Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:01:40
Deutsch
29.Mar 2024

Polynom = x^2-276x+11

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) = 11 = 11
f(1) = 33 = 3*11
f(2) = 537 = 3*179
f(3) = 101 = 101
f(4) = 1077 = 3*359
f(5) = 21 = 3*7
f(6) = 1609 = 1609
f(7) = 117 = 3*3*13
f(8) = 2133 = 3*3*3*79
f(9) = 299 = 13*23
f(10) = 2649 = 3*883
f(11) = 363 = 3*11*11
f(12) = 3157 = 7*11*41
f(13) = 213 = 3*71
f(14) = 3657 = 3*23*53
f(15) = 61 = 61
f(16) = 4149 = 3*3*461
f(17) = 549 = 3*3*61
f(18) = 4633 = 41*113
f(19) = 609 = 3*7*29
f(20) = 5109 = 3*13*131
f(21) = 167 = 167
f(22) = 5577 = 3*11*13*13
f(23) = 363 = 3*11*11
f(24) = 6037 = 6037
f(25) = 783 = 3*3*3*29
f(26) = 6489 = 3*3*7*103
f(27) = 839 = 839
f(28) = 6933 = 3*2311
f(29) = 447 = 3*149
f(30) = 7369 = 7369
f(31) = 237 = 3*79
f(32) = 7797 = 3*23*113
f(33) = 1001 = 7*11*13
f(34) = 8217 = 3*3*11*83
f(35) = 1053 = 3*3*3*3*13
f(36) = 8629 = 8629
f(37) = 69 = 3*23
f(38) = 9033 = 3*3011
f(39) = 577 = 577
f(40) = 9429 = 3*7*449
f(41) = 1203 = 3*401
f(42) = 9817 = 9817
f(43) = 1251 = 3*3*139
f(44) = 10197 = 3*3*11*103
f(45) = 649 = 11*59
f(46) = 10569 = 3*13*271
f(47) = 21 = 3*7
f(48) = 10933 = 13*29*29
f(49) = 1389 = 3*463
f(50) = 11289 = 3*53*71
f(51) = 1433 = 1433
f(52) = 11637 = 3*3*3*431
f(53) = 369 = 3*3*41
f(54) = 11977 = 7*29*59
f(55) = 759 = 3*11*23
f(56) = 12309 = 3*11*373
f(57) = 1559 = 1559
f(58) = 12633 = 3*4211
f(59) = 1599 = 3*13*41
f(60) = 12949 = 23*563
f(61) = 819 = 3*3*7*13
f(62) = 13257 = 3*3*3*491
f(63) = 419 = 419
f(64) = 13557 = 3*4519
f(65) = 1713 = 3*571
f(66) = 13849 = 11*1259
f(67) = 1749 = 3*11*53
f(68) = 14133 = 3*7*673
f(69) = 223 = 223
f(70) = 14409 = 3*3*1601
f(71) = 909 = 3*3*101
f(72) = 14677 = 13*1129
f(73) = 1851 = 3*617
f(74) = 14937 = 3*13*383
f(75) = 1883 = 7*269
f(76) = 15189 = 3*61*83
f(77) = 957 = 3*11*29
f(78) = 15433 = 11*23*61
f(79) = 243 = 3*3*3*3*3
f(80) = 15669 = 3*3*1741
f(81) = 1973 = 1973
f(82) = 15897 = 3*7*757
f(83) = 2001 = 3*23*29
f(84) = 16117 = 71*227
f(85) = 507 = 3*13*13
f(86) = 16329 = 3*5443
f(87) = 1027 = 13*79
f(88) = 16533 = 3*3*11*167
f(89) = 2079 = 3*3*3*7*11
f(90) = 16729 = 16729
f(91) = 2103 = 3*701
f(92) = 16917 = 3*5639
f(93) = 1063 = 1063
f(94) = 17097 = 3*41*139
f(95) = 537 = 3*179
f(96) = 17269 = 7*2467
f(97) = 2169 = 3*3*241
f(98) = 17433 = 3*3*13*149
f(99) = 2189 = 11*199
f(100) = 17589 = 3*11*13*41

