Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:57:42
Deutsch
29.Mar 2024

Polynom = x^2+136x+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) = 5 = 5
f(2) = 299 = 13*23
f(3) = 55 = 5*11
f(4) = 583 = 11*53
f(5) = 91 = 7*13
f(6) = 875 = 5*5*5*7
f(7) = 1 = 1
f(8) = 1175 = 5*5*47
f(9) = 83 = 83
f(10) = 1483 = 1483
f(11) = 205 = 5*41
f(12) = 1799 = 7*257
f(13) = 245 = 5*7*7
f(14) = 2123 = 11*193
f(15) = 143 = 11*13
f(16) = 2455 = 5*491
f(17) = 41 = 41
f(18) = 2795 = 5*13*43
f(19) = 371 = 7*53
f(20) = 3143 = 7*449
f(21) = 415 = 5*83
f(22) = 3499 = 3499
f(23) = 115 = 5*23
f(24) = 3863 = 3863
f(25) = 253 = 11*23
f(26) = 4235 = 5*7*11*11
f(27) = 553 = 7*79
f(28) = 4615 = 5*13*71
f(29) = 601 = 601
f(30) = 5003 = 5003
f(31) = 325 = 5*5*13
f(32) = 5399 = 5399
f(33) = 175 = 5*5*7
f(34) = 5803 = 7*829
f(35) = 751 = 751
f(36) = 6215 = 5*11*113
f(37) = 803 = 11*73
f(38) = 6635 = 5*1327
f(39) = 107 = 107
f(40) = 7063 = 7*1009
f(41) = 455 = 5*7*13
f(42) = 7499 = 7499
f(43) = 965 = 5*193
f(44) = 7943 = 13*13*47
f(45) = 1021 = 1021
f(46) = 8395 = 5*23*73
f(47) = 539 = 7*7*11
f(48) = 8855 = 5*7*11*23
f(49) = 71 = 71
f(50) = 9323 = 9323
f(51) = 1195 = 5*239
f(52) = 9799 = 41*239
f(53) = 1255 = 5*251
f(54) = 10283 = 7*13*113
f(55) = 329 = 7*47
f(56) = 10775 = 5*5*431
f(57) = 689 = 13*53
f(58) = 11275 = 5*5*11*41
f(59) = 1441 = 11*131
f(60) = 11783 = 11783
f(61) = 1505 = 5*7*43
f(62) = 12299 = 7*7*251
f(63) = 785 = 5*157
f(64) = 12823 = 12823
f(65) = 409 = 409
f(66) = 13355 = 5*2671
f(67) = 1703 = 13*131
f(68) = 13895 = 5*7*397
f(69) = 1771 = 7*11*23
f(70) = 14443 = 11*13*101
f(71) = 115 = 5*23
f(72) = 14999 = 53*283
f(73) = 955 = 5*191
f(74) = 15563 = 79*197
f(75) = 1981 = 7*283
f(76) = 16135 = 5*7*461
f(77) = 2053 = 2053
f(78) = 16715 = 5*3343
f(79) = 1063 = 1063
f(80) = 17303 = 11*11*11*13
f(81) = 275 = 5*5*11
f(82) = 17899 = 7*2557
f(83) = 2275 = 5*5*7*13
f(84) = 18503 = 18503
f(85) = 2351 = 2351
f(86) = 19115 = 5*3823
f(87) = 607 = 607
f(88) = 19735 = 5*3947
f(89) = 1253 = 7*179
f(90) = 20363 = 7*2909
f(91) = 2585 = 5*11*47
f(92) = 20999 = 11*23*83
f(93) = 2665 = 5*13*41
f(94) = 21643 = 23*941
f(95) = 1373 = 1373
f(96) = 22295 = 5*7*7*7*13
f(97) = 707 = 7*101
f(98) = 22955 = 5*4591
f(99) = 2911 = 41*71
f(100) = 23623 = 23623

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+136x+23

f(0)=23
f(1)=5
f(2)=13
f(3)=11
f(4)=53
f(5)=7
f(6)=1
f(7)=1
f(8)=47
f(9)=83
f(10)=1483
f(11)=41
f(12)=257
f(13)=1
f(14)=193
f(15)=1
f(16)=491
f(17)=1
f(18)=43
f(19)=1
f(20)=449
f(21)=1
f(22)=3499
f(23)=1
f(24)=3863
f(25)=1
f(26)=1
f(27)=79
f(28)=71
f(29)=601
f(30)=5003
f(31)=1
f(32)=5399
f(33)=1
f(34)=829
f(35)=751
f(36)=113
f(37)=73
f(38)=1327
f(39)=107
f(40)=1009
f(41)=1
f(42)=7499
f(43)=1
f(44)=1
f(45)=1021
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=9323
