1, 192407, 1, 1, 1, 1, 6151, 1327, 1, 1, 1, 201307, 1, 2011, 1, 204923, 2339, 2617, 1, 1, 13093, 210407, 1, 1, 26647, 1, 26879, 1, 3389, 3571, 1, 219707, 27581, 3121, 27817, 17191, 1, 225383, 1, 6143, 1, 1, 28771, 231131, 7253, 6299, 14627, 235003, 2269, 1657, 29741, 5827, 1, 3049, 3779, 5923, 1051, 5209, 30727, 246811, 1, 19139, 1201, 1, 31477, 252823, 31729, 3491, 15991, 4211, 8059, 23537, 32491, 260951, 1, 20231, 1, 3631, 16631, 267131, 33521, 9283, 1, 3821, 17021, 1, 1, 21191, 2659, 3119, 1, 279707, 1, 25621, 17681, 1, 1, 286103, 35897, 6133, 1, 1, 9109, 1, 1, 1223, 36979, 26993, 1, 2647, 1, 1, 1, 1, 1, 27793, 1, 1, 1, 1, 1, 312407, 39191, 314651, 2467, 316903, 1, 24551, 40037, 29221, 1, 323707, 1, 7951, 10223, 328283, 41179, 1, 41467, 1, 1, 335207, 21023, 1, 42337, 339863, 1, 342203, 1951, 2713, 1, 1, 3347, 1, 1, 351643, 1, 354023, 1, 32401, 44701, 1609, 4091, 361211, 22651, 1, 1, 1571, 1583, 368471, 4201, 2221, 1, 33941, 1, 1, 47129, 1291, 1, 29287, 23873, 3167, 6007, 1, 4397, 388183, 48679, 10559, 1531, 393191, 1, 1, 1, 1, 49937, 36433, 25127, 13907, 1, 3793, 50891, 14083, 1, 37361, 1, 2447, 2357, 3889, 52177, 1, 52501, 5333, 1, 5807, 1, 1, 53479, 3001, 4139, 10531, 1, 434407, 1, 5987, 1481, 1, 1901, 442363, 2521, 445031, 1, 34439, 56131, 450391, 56467, 1601, 1, 455783, 28571, 41681, 1, 461207, 1, 1, 2237, 9929, 1, 1, 1, 472151, 5381, 474907, 1, 477671, 1, 16567, 1, 37171, 5507, 16759, 1, 1, 15319, 491611, 61627, 3557, 61979, 1, 15583, 1, 2411, 1, 1, 8291, 63397, 1979, 1, 511463, 4007, 5779, 5861, 517207, 4987, 3637, 1, 6301, 1, 1, 2273, 528791, 1, 48337, 1, 534631, 1523, 41351, 1, 13183, 1831, 2437, 17029, 13327, 1, 2399, 1861, 1, 1, 3727, 2677, 1, 8747, 561307, 70351, 1, 1, 567323, 1, 1, 1, 573371, 71861, 1, 5557, 579451, 3301, 1873, 18251, 53233, 73387, 588631, 73771, 7129, 18539, 7529, 1, 1, 74929, 600983, 1, 12853, 37853, 16411, 1, 610331, 76487, 613463, 1, 47431, 1, 1, 38833, 1, 1, 2687, 1, 629243, 1, 57493, 19813, 8707, 1, 49139, 80051, 642011, 20113, 4273, 40427, 9133, 1, 1, 1, 2053, 1, 50627, 1, 22807, 1, 664663, 2251, 1, 5231, 14281, 3823, 16451, 1, 677783, 1, 52391, 1, 2663, 1949, 687707, 86171, 5711, 86587, 11383, 21751, 697703, 43711, 53927, 1, 64037, 88261, 5573, 1, 711143, 5569, 1747, 89527, 19403, 1, 721307, 1, 1, 1, 1789, 91229, 7243, 1, 734971, 1, 1, 1, 6131, 2267, 57331, 1, 748763, 1, 25939, 47123, 755707, 1, 1, 8647, 12503, 47777, 69653, 1, 59207, 96431, 773207, 1, 11593, 3041, 1, 1, 783803, 1, 1, 98641, 1, 1, 794471, 1, 798043, 1, 9007, 3463, 1, 25219, 19727, 1, 3371, 7829, 1, 102229, 1, 51341, 823271, 1, 826907, 103591, 830551, 104047, 17749, 13063, 64451, 1, 2293, 2243, 76837, 1, 22943, 53171, 852583, 26701, 29527, 1, 2113, 8287, 1, 2459, 867431, 54331, 871163, 1, 1, 109597, 878651, 5003, 1, 1, 6197, 8539, 1, 2719, 893723, 6997, 12641, 1, 81937, 112901, 905111, 1, 12451, 1, 1, 1, 916571, 1, 920407, 1, 9151, 1, 1, 1,