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^2-276x+11

f(0)=11
f(1)=3
f(2)=179
f(3)=101
f(4)=359
f(5)=7
f(6)=1609
f(7)=13
f(8)=79
f(9)=23
f(10)=883
f(11)=1
f(12)=41
f(13)=71
f(14)=53
f(15)=61
f(16)=461
f(17)=1
f(18)=113
f(19)=29
f(20)=131
f(21)=167
f(22)=1
f(23)=1
f(24)=6037
f(25)=1
f(26)=103
f(27)=839
f(28)=2311
f(29)=149
f(30)=7369
f(31)=1
f(32)=1
f(33)=1
f(34)=83
f(35)=1
f(36)=8629
f(37)=1
f(38)=3011
f(39)=577
f(40)=449
f(41)=401
f(42)=9817
f(43)=139
f(44)=1
f(45)=59
f(46)=271
f(47)=1
f(48)=1
f(49)=463
f(50)=1
f(51)=1433
f(52)=431
f(53)=1
f(54)=1
f(55)=1
f(56)=373
f(57)=1559
f(58)=4211
f(59)=1
f(60)=563
f(61)=1
f(62)=491
f(63)=419
f(64)=4519
f(65)=571
f(66)=1259
f(67)=1
f(68)=673
f(69)=223
f(70)=1601
f(71)=1
f(72)=1129
f(73)=617
f(74)=383
f(75)=269
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1741
f(81)=1973
f(82)=757
f(83)=1
f(84)=227
f(85)=1
f(86)=5443
f(87)=1
f(88)=1
f(89)=1
f(90)=16729
f(91)=701
f(92)=5639
f(93)=1063
f(94)=1
f(95)=1
f(96)=2467
f(97)=241
f(98)=1
f(99)=199

b) Substitution of the polynom
The polynom f(x)=x^2-276x+11 could be written as f(y)= y^2-19033 with x=y+138

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-138
f'(x)>2x-277 with x > 138