f(51)=239
f(52)=1
f(53)=251
f(54)=1
f(55)=1
f(56)=431
f(57)=1
f(58)=1
f(59)=131
f(60)=11783
f(61)=1
f(62)=1
f(63)=157
f(64)=12823
f(65)=409
f(66)=2671
f(67)=1
f(68)=397
f(69)=1
f(70)=101
f(71)=1
f(72)=283
f(73)=191
f(74)=197
f(75)=1
f(76)=461
f(77)=2053
f(78)=3343
f(79)=1063
f(80)=1
f(81)=1
f(82)=2557
f(83)=1
f(84)=18503
f(85)=2351
f(86)=3823
f(87)=607
f(88)=3947
f(89)=179
f(90)=2909
f(91)=1
f(92)=1
f(93)=1
f(94)=941
f(95)=1373
f(96)=1
f(97)=1
f(98)=4591
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+136x+23 could be written as f(y)= y^2-4601 with x=y-68

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+68
f'(x)>2x+135

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, 5, 13, 11, 53, 7, 1, 1, 47, 83, 1483, 41, 257, 1, 193, 1, 491, 1, 43, 1, 449, 1, 3499, 1, 3863, 1, 1, 79, 71, 601, 5003, 1, 5399, 1, 829, 751, 113, 73, 1327, 107, 1009, 1, 7499, 1, 1, 1021, 1, 1, 1, 1, 9323, 239, 1, 251, 1, 1, 431, 1, 1, 131, 11783, 1, 1, 157, 12823, 409, 2671, 1, 397, 1, 101, 1, 283, 191, 197, 1, 461, 2053, 3343, 1063, 1, 1, 2557, 1, 18503, 2351, 3823, 607, 3947, 179, 2909, 1, 1, 1, 941, 1373, 1, 1, 4591, 1, 23623, 599, 1, 1, 1, 1583, 1, 3253, 211, 1, 1, 1, 27799, 1, 2593, 1, 5851, 1, 857, 1, 433, 389, 2423, 797, 419, 1, 6607, 2089, 6763, 1069, 34603, 1, 1, 1, 883, 1, 673, 2339, 1, 683, 1, 977, 39499, 499, 40343, 1, 1, 1, 647, 1, 42923, 271, 6257, 1, 44683, 5641, 1823, 523, 1, 1, 967, 1, 48299, 1, 49223, 6211, 1433, 1, 929, 293, 733, 1, 52999, 1, 593, 1, 10987, 1, 1, 641, 739, 1, 57899, 1, 1, 1, 521, 1, 1741, 7681, 1, 1, 863, 1, 1307, 1153, 277, 631, 1, 1, 5171, 1, 887, 1721, 69383, 8741, 2819, 1, 1, 1, 5591, 1831, 6709, 1, 74923, 337, 1, 4789, 15439, 9721, 1, 1973, 1, 1, 7333, 2539, 1259, 10303, 16603, 1493, 1, 1, 1, 1, 7873, 991, 1, 1579, 17807, 1, 90263, 1, 91499, 1, 13249, 1061, 1709, 2957, 19051, 1, 13789, 347, 7523, 1, 99083, 1, 1, 1, 1, 12791, 102983, 2591, 1, 1, 1, 1, 21391, 1223, 1, 1, 1321, 1, 1, 349, 112363, 1087, 1, 14303, 1, 1, 1031, 1, 117899, 1, 2251, 2143, 3449, 7589, 1879, 1, 1, 1, 2551, 1, 126443, 1987, 25579, 8039, 1, 1, 1699, 1, 132299, 1663, 1, 1051, 773, 1, 5471, 17191, 138283, 1, 1, 1, 1553, 17761, 28571, 1381, 28879, 1, 20849, 1, 1, 1, 6481, 18731, 1, 1, 4349, 1, 11831, 1, 155399, 1, 2039, 1409, 31723, 1, 439, 1, 12451, 1, 23357, 1, 15013, 1, 33359, 911, 4813, 3023, 170123, 2137, 171799, 1, 173483, 1, 1, 22003, 1, 2777, 1669, 2243, 1, 1, 953, 22861, 1, 1049, 1613, 1, 1163, 4703, 188999, 1, 190763, 1, 5501, 1, 3533, 1877, 4783, 1, 1171, 1, 1, 6269, 983, 25303, 3697, 1, 1, 1, 15923, 1, 208843, 2017, 42139, 3779, 6073, 1213, 1, 1, 2963, 5431, 1, 1, 8803, 6907, 1, 13933, 20353, 1, 32257, 5669, 9901, 14293, 1997, 7207, 509, 4153, 4969, 1, 1, 1, 2861, 2129, 6841, 1, 1, 1, 243403, 1, 3187, 1, 19031, 31051, 1, 1361, 50287, 