6. Sequence of the polynom (only primes)

37, 13, 29, 293, 11, 409, 47, 71, 617, 83, 709, 61, 79, 113, 937, 997, 1049, 67, 1093, 139, 1129, 89, 73, 107, 41, 149, 1193, 103, 251, 571, 743, 127, 101, 151, 1307, 1511, 1723, 229, 257, 167, 241, 347, 2903, 379, 3163, 223, 337, 307, 4283, 277, 4583, 631, 5531, 6203, 797, 6551, 6907, 443, 661, 233, 7643, 647, 269, 8807, 563, 9623, 1229, 641, 283, 839, 11351, 1447, 12263, 439, 1621, 281, 13691, 1091, 14683, 1867, 1381, 1931, 499, 16231, 1031, 16763, 2129, 17851, 311, 19543, 20123, 21911, 2777, 1427, 23143, 733, 2161, 3011, 24407, 25703, 1627, 643, 27031, 1753, 449, 3767, 30491, 2837, 1973, 863, 367, 4129, 33403, 2111, 521, 401, 35671, 4507, 3313, 1151, 37223, 2351, 38011, 4801, 40423, 41243, 42071, 677, 653, 433, 4133, 5737, 47207, 1489, 48091, 6067, 601, 3907, 3203, 1783, 6521, 52631, 6637, 1847, 54503, 859, 6991, 4339, 547, 557, 1423, 3677, 5393, 7477, 1471, 691, 607, 3863, 62311, 4871, 65371, 6037, 1823, 1123, 8629, 5351, 6421, 71707, 821, 877, 73883, 1163, 4721, 76091, 5939, 9721, 7121, 4931, 80603, 2819, 7537, 2609, 85243, 10729, 86423, 1109, 88807, 90011, 8293, 883, 727, 1993, 94907, 11941, 8741, 12097, 98663, 7687, 967, 101207, 907, 1549, 13217, 106391, 13381, 8387, 857, 13879, 3019, 1277, 113051, 1777, 114407, 7193, 115771, 14557, 9011, 7451, 991, 3769, 15427, 124123, 9767, 1451, 1759, 16141, 1459, 8161, 131303, 2063, 132763, 16871, 137191, 138683, 1361, 1171, 2017, 144731, 1399, 11251, 18379, 2423, 4643, 149351, 853, 150907, 19157, 154043, 9677, 11971, 157211, 19751, 14437, 3413, 10177, 163643, 1597, 953, 4111, 15473, 21379, 171863, 21587, 173531, 5449, 11003, 1237, 1709, 1669, 2039, 1297, 11321, 1433, 2857, 183707, 23071, 1733, 187163, 1321, 2297, 192407, 6151, 1327, 201307, 2011, 204923, 2339, 2617, 13093, 210407, 26647, 26879, 3389, 3571, 219707, 27581, 3121, 27817, 17191, 225383, 6143, 28771, 231131, 7253, 6299, 14627, 235003, 2269, 1657, 29741, 5827, 3049, 3779, 5923, 1051, 5209, 30727, 246811, 19139, 1201, 31477, 252823, 31729, 3491, 15991, 4211, 8059, 23537, 32491, 260951, 20231, 3631, 16631, 267131, 33521, 9283, 3821, 17021, 21191, 2659, 3119, 279707, 25621, 17681, 286103, 35897, 6133, 9109, 1223, 36979, 26993, 2647, 27793, 312407, 39191, 314651, 2467, 316903, 24551, 40037, 29221, 323707, 7951, 10223, 328283, 41179, 41467, 335207, 21023, 42337, 339863, 342203, 1951, 2713, 3347, 351643, 354023, 32401, 44701, 1609, 4091, 361211, 22651, 1571, 1583, 368471, 4201, 2221, 33941, 47129, 1291, 29287, 23873, 3167, 6007, 4397, 388183, 48679, 10559, 1531, 393191, 49937, 36433, 25127, 13907, 3793, 50891, 14083, 37361, 2447, 2357, 3889, 52177, 52501, 5333, 5807, 53479, 3001, 4139, 10531, 434407, 5987, 1481, 1901, 442363, 2521, 445031, 34439, 56131, 450391, 56467, 1601, 455783, 28571, 41681, 461207, 2237, 9929, 472151, 5381, 474907, 477671, 16567, 37171, 5507, 16759, 15319, 491611, 61627, 3557, 61979, 15583, 2411, 8291, 63397, 