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

11, 3, 179, 101, 359, 7, 1609, 13, 79, 23, 883, 1, 41, 71, 53, 61, 461, 1, 113, 29, 131, 167, 1, 1, 6037, 1, 103, 839, 2311, 149, 7369, 1, 1, 1, 83, 1, 8629, 1, 3011, 577, 449, 401, 9817, 139, 1, 59, 271, 1, 1, 463, 1, 1433, 431, 1, 1, 1, 373, 1559, 4211, 1, 563, 1, 491, 419, 4519, 571, 1259, 1, 673, 223, 1601, 1, 1129, 617, 383, 269, 1, 1, 1, 1, 1741, 1973, 757, 1, 227, 1, 5443, 1, 1, 1, 16729, 701, 5639, 1063, 1, 1, 2467, 241, 1, 199, 1, 1, 17737, 1, 1, 2243, 1, 251, 18133, 379, 1, 1, 211, 1, 18457, 257, 229, 1, 6211, 389, 353, 1, 569, 2351, 1, 1, 1453, 197, 6311, 1, 6323, 1, 157, 1, 2113, 1, 6343, 1, 2719, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 761, 1, 1, 1, 3467, 1, 1, 547, 1, 1, 5303, 1, 659, 1, 1, 1, 7211, 1, 2621, 1, 947, 1, 1, 397, 1, 1, 3517, 1, 11243, 1, 1327, 1, 4217, 1, 13367, 1, 1, 1, 1, 1, 1, 1, 5437, 1, 5689, 727, 1621, 1, 1, 1187, 1, 1, 877, 857, 1, 2671, 1, 1, 2053, 479, 1, 1, 1, 1, 3581, 1, 2879, 823, 811, 1, 1201, 1, 9497, 1, 1, 1, 1, 1279, 1483, 1, 971, 677, 1, 1, 1, 613, 11597, 1, 1, 1, 1, 1, 1, 1, 38567, 1627, 13177, 1, 587, 1, 3191, 1, 1, 1, 14489, 1, 6353, 937, 15161, 5749, 5167, 653, 1, 1, 16189, 1, 719, 2089, 1747, 1, 821, 1, 17597, 1, 1, 2267, 1409, 991, 1, 1, 57143, 601, 19417, 7351, 1, 1, 1, 1, 1, 1, 20921, 1, 809, 2687, 1, 1, 1, 1, 9629, 2833, 22861, 8647, 1789, 733, 1, 1, 1, 827, 1, 3083, 1223, 1567, 1, 2389, 8563, 1, 78311, 1, 2411, 1, 929, 1697, 797, 1, 1, 10501, 1, 1777, 1, 1, 2237, 10987, 9839, 1, 89867, 1, 1, 5741, 30841, 1, 1, 1, 1, 1, 32189, 1013, 97931, 1, 1, 1, 1, 1, 102071, 2141, 1, 1, 2689, 1, 4621, 743, 3989, 1, 3307, 1, 110567, 4637, 37337, 1, 1801, 1, 2803, 1, 3527, 1, 1, 1, 1, 1, 1, 15199, 3137, 1, 1361, 1, 1, 7883, 1, 1, 769, 1, 43321, 8171, 1, 1, 12097, 1, 2137, 16927, 1, 1427, 1, 2887, 46457, 2503, 1, 1, 997, 1, 3697, 1, 1, 1, 1, 1, 5521, 1171, 4567, 1, 152363, 1, 51341, 1489, 17299, 1087, 22481, 1, 53017, 1, 4871, 6733, 2663, 1, 1, 10313, 4253, 6947, 5779, 7019, 1, 967, 1, 1, 1, 7237, 58189, 1, 1109, 1, 178103, 1, 1, 1, 1, 7607, 1, 1, 61757, 2909, 1, 1, 1321, 1, 63577, 1, 64189, 1, 27773, 2713, 2423, 1, 66041, 1, 18181, 1, 67289, 1, 22639, 2843, 8941, 1, 9883, 1, 1, 1, 1, 983, 1823, 1913, 71741, 1, 217163, 1, 73037, 1, 1, 1, 9697, 1, 1, 1, 2609, 1, 1, 1597, 2333, 1, 1, 9749, 33569, 9833, 1, 3719, 1, 1667, 241067, 1, 1, 30517, 1151, 1, 247223, 1, 1319, 1, 3643, 1, 23041, 1, 1, 1, 28627, 3593, 259751, 10867, 2129, 16433, 1, 1, 1861, 1237, 1, 1, 3919, 1, 272567, 5701, 91577, 1499, 2797, 1, 39869, 1, 1, 1103, 1549, 1, 1, 1, 1, 1, 1, 6067, 1, 12227, 1, 1607, 10993, 2069, 299063, 1, 1, 1, 101197, 12697, 305867, 1, 1, 1487, 14783, 12983, 312743, 1, 9547, 19763, 1, 1, 319691, 1, 1, 1, 8317, 3391, 4243, 1, 36563, 41281, 110477, 13859, 333803, 1, 112061, 1, 37619, 1, 1, 1, 1, 1, 1459, 7229, 12007, 1, 38959, 1, 1, 1, 1, 1, 119321, 1, 1907, 5023, 1, 3793, 121789, 2083, 1, 1, 6277, 1, 4603, 1, 125117, 3923, 2347, 15797, 11527, 4337, 1, 1, 1, 1, 9949, 48661, 1289, 1, 3001, 1, 1, 6203, 132761, 16649, 400871, 1, 134489, 1, 45119, 1, 1, 1553, 1931, 51577, 1, 1, 5273, 1451, 3583, 52567, 1, 1, 38593, 1, 1, 26783, 1, 1, 1, 1, 2459, 2099, 1, 1, 440651, 1, 16421, 2417, 1, 1, 1, 1, 21503, 5147, 1, 6329, 35159, 9551, 153277, 1, 3761, 19333, 465383, 2161, 4729, 1, 1, 1, 1, 1523, 6907, 1, 53267, 1, 482231, 1, 14699, 60811, 1579, 20389, 5393, 1709, 4219, 30941, 1, 20747, 4127, 1, 1, 31481, 1, 1759, 6121, 1, 1, 1, 1663, 1, 4271, 1, 19249, 1, 174221, 21839, 2357, 1, 13553, 1, 1, 1, 1987, 1, 179161, 16843, 1, 1613, 23629, 1, 60719, 1, 16651, 1, 78929, 1, 185177, 1, 1, 2593, 7109, 1, 1, 3217, 14557, 23719, 1, 7949, 9109, 35963, 192317, 1, 5743, 2203, 1, 1, 21713, 1, 20323, 1, 4817, 74257, 1, 1, 1877, 1, 2477, 18859, 1, 1, 1, 25409, 1, 4789, 2969, 1, 1, 1, 1, 1, 208057, 13037, 627383, 1, 70067, 6079, 19207, 1, 637067, 6653, 30491, 1, 3109, 1, 646823, 1, 1, 1, 1, 6823, 656651, 1, 73327, 82699, 31583, 3463, 1787, 13921, 1, 1, 8311, 1, 676523, 1, 226621, 1, 7853, 28537, 98081, 1, 1, 21613, 1, 14479, 53591, 4157, 233357, 87721, 78163, 1, 1, 1, 1, 1, 237901, 2293, 17491, 1, 1, 6449,