1, 36209, 3181, 255499, 1, 1801, 32321, 7417, 1, 1217, 8209, 263723, 6619, 2029, 1, 1, 1, 10799, 1303, 10883, 34141, 1, 1, 1, 3467, 547, 1, 5101, 1, 1, 1, 21911, 1787, 286999, 1, 1, 1, 5297, 3323, 1249, 18413, 295703, 1, 42557, 1, 4111, 1637, 4651, 1, 1, 2729, 306763, 1, 308999, 7753, 569, 2789, 1, 1229, 5741, 1, 318023, 1, 45757, 1, 322583, 20233, 1, 1, 1, 1, 1, 4133, 25523, 2081, 7109, 41911, 9613, 6029, 67759, 1, 31013, 1, 26423, 1231, 49409, 1, 1, 21839, 70123, 1, 1, 1, 32309, 1783, 1, 1, 72043, 1, 1, 45481, 365063, 9157, 33409, 1, 1289, 1657, 1, 46703, 1, 1, 9203, 1, 1, 1, 1511, 1, 15391, 661, 2213, 3469, 389783, 1, 392299, 9839, 1, 643, 11353, 12457, 79979, 25073, 402443, 10093, 1, 1451, 1, 1, 7457, 1, 1, 7393, 1, 2083, 417899, 1, 420503, 1, 1, 1, 85147, 53381, 1, 1, 430999, 1, 61949, 54371, 1, 4973, 1, 13759, 1, 1, 1, 1, 446983, 56041, 17987, 4027, 1, 1, 1, 11411, 457799, 1, 65789, 1, 7127, 1, 93199, 1, 1, 1, 1, 1, 474263, 1, 1, 757, 13709, 1, 1, 1, 11839, 1, 488203, 1, 14029, 61553, 1, 2381, 9371, 1, 1, 1789, 21841, 62971, 1, 1979, 101611, 4549, 10427, 12809, 3593, 1, 1489, 1, 2969, 1, 20899, 1, 1559, 13171, 528299, 1, 6899, 1, 2273, 1, 8263, 2927, 1, 1, 542999, 1, 49633, 6221, 1, 9829, 1, 8647, 1, 1, 557899, 2797, 11447, 1, 10253, 35339, 1597, 4441, 24781, 1, 3559, 1, 1, 18049, 10529, 3299, 16633, 1, 1, 14669, 12517, 1, 45491, 2647, 1, 1, 1, 74891, 1, 1, 86257, 1, 1, 3307, 122011, 5881, 11149, 1, 1, 1931, 4729, 15527, 622663, 78031, 17881, 2801, 125803, 3583, 4421, 3169, 5623, 1, 91229, 40013, 5581, 20107, 1, 80831, 8419, 1, 1, 1, 654743, 41023, 1, 11779, 2699, 1, 2399, 1, 5519, 1, 95869, 1, 1, 84503, 27107, 1, 14489, 1, 8887, 1, 687623, 86161, 138191, 1, 1, 1, 697643, 17483, 53923, 1, 64033, 1, 1, 1, 142223, 89101, 714503, 3581, 1, 1, 2851, 1, 1, 90803, 1, 13033, 2549, 4583, 17093, 9209, 738443, 1, 1, 1, 11467, 881, 748823, 4691, 752299, 2693, 1, 1, 1, 1, 2347, 1, 109469, 1, 769799, 19289, 14591, 1, 155371, 1, 2027, 97771, 1, 1, 1, 4933, 8693, 7079, 158927, 1, 1, 9091, 1, 1, 115057, 1, 4787, 2357, 162523, 1, 23321, 1, 1, 10271, 823499, 1, 827143, 1, 3391, 4003, 1, 1, 5861, 1, 2269, 1, 17989, 26479, 33967, 53189, 6823, 15263, 9413, 1951, 1, 1, 10937, 27059, 24793, 1, 174299, 109171, 875243, 2741, 1949, 1, 5483, 110581, 1, 111053, 1, 55763, 1, 1, 897899, 1, 81973, 112951, 13931, 4051, 25981, 1, 19429, 22877, 916999, 22973, 11959, 1, 2341, 1, 185711, 1, 5939, 1, 10289, 1, 940183, 1, 3433, 118253, 5417, 16963, 2447, 11923, 73523, 1, 959723, 1, 2503, 10973, 1, 1, 1, 1, 1, 3491, 3461, 9437, 196687, 1, 1, 1, 141629, 4967, 995399, 4987, 12041, 15647, 28669, 1, 1, 11471, 8951, 1, 1015499, 1, 20807, 1, 204719, 128203, 4373, 11701, 13399, 1, 4909, 12973, 1, 130241, 41759, 18679, 1, 65633, 2333, 1, 2521, 26459, 1, 18973, 9257, 1, 9293, 66923, 1072843, 1, 1, 1, 1081163, 1, 1, 1, 4447, 1, 84131, 5479, 99809, 1, 1, 9859, 1, 1, 222107, 6047, 85751, 1, 159857, 1, 9283, 1, 225499, 1, 1, 2531,