1979, 511463, 4007, 5779, 5861, 517207, 4987, 3637, 6301, 2273, 528791, 48337, 534631, 1523, 41351, 13183, 1831, 2437, 17029, 13327, 2399, 1861, 3727, 2677, 8747, 561307, 70351, 567323, 573371, 71861, 5557, 579451, 3301, 1873, 18251, 53233, 73387, 588631, 73771, 7129, 18539, 7529, 74929, 600983, 12853, 37853, 16411, 610331, 76487, 613463, 47431, 38833, 2687, 629243, 57493, 19813, 8707, 49139, 80051, 642011, 20113, 4273, 40427, 9133, 2053, 50627, 22807, 664663, 2251, 5231, 14281, 3823, 16451, 677783, 52391, 2663, 1949, 687707, 86171, 5711, 86587, 11383, 21751, 697703, 43711, 53927, 64037, 88261, 5573, 711143, 5569, 1747, 89527, 19403, 721307, 1789, 91229, 7243, 734971, 6131, 2267, 57331, 748763, 25939, 47123, 755707, 8647, 12503, 47777, 69653, 59207, 96431, 773207, 11593, 3041, 783803, 98641, 794471, 798043, 9007, 3463, 25219, 19727, 3371, 7829, 102229, 51341, 823271, 826907, 103591, 830551, 104047, 17749, 13063, 64451, 2293, 2243, 76837, 22943, 53171, 852583, 26701, 29527, 2113, 8287, 2459, 867431, 54331, 871163, 109597, 878651, 5003, 6197, 8539, 2719, 893723, 6997, 12641, 81937, 112901, 905111, 12451, 916571, 920407, 9151,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-68x-37 and
the reducible primes which appear as divisor for the first time
p | x^2-68x-37 and p < x^2-68x-37

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)
1109810.9000000.8000000.1000000.0000000.0000000.000000
2100393360.3900000.3300000.0600004.3333334.1250006.000000
31.0006023532490.6020000.3530000.24900015.43589810.69697041.500000
410.0006.5782.6143.9640.6578000.2614000.39640010.9269107.40509915.919679
5100.00067.12719.93647.1910.6712700.1993600.47191010.2047747.62662611.904894
61.000.000675.173160.325514.8480.6751730.1603250.51484810.0581448.04198510.909877
710.000.0006.779.3711.343.8805.435.4910.6779370.1343880.54354910.0409398.38222410.557467
8100.000.00067.986.42811.565.86356.420.5650.6798640.1156590.56420610.0284288.60632110.380031
91.000.000.000681.300.868101.497.924579.802.9440.6813010.1014980.57980310.0211308.77564610.276447
1010.000.000.0006.824.695.436904.399.5605.920.295.8760.6824700.0904400.59203010.0171548.91052310.210876


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
244401.0000001.0000000.0000002.0000002.000000-nan
387610.8750000.7500000.1250001.7500001.500000inf
416131030.8125000.6250000.1875001.8571431.6666673.000000
532241950.7500000.5937500.1562501.8461541.9000001.666667
664262150.4062500.3281250.0781251.0833331.1052631.000000
7128544590.4218750.3515620.0703122.0769232.1428571.800000
8256122100220.4765620.3906250.0859382.2592592.2222222.444444
9512287193940.5605470.3769530.1835942.3524591.9300004.272727
101.0246173602570.6025390.3515620.2509772.1498261.8652852.734043
112.0481.2786626160.6240230.3232420.3007812.0713131.8388892.396887
124.0962.6411.2031.4380.6447750.2937010.3510742.0665101.8172212.334416
138.1925.3682.1993.1690.6552730.2684330.3868412.0325631.8279302.203755