6. Sequence of the polynom (only primes)

11, 3, 179, 101, 359, 7, 1609, 13, 79, 23, 883, 41, 71, 53, 61, 461, 113, 29, 131, 167, 6037, 103, 839, 2311, 149, 7369, 83, 8629, 3011, 577, 449, 401, 9817, 139, 59, 271, 463, 1433, 431, 373, 1559, 4211, 563, 491, 419, 4519, 571, 1259, 673, 223, 1601, 1129, 617, 383, 269, 1741, 1973, 757, 227, 5443, 16729, 701, 5639, 1063, 2467, 241, 199, 17737, 2243, 251, 18133, 379, 211, 18457, 257, 229, 6211, 389, 353, 569, 2351, 1453, 197, 6311, 6323, 157, 2113, 6343, 2719, 761, 3467, 547, 5303, 659, 7211, 2621, 947, 397, 3517, 11243, 1327, 4217, 13367, 5437, 5689, 727, 1621, 1187, 877, 857, 2671, 2053, 479, 3581, 2879, 823, 811, 1201, 9497, 1279, 1483, 971, 677, 613, 11597, 38567, 1627, 13177, 587, 3191, 14489, 6353, 937, 15161, 5749, 5167, 653, 16189, 719, 2089, 1747, 821, 17597, 2267, 1409, 991, 57143, 601, 19417, 7351, 20921, 809, 2687, 9629, 2833, 22861, 8647, 1789, 733, 827, 3083, 1223, 1567, 2389, 8563, 78311, 2411, 929, 1697, 797, 10501, 1777, 2237, 10987, 9839, 89867, 5741, 30841, 32189, 1013, 97931, 102071, 2141, 2689, 4621, 743, 3989, 3307, 110567, 4637, 37337, 1801, 2803, 3527, 15199, 3137, 1361, 7883, 769, 43321, 8171, 12097, 2137, 16927, 1427, 2887, 46457, 2503, 997, 3697, 5521, 1171, 4567, 152363, 51341, 1489, 17299, 1087, 22481, 53017, 4871, 6733, 2663, 10313, 4253, 6947, 5779, 7019, 967, 7237, 58189, 1109, 178103, 7607, 61757, 2909, 1321, 63577, 64189, 27773, 2713, 2423, 66041, 18181, 67289, 22639, 2843, 8941, 9883, 983, 1823, 1913, 71741, 217163, 73037, 9697, 2609, 1597, 2333, 9749, 33569, 9833, 3719, 1667, 241067, 30517, 1151, 247223, 1319, 3643, 23041, 28627, 3593, 259751, 10867, 2129, 16433, 1861, 1237, 3919, 272567, 5701, 91577, 1499, 2797, 39869, 1103, 1549, 6067, 12227, 1607, 10993, 2069, 299063, 101197, 12697, 305867, 1487, 14783, 12983, 312743, 9547, 19763, 319691, 8317, 3391, 4243, 36563, 41281, 110477, 13859, 333803, 112061, 37619, 1459, 7229, 12007, 38959, 119321, 1907, 5023, 3793, 121789, 2083, 6277, 4603, 125117, 3923, 2347, 15797, 11527, 4337, 9949, 48661, 1289, 3001, 6203, 132761, 16649, 400871, 134489, 45119, 1553, 1931, 51577, 5273, 1451, 3583, 52567, 38593, 26783, 2459, 2099, 440651, 16421, 2417, 21503, 5147, 6329, 35159, 9551, 153277, 3761, 19333, 465383, 2161, 4729, 1523, 6907, 53267, 482231, 14699, 60811, 1579, 20389, 5393, 1709, 4219, 30941, 20747, 4127, 31481, 1759, 6121, 1663, 4271, 19249, 174221, 21839, 2357, 13553, 1987, 179161, 16843, 1613, 23629, 60719, 16651, 78929, 185177, 2593, 7109, 3217, 14557, 23719, 7949, 9109, 35963, 192317, 5743, 2203, 21713, 20323, 4817, 74257, 1877, 2477, 18859, 25409, 4789, 2969, 208057, 13037, 627383, 70067, 6079, 19207, 637067, 6653, 30491, 3109, 646823, 6823, 656651, 73327, 82699, 31583, 3463, 1787, 13921, 8311, 676523, 226621, 7853, 28537, 98081, 21613, 14479, 53591, 4157, 233357, 87721, 78163, 237901, 2293, 17491, 6449,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-276x+11 and
the reducible primes which appear as divisor for the first time
p | x^2-276x+11 and p < x^2-276x+11