6. Sequence of the polynom (only primes)

23, 5, 13, 11, 53, 7, 47, 83, 1483, 41, 257, 193, 491, 43, 449, 3499, 3863, 79, 71, 601, 5003, 5399, 829, 751, 113, 73, 1327, 107, 1009, 7499, 1021, 9323, 239, 251, 431, 131, 11783, 157, 12823, 409, 2671, 397, 101, 283, 191, 197, 461, 2053, 3343, 1063, 2557, 18503, 2351, 3823, 607, 3947, 179, 2909, 941, 1373, 4591, 23623, 599, 1583, 3253, 211, 27799, 2593, 5851, 857, 433, 389, 2423, 797, 419, 6607, 2089, 6763, 1069, 34603, 883, 673, 2339, 683, 977, 39499, 499, 40343, 647, 42923, 271, 6257, 44683, 5641, 1823, 523, 967, 48299, 49223, 6211, 1433, 929, 293, 733, 52999, 593, 10987, 641, 739, 57899, 521, 1741, 7681, 863, 1307, 1153, 277, 631, 5171, 887, 1721, 69383, 8741, 2819, 5591, 1831, 6709, 74923, 337, 4789, 15439, 9721, 1973, 7333, 2539, 1259, 10303, 16603, 1493, 7873, 991, 1579, 17807, 90263, 91499, 13249, 1061, 1709, 2957, 19051, 13789, 347, 7523, 99083, 12791, 102983, 2591, 21391, 1223, 1321, 349, 112363, 1087, 14303, 1031, 117899, 2251, 2143, 3449, 7589, 1879, 2551, 126443, 1987, 25579, 8039, 1699, 132299, 1663, 1051, 773, 5471, 17191, 138283, 1553, 17761, 28571, 1381, 28879, 20849, 6481, 18731, 4349, 11831, 155399, 2039, 1409, 31723, 439, 12451, 23357, 15013, 33359, 911, 4813, 3023, 170123, 2137, 171799, 173483, 22003, 2777, 1669, 2243, 953, 22861, 1049, 1613, 1163, 4703, 188999, 190763, 5501, 3533, 1877, 4783, 1171, 6269, 983, 25303, 3697, 15923, 208843, 2017, 42139, 3779, 6073, 1213, 2963, 5431, 8803, 6907, 13933, 20353, 32257, 5669, 9901, 14293, 1997, 7207, 509, 4153, 4969, 2861, 2129, 6841, 243403, 3187, 19031, 31051, 1361, 50287, 36209, 3181, 255499, 1801, 32321, 7417, 1217, 8209, 263723, 6619, 2029, 10799, 1303, 10883, 34141, 3467, 547, 5101, 21911, 1787, 286999, 5297, 3323, 1249, 18413, 295703, 42557, 4111, 1637, 4651, 2729, 306763, 308999, 7753, 569, 2789, 1229, 5741, 318023, 45757, 322583, 20233, 4133, 25523, 2081, 7109, 41911, 9613, 6029, 67759, 31013, 26423, 1231, 49409, 21839, 70123, 32309, 1783, 72043, 45481, 365063, 9157, 33409, 1289, 1657, 46703, 9203, 1511, 15391, 661, 2213, 3469, 389783, 392299, 9839, 643, 11353, 12457, 79979, 25073, 402443, 10093, 1451, 7457, 7393, 2083, 417899, 420503, 85147, 53381, 430999, 61949, 54371, 4973, 13759, 446983, 56041, 17987, 4027, 11411, 457799, 65789, 7127, 93199, 474263, 757, 13709, 11839, 488203, 14029, 61553, 2381, 9371, 1789, 21841, 62971, 1979, 101611, 4549, 10427, 12809, 3593, 1489, 2969, 20899, 1559, 13171, 528299, 6899, 2273, 8263, 2927, 542999, 49633, 6221, 9829, 8647, 557899, 2797, 11447, 10253, 35339, 1597, 4441, 24781, 3559, 18049, 10529, 3299, 16633, 14669, 12517, 45491, 2647, 74891, 86257, 3307, 122011, 5881, 11149, 1931, 4729, 15527, 622663, 78031, 17881, 2801, 125803, 3583, 4421, 3169, 5623, 91229, 40013, 5581, 20107, 80831, 8419, 654743, 41023, 11779, 2699, 2399, 5519, 95869, 84503, 27107, 14489, 8887, 687623, 86161, 138191, 697643, 17483, 53923, 64033, 142223, 89101, 714503, 3581, 2851, 90803, 13033, 2549, 4583, 17093, 9209, 738443, 11467, 881, 748823, 4691, 752299, 2693, 2347, 109469, 769799, 19289, 14591, 155371, 2027, 97771, 4933, 8693, 7079, 158927, 9091, 115057, 4787, 2357, 162523, 23321, 10271, 823499, 827143, 3391, 4003, 5861, 2269, 17989, 26479, 33967, 53189, 6823, 15263, 9413, 1951, 10937, 27059, 24793, 174299, 109171, 875243, 2741, 1949, 5483, 110581, 111053, 55763, 897899, 81973, 112951, 13931, 4051, 25981, 19429, 22877, 916999, 22973, 11959, 2341, 185711, 5939, 10289, 940183, 3433, 118253, 5417, 16963, 2447, 11923, 73523, 959723, 2503, 10973, 3491, 3461, 9437, 196687, 141629, 4967, 995399, 4987, 12041, 15647, 28669, 11471, 8951, 1015499, 20807, 204719, 128203, 4373, 11701, 13399, 4909, 12973, 130241, 41759, 18679, 65633, 2333, 2521, 26459, 18973, 9257, 9293, 66923, 1072843, 1081163, 4447, 84131, 5479, 99809, 9859, 222107, 6047, 85751, 159857, 9283, 225499, 2531,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+136x+23 and
the reducible primes which appear as divisor for the first time
p | x^2+136x+23 and p < x^2+136x+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)
1108260.8000000.2000000.8000000.0000000.0000000.000000
21005913460.5900000.1300000.5900007.3750006.5000007.666667
31.000577844930.5770000.0840000.5770009.7796616.46153810.717391
410.0006.0395675.4720.6039000.0567000.60390010.4662056.75000011.099392
5100.00062.3834.35158.0320.6238300.0435100.62383010.3300227.67372110.605263
61.000.000636.29235.919600.3730.6362920.0359190.63629210.1997668.25534310.345551
710.000.0006.447.934303.6116.144.3230.6447930.0303610.64479310.1336098.45265810.234176
8100.000.00065.110.0262.627.88262.482.1440.6511000.0262790.65110010.0978128.65542410.169086
91.000.000.000655.913.90523.189.823632.724.0820.6559140.0231900.65591410.0739328.82453110.126478
1010.000.000.0006.597.446.811207.531.1446.389.915.6670.6597450.0207530.65974510.0584038.94923310.099055


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
386150.7500000.1250000.6250001.2000001.0000001.250000
416122100.7500000.1250000.6250002.0000002.0000002.000000
532217140.6562500.2187500.4375001.7500003.5000001.400000
6643711260.5781250.1718750.4062501.7619051.5714291.857143
71287414600.5781250.1093750.4687502.0000001.2727272.307692
8256145281170.5664060.1093750.4570311.9594592.0000001.950000
9512292502420.5703120.0976560.4726562.0137931.7857142.068376
101.024590855050.5761720.0830080.4931642.0205481.7000002.086777
112.0481.1961561.0400.5839840.0761720.5078122.0271191.8352942.059406