1416.38410.8644.0586.8060.6630860.2476810.4154052.0238451.8453842.147681
1532.76821.8517.43414.4170.6668400.2268680.4399722.0113221.8319372.118278
1665.53643.90113.69830.2030.6698760.2090150.4608612.0091071.8426152.094957
17131.07288.05325.49262.5610.6717910.1944890.4773032.0057181.8610022.071351
18262.144176.48047.567128.9130.6732180.1814540.4917642.0042471.8659582.060597
19524.288353.48689.093264.3930.6742210.1699310.5042902.0029801.8730002.050941
201.048.576707.955167.478540.4770.6751590.1597190.5154392.0027811.8798112.044218
212.097.1521.417.860316.9281.100.9320.6760880.1511230.5249652.0027541.8923562.036964
224.194.3042.839.530600.7312.238.7990.6769970.1432250.5337712.0026871.8954812.033549
238.388.6085.685.3091.141.7014.543.6080.6777420.1361010.5416402.0022011.9005192.029485
2416.777.21611.381.5092.175.3859.206.1240.6783910.1296630.5487282.0019161.9053892.026170
2533.554.43222.783.1434.156.31018.626.8330.6789910.1238680.5551232.0017681.9106092.023309
2667.108.86445.603.8567.953.95537.649.9010.6795500.1185230.5610272.0016491.9137062.021272
27134.217.72891.275.69515.249.95176.025.7440.6800570.1136210.5664362.0014911.9172792.019281
28268.435.456182.675.11529.290.033153.385.0820.6805180.1091140.5714042.0013561.9206642.017541
29536.870.912365.576.77856.351.400309.225.3780.6809400.1049630.5759772.0012401.9239102.016007
301.073.741.824731.586.096108.572.180623.013.9160.6813430.1011160.5802272.0011831.9266992.014757
312.147.483.6481.463.972.708209.454.7831.254.517.9250.6817150.0975350.5841802.0010941.9291752.013628
324.294.967.2962.929.472.776404.594.9232.524.877.8530.6820710.0942020.5878692.0010431.9316582.012628
338.589.934.5925.861.769.273782.491.3335.079.277.9400.6824000.0910940.5913062.0009641.9340122.011693
3417.179.869.18411.728.909.0641.514.989.72110.213.919.3430.6827120.0881840.5945282.0009161.9361102.010900


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
122200020
244220040
386331140
41610462251
532191187471
6642111108481
7128452025917118
8256100435718351730
95121938311035713057
101.0243601611995213852118
112.0486622913719623694236
124.0961.203536667169435183416
138.1922.1999751.224319791305784
1416.3844.0581.8192.2395561.4855501.467
1532.7687.4343.3414.0931.0252.7299962.684
1665.53613.6986.1467.5521.8414.9901.8365.031
17131.07225.49211.50113.9913.4219.3143.4389.319
18262.14447.56721.43026.1376.38717.4546.28617.440
19524.28889.09340.02749.06611.79632.75711.75832.782
201.048.576167.47875.31992.15922.20461.64422.04161.589
212.097.152316.928142.596174.33241.777116.93341.731116.487
224.194.304600.731270.080330.65179.017221.31178.982221.421
238.388.6081.141.701513.249628.452149.903420.718150.068421.012
2416.777.2162.175.385977.0041.198.381285.078802.158285.505802.644
2533.554.4324.156.3101.865.2012.291.109543.1111.534.436544.1901.534.573
2667.108.8647.953.9553.567.4454.386.5101.037.3932.939.0911.037.7862.939.685