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)
11010281.0000000.2000001.0000000.0000000.0000000.000000
2100647570.6400000.0700000.6400006.4000003.5000007.125000
31.000419433760.4190000.0430000.4190006.5468756.1428576.596491
410.0005.7083325.3760.5708000.0332000.57080013.6229117.72093014.297873
5100.00060.7772.66658.1110.6077700.0266600.60777010.6476888.03012110.809338
61.000.000625.02921.891603.1380.6250290.0218910.62502910.2839738.21117810.379067
710.000.0006.355.646184.9516.170.6950.6355650.0184950.63556510.1685628.44872310.230984
8100.000.00064.317.3311.601.06562.716.2660.6431730.0160110.64317310.1197168.65669810.163566
91.000.000.000648.940.62714.132.036634.808.5910.6489410.0141320.64894110.0896698.82664710.121913
1010.000.000.0006.535.220.056126.469.8926.408.750.1640.6535220.0126470.65352210.0705998.94916210.095563


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)
123121.5000000.5000001.0000000.0000000.0000000.000000
245141.2500000.2500001.0000001.6666671.0000002.000000
388261.0000000.2500000.7500001.6000002.0000001.500000
416152130.9375000.1250000.8125001.8750001.0000002.166667
532244200.7500000.1250000.6250001.6000002.0000001.538462
664436370.6718750.0937500.5781251.7916671.5000001.850000
71287810680.6093750.0781250.5312501.8139541.6666671.837838
82568210720.3203120.0390620.2812501.0512821.0000001.058824
9512173221510.3378910.0429690.2949222.1097562.2000002.097222
101.024434443900.4238280.0429690.3808592.5086712.0000002.582782
112.0481.000819190.4882810.0395510.4487302.3041471.8409092.356410
124.0962.1931522.0410.5354000.0371090.4982912.1930001.8765432.220892
138.1924.6332804.3530.5655520.0341800.5313722.1126311.8421052.132778
1416.3849.5545379.0170.5831300.0327760.5503542.0621631.9178572.071445
1532.76819.48196418.5170.5945130.0294190.5650942.0390411.7951582.053566
1665.53639.5591.82037.7390.6036220.0277710.5758512.0306451.8879672.038073
17131.07280.0133.39676.6170.6104510.0259090.5845412.0226241.8659342.030181
18262.144161.6716.383155.2880.6167260.0243490.5923772.0205591.8795642.026809
19524.288325.81612.111313.7050.6214450.0231000.5983452.0153031.8973842.020150
201.048.576655.75922.904632.8550.6253810.0218430.6035382.0126671.8911732.017357
212.097.1521.318.88243.4441.275.4380.6288920.0207160.6081762.0112301.8967872.015372
224.194.3042.651.18582.3922.568.7930.6320920.0196440.6124482.0101761.8965102.014048
238.388.6085.326.129156.8545.169.2750.6349240.0186980.6162262.0089621.9037532.012336
2416.777.21610.695.151299.24910.395.9020.6374810.0178370.6196442.0080531.9078192.011095
2533.554.43221.469.945573.14020.896.8050.6398540.0170810.6227732.0074471.9152612.010100
2667.108.86443.083.6101.099.16841.984.4420.6419960.0163790.6256172.0066941.9178002.009132
27134.217.72886.432.4292.113.42784.319.0020.6439720.0157460.6282262.0061561.9227522.008339
28268.435.456173.358.0484.066.184169.291.8640.6458090.0151480.6306612.0057061.9239772.007755
29536.870.912347.631.5837.836.982339.794.6010.6475140.0145980.6329172.0052811.9273562.007152
301.073.741.824696.962.78715.120.558681.842.2290.6490970.0140820.6350152.0048891.9293852.006631
312.147.483.6481.397.090.56129.209.4641.367.881.0970.6505710.0136020.6369692.0045411.9317722.006155
324.294.967.2962.800.122.70156.499.8412.743.622.8600.6519540.0131550.6387992.0042531.9342992.005747
338.589.934.5925.611.373.165109.393.5585.501.979.6070.6532500.0127350.6405152.0039741.9361752.005370
3417.179.869.18411.243.593.803212.037.82411.031.555.9790.6544630.0123420.6421212.0037151.9383032.005016