124.0962.4342702.1640.5942380.0659180.5283202.0351171.7307692.080769
138.1924.9314744.4570.6019290.0578610.5440672.0258831.7555562.059612
1416.38410.0008679.1330.6103520.0529170.5574342.0279861.8291142.049136
1532.76820.1761.60418.5720.6157230.0489500.5667722.0176001.8500582.033505
1665.53640.6522.99537.6570.6203000.0457000.5746002.0148691.8672072.027622
17131.07282.0505.60176.4490.6259920.0427320.5832602.0183511.8701172.030140
18262.144165.03510.615154.4200.6295590.0404930.5890662.0113951.8951972.019909
19524.288332.07719.849312.2280.6333870.0378590.5955282.0121611.8699012.021940
201.048.576667.40437.460629.9440.6364860.0357250.6007612.0097871.8872492.017577
212.097.1521.340.77671.2311.269.5450.6393320.0339660.6053662.0089421.9015222.015330
224.194.3042.692.276135.3782.556.8980.6418890.0322770.6096122.0079981.9005492.014027
238.388.6085.404.452257.5965.146.8560.6442610.0307080.6135532.0073911.9027912.012930
2416.777.21610.844.076492.19810.351.8780.6463570.0293370.6170202.0065081.9107362.011301
2533.554.43221.753.750941.23720.812.5130.6483120.0280510.6202612.0060491.9123142.010506
2667.108.86443.630.0961.805.65841.824.4380.6501390.0269060.6232332.0056361.9183882.009582
27134.217.72887.482.1013.468.55684.013.5450.6517920.0258430.6259502.0050861.9209372.008719
28268.435.456175.371.4596.672.927168.698.5320.6533100.0248590.6284512.0046551.9238342.007992
29536.870.912351.502.75112.856.735338.646.0160.6547250.0239480.6307772.0043331.9267012.007404
301.073.741.824704.423.28924.808.875679.614.4140.6560450.0231050.6329402.0040341.9296402.006858
312.147.483.6481.411.493.70747.933.5941.363.560.1130.6572780.0223210.6349572.0037581.9321152.006373
324.294.967.2962.827.953.32492.708.4552.735.244.8690.6584340.0215850.6368492.0035181.9341022.005958
338.589.934.5925.665.194.836179.515.5835.485.679.2530.6595160.0208980.6386172.0032841.9363452.005553
3417.179.869.18411.347.844.465347.941.49310.999.902.9720.6605310.0202530.6402792.0030811.9382252.005203


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
121010001
241010001
381010001
4162110101
5327250403
66411380605
712814590608
8256281018015013
9512501832028022
101.024853055044041
112.04815653103082074
124.0962708718301350135
138.19247415332102370237
1416.38486728658104230444
1532.7681.6045291.07507870817
1665.5362.9959822.01301.47801.517
17131.0725.6011.8463.75502.77702.824
18262.14410.6153.5357.08005.31605.299
19524.28819.8496.67013.17909.90109.948
201.048.57637.46012.48024.980018.762018.698
212.097.15271.23123.79147.440035.664035.567
224.194.304135.37845.31190.067067.768067.610
238.388.608257.59685.975171.6210129.1160128.480
2416.777.216492.198164.410327.7880246.6310245.567
2533.554.432941.237313.436627.8010470.6020470.635
2667.108.8641.805.658601.4421.204.2160902.7110902.947
27134.217.7283.468.5561.155.0642.313.49201.734.92801.733.628
28268.435.4566.672.9272.222.9894.449.93803.337.07203.335.855
29536.870.91212.856.7354.284.5868.572.14906.429.86706.426.868
301.073.741.82424.808.8758.268.98816.539.887012.405.559012.403.316
312.147.483.64847.933.59415.974.33731.959.257023.969.351023.964.243
324.294.967.29692.708.45530.896.15061.812.305046.356.314046.352.141
338.589.934.592179.515.58359.830.168119.685.415089.760.147089.755.436
3417.179.869.184347.941.493115.967.337231.974.1560173.968.2450173.973.248


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
122110020
244130130
385140131
41610283331
532144105333
6642610167667
712860283212121719
8256117585927293031
951224212112159576363
101.024505247258114127136128
112.0481.040529511247259280254
124.0962.1641.0911.073518555554537
138.1924.4572.2632.1941.1171.1381.0951.107
1416.3849.1334.6424.4912.2752.3242.2432.291
1532.76818.5729.4309.1424.6094.7064.6184.639
1665.53637.65719.08118.5769.3109.4779.4519.419
17131.07276.44938.71437.73519.02219.20219.09919.126
18262.144154.42078.35676.06438.44438.77238.61738.587
19524.288312.228158.159154.06977.82578.04378.24078.120
201.048.576629.944318.841311.103157.508157.767157.041157.628
212.097.1521.269.545642.426627.119316.925317.822316.911317.887
224.194.3042.556.8981.292.0301.264.868638.311640.065639.010639.512
238.388.6085.146.8562.600.7832.546.0731.285.6011.286.9331.285.9981.288.324
2416.777.21610.351.8785.227.7305.124.1482.585.6162.589.3062.585.8762.591.080
2533.554.43220.812.51310.507.09810.305.4155.198.7005.204.3585.199.9595.209.496
2667.108.86441.824.43821.105.91520.718.52310.450.32010.459.59110.450.28210.464.245
27134.217.72884.013.54542.382.41541.631.13020.986.84021.016.52820.996.13621.014.041
28268.435.456168.698.53285.069.12583.629.40742.142.56842.205.04742.155.13742.195.780
29536.870.912338.646.016170.703.405167.942.61184.604.11784.715.01884.612.84284.714.039
301.073.741.824679.614.414342.459.456337.154.958169.797.807170.008.965169.797.853170.009.789
312.147.483.6481.363.560.113686.909.480676.650.633340.687.460341.099.522340.687.672341.085.459
324.294.967.2962.735.244.8691.377.509.8181.357.735.051683.416.249684.229.213683.384.500684.214.907
338.589.934.5925.485.679.2532.762.031.4552.723.647.7981.370.628.2891.372.238.9521.370.594.2461.372.217.766
3417.179.869.18410.999.902.9725.537.083.6595.462.819.3132.748.446.7092.751.552.0942.748.372.3642.751.531.805


8. Check for existing Integer Sequences by OEIS

Found in Database : 23, 5, 13, 11, 53, 7, 1, 1, 47, 83, 1483, 41, 257, 1, 193, 1, 491, 1, 43, 1,
Found in Database : 23, 5, 13, 11, 53, 7, 47, 83, 1483, 41, 257, 193, 491, 43, 449, 3499, 3863, 79, 71, 601, 5003, 5399, 829, 751, 113, 73, 1327, 107,
Found in Database : 5, 7, 11, 13, 23, 41, 43, 47, 53, 71, 73, 79, 83, 101, 107, 113, 131,