27134.217.72815.249.9516.835.8838.414.0681.985.5795.638.9381.985.1455.640.289
28268.435.45629.290.03313.123.11716.166.9163.806.84510.837.6653.806.71910.838.804
29536.870.91256.351.40025.240.93031.110.4707.314.05820.861.7787.312.75420.862.810
301.073.741.824108.572.18048.615.20559.956.97514.071.35440.217.39314.072.02440.211.409
312.147.483.648209.454.78393.766.592115.688.19127.112.24477.618.25627.115.06677.609.217
324.294.967.296404.594.923181.077.903223.517.02052.309.916149.991.40752.309.697149.983.903
338.589.934.592782.491.333350.133.183432.358.150101.056.122290.193.686101.056.353290.185.172
3417.179.869.1841.514.989.721677.764.987837.224.734195.450.753562.040.740195.468.863562.029.365


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
240000000
381010001
4163210012
5325231112
6645231112
71289633222
82562211116466
951294544021232426
101.02425713911856686370
112.048616322294149151154162
124.0961.438732706353364344377
138.1923.1691.5871.582776830768795
1416.3846.8063.4303.3761.6461.7531.6951.712
1532.76814.4177.2647.1533.5643.6493.5743.630
1665.53630.20315.10315.1007.4517.5857.5117.656
17131.07262.56131.41231.14915.51215.77715.41515.857
18262.144128.91364.63764.27631.93032.45132.00032.532
19524.288264.393132.518131.87565.62466.43565.55366.781
201.048.576540.477270.980269.497134.393135.760134.222136.102
212.097.1521.100.932551.852549.080274.052276.411273.567276.902
224.194.3042.238.7991.122.1091.116.690557.089561.861557.323562.526
238.388.6084.543.6082.278.0062.265.6021.130.4791.140.9951.131.6951.140.439
2416.777.2169.206.1244.614.9014.591.2232.291.8652.310.7952.292.3612.311.103
2533.554.43218.626.8339.334.6379.292.1964.637.6614.676.2144.638.7914.674.167
2667.108.86437.649.90118.869.53618.780.3659.376.0239.447.8949.380.7739.445.211
27134.217.72876.025.74438.098.58537.927.15918.932.75819.076.61718.945.63219.070.737
28268.435.456153.385.08276.856.88276.528.20038.216.37838.476.33638.219.42138.472.947
29536.870.912309.225.378154.931.984154.293.39477.057.95177.565.57577.045.13477.556.718
301.073.741.824623.013.916312.113.480310.900.436155.259.390156.244.490155.266.662156.243.374
312.147.483.6481.254.517.925628.404.807626.113.118312.672.410314.561.915312.707.755314.575.845
324.294.967.2962.524.877.8531.264.671.3001.260.206.553629.371.773633.040.781629.420.997633.044.302
338.589.934.5925.079.277.9402.543.950.8052.535.327.1351.266.256.3911.273.351.4061.266.312.7371.273.357.406
3417.179.869.18410.213.919.3435.115.310.1905.098.609.1532.546.655.7122.560.325.7262.546.653.2332.560.284.672


8. Check for existing Integer Sequences by OEIS

Found in Database : 37, 13, 1, 29, 293, 11, 409, 1, 47, 71, 617, 83, 709, 1, 61, 1, 79, 113, 937, 1,
Found in Database : 37, 13, 29, 293, 11, 409, 47, 71, 617, 83, 709, 61, 79, 113, 937, 997, 1049, 67, 1093, 139, 1129, 89, 73, 107, 41, 149, 1193,
Found in Database : 11, 13, 29, 37, 41, 47, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 139, 149,