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
121010100
241010100
382111100
4162111100
5324312110
6646513120
712810916130
825610916130
9512229136637
101.02444935616319
112.04881972636336
124.0961529143666377
138.192280927161293142
1416.384537952862633265
1532.768964995564883467
1665.5361.82091.81169153896
17131.0723.39693.38761.72431.663
18262.1446.38396.37463.21933.155
19524.28812.111912.10266.12235.980
201.048.57622.904922.895611.534311.361
212.097.15243.444943.435621.706321.729
224.194.30482.392982.383641.316341.067
238.388.608156.8549156.845678.617378.228
2416.777.216299.2499299.2406149.8723149.368
2533.554.432573.1409573.1316286.9583286.173
2667.108.8641.099.16891.099.1596550.0923549.067
27134.217.7282.113.42792.113.41861.057.18731.056.231
28268.435.4564.066.18494.066.17562.033.29032.032.885
29536.870.9127.836.98297.836.97363.918.67733.918.296
301.073.741.82415.120.558915.120.54967.560.24637.560.303
312.147.483.64829.209.464929.209.455614.603.110314.606.345
324.294.967.29656.499.841956.499.832628.249.544328.250.288
338.589.934.592109.393.5589109.393.549654.697.133354.696.416
3417.179.869.184212.037.8249212.037.8156106.019.1463106.018.669


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
122010200
244030211
386130222
41613381354
532205132468
664371124611713
712868254114201420
825672284215211422
9512151698038363938
101.0243901852031109110584
112.048919472445261218240200
124.0962.0411.052987532481553475
138.1924.3532.2532.0981.1471.0411.1551.010
1416.3849.0174.6164.3992.3362.1472.3822.152
1532.76818.5179.5688.9474.8954.3924.8484.382
1665.53637.73919.31118.42610.0088.9269.7869.019
17131.07276.61739.06537.55019.99018.35419.97818.295
18262.144155.28879.24576.04140.36337.20240.43737.286
19524.288313.705159.740153.96381.47675.16381.64475.422
201.048.576632.855322.369310.484164.146152.104164.179152.426
212.097.1521.275.438648.775626.661329.903308.152329.558307.825
224.194.3042.568.7931.305.6041.263.187663.208621.119662.963621.503
238.388.6085.169.2752.624.4192.544.8541.331.8151.252.0831.332.6141.252.763
2416.777.21610.395.9025.274.6925.121.2082.674.3272.521.9832.675.5642.524.028
2533.554.43220.896.80510.594.78110.302.0225.367.0115.078.1085.371.3895.080.297
2667.108.86441.984.44221.273.00420.711.43610.772.14910.217.13110.777.79310.217.369
27134.217.72884.319.00242.700.05441.618.94621.614.80320.546.69521.617.17820.540.326
28268.435.456169.291.86485.683.94883.607.91443.351.77941.291.99143.356.67141.291.423
29536.870.912339.794.601171.890.384167.904.21586.933.17182.952.65686.951.52082.957.254
301.073.741.824681.842.229344.758.695337.083.532174.293.769166.612.370174.318.673166.617.417
312.147.483.6481.367.881.097691.337.575676.543.520349.371.892334.552.470349.394.909334.561.826
324.294.967.2962.743.622.8601.386.104.3511.357.518.507700.228.185671.572.602700.245.399671.576.674
338.589.934.5925.501.979.6072.778.632.8562.723.346.7491.403.206.6291.347.761.2421.403.250.9371.347.760.799
3417.179.869.18411.031.555.9795.569.257.5795.462.298.3982.811.676.1352.704.078.8402.811.668.1422.704.132.862


8. Check for existing Integer Sequences by OEIS

Found in Database : 11, 3, 179, 101, 359, 7, 1609, 13, 79, 23, 883, 1, 41, 71, 53, 61, 461, 1, 113, 29,
Found in Database : 11, 3, 179, 101, 359, 7, 1609, 13, 79, 23, 883, 41, 71, 53, 61, 461, 113, 29, 131, 167, 6037, 103, 839, 2311, 149, 7369, 83, 8629, 3011, 577,
Found in Database : 3, 7, 11, 13, 23, 29, 41, 53, 59, 61, 71, 79, 83, 101, 103, 113, 131, 139, 149,