Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:58:29
Deutsch
28.Mar 2024

Polynom = x^2+3x+1

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) = 1 = 1
f(1) = 5 = 5
f(2) = 11 = 11
f(3) = 19 = 19
f(4) = 29 = 29
f(5) = 41 = 41
f(6) = 55 = 5*11
f(7) = 71 = 71
f(8) = 89 = 89
f(9) = 109 = 109
f(10) = 131 = 131
f(11) = 155 = 5*31
f(12) = 181 = 181
f(13) = 209 = 11*19
f(14) = 239 = 239
f(15) = 271 = 271
f(16) = 305 = 5*61
f(17) = 341 = 11*31
f(18) = 379 = 379
f(19) = 419 = 419
f(20) = 461 = 461
f(21) = 505 = 5*101
f(22) = 551 = 19*29
f(23) = 599 = 599
f(24) = 649 = 11*59
f(25) = 701 = 701
f(26) = 755 = 5*151
f(27) = 811 = 811
f(28) = 869 = 11*79
f(29) = 929 = 929
f(30) = 991 = 991
f(31) = 1055 = 5*211
f(32) = 1121 = 19*59
f(33) = 1189 = 29*41
f(34) = 1259 = 1259
f(35) = 1331 = 11*11*11
f(36) = 1405 = 5*281
f(37) = 1481 = 1481
f(38) = 1559 = 1559
f(39) = 1639 = 11*149
f(40) = 1721 = 1721
f(41) = 1805 = 5*19*19
f(42) = 1891 = 31*61
f(43) = 1979 = 1979
f(44) = 2069 = 2069
f(45) = 2161 = 2161
f(46) = 2255 = 5*11*41
f(47) = 2351 = 2351
f(48) = 2449 = 31*79
f(49) = 2549 = 2549
f(50) = 2651 = 11*241
f(51) = 2755 = 5*19*29
f(52) = 2861 = 2861
f(53) = 2969 = 2969
f(54) = 3079 = 3079
f(55) = 3191 = 3191
f(56) = 3305 = 5*661
f(57) = 3421 = 11*311
f(58) = 3539 = 3539
f(59) = 3659 = 3659
f(60) = 3781 = 19*199
f(61) = 3905 = 5*11*71
f(62) = 4031 = 29*139
f(63) = 4159 = 4159
f(64) = 4289 = 4289
f(65) = 4421 = 4421
f(66) = 4555 = 5*911
f(67) = 4691 = 4691
f(68) = 4829 = 11*439
f(69) = 4969 = 4969
f(70) = 5111 = 19*269
f(71) = 5255 = 5*1051
f(72) = 5401 = 11*491
f(73) = 5549 = 31*179
f(74) = 5699 = 41*139
f(75) = 5851 = 5851
f(76) = 6005 = 5*1201
f(77) = 6161 = 61*101
f(78) = 6319 = 71*89
f(79) = 6479 = 11*19*31
f(80) = 6641 = 29*229
f(81) = 6805 = 5*1361
f(82) = 6971 = 6971
f(83) = 7139 = 11*11*59
f(84) = 7309 = 7309
f(85) = 7481 = 7481
f(86) = 7655 = 5*1531
f(87) = 7831 = 41*191
f(88) = 8009 = 8009
f(89) = 8189 = 19*431
f(90) = 8371 = 11*761
f(91) = 8555 = 5*29*59
f(92) = 8741 = 8741
f(93) = 8929 = 8929
f(94) = 9119 = 11*829
f(95) = 9311 = 9311
f(96) = 9505 = 5*1901
f(97) = 9701 = 89*109
f(98) = 9899 = 19*521
f(99) = 10099 = 10099
f(100) = 10301 = 10301

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+3x+1

f(0)=1
f(1)=5
f(2)=11
f(3)=19
f(4)=29
f(5)=41
f(6)=1
f(7)=71
f(8)=89
f(9)=109
f(10)=131
f(11)=31
f(12)=181
f(13)=1
f(14)=239
f(15)=271
f(16)=61
f(17)=1
f(18)=379
f(19)=419
f(20)=461
f(21)=101
f(22)=1
f(23)=599
f(24)=59
f(25)=701
f(26)=151
f(27)=811
f(28)=79
f(29)=929
f(30)=991
f(31)=211
f(32)=1
f(33)=1
f(34)=1259
f(35)=1
f(36)=281
f(37)=1481
f(38)=1559
f(39)=149
f(40)=1721
f(41)=1
f(42)=1
f(43)=1979
f(44)=2069
f(45)=2161
f(46)=1
f(47)=2351
f(48)=1
f(49)=2549
f(50)=241
f(51)=1
f(52)=2861
f(53)=2969
f(54)=3079
f(55)=3191
f(56)=661
f(57)=311
f(58)=3539
f(59)=3659
f(60)=199
f(61)=1
f(62)=139
f(63)=4159
f(64)=4289
f(65)=4421
f(66)=911
f(67)=4691
f(68)=439
f(69)=4969
f(70)=269
f(71)=1051
f(72)=491
f(73)=179
f(74)=1
f(75)=5851
f(76)=1201
f(77)=1
f(78)=1
f(79)=1
f(80)=229
f(81)=1361
f(82)=6971
f(83)=1
f(84)=7309
f(85)=7481
f(86)=1531
f(87)=191
f(88)=8009
f(89)=431
f(90)=761
f(91)=1
f(92)=8741
f(93)=8929
f(94)=829
f(95)=9311
f(96)=1901
f(97)=1
f(98)=521
f(99)=10099

b) Substitution of the polynom
The polynom f(x)=x^2+3x+1 could be written as f(y)= y^2-1.25 with x=y-1.5

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+1.5
f'(x)>2x+2

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

1, 5, 11, 19, 29, 41, 1, 71, 89, 109, 131, 31, 181, 1, 239, 271, 61, 1, 379, 419, 461, 101, 1, 599, 59, 701, 151, 811, 79, 929, 991, 211, 1, 1, 1259, 1, 281, 1481, 1559, 149, 1721, 1, 1, 1979, 2069, 2161, 1, 2351, 1, 2549, 241, 1, 2861, 2969, 3079, 3191, 661, 311, 3539, 3659, 199, 1, 139, 4159, 4289, 4421, 911, 4691, 439, 4969, 269, 1051, 491, 179, 1, 5851, 1201, 1, 1, 1, 229, 1361, 6971, 1, 7309, 7481, 1531, 191, 8009, 431, 761, 1, 8741, 8929, 829, 9311, 1901, 1, 521, 10099, 10301, 1, 10711, 1, 359, 1031, 2311, 1, 631, 421, 401, 2531, 1171, 13109, 13339, 331, 251, 739, 1, 14519, 509, 3001, 1, 1409, 15749, 16001, 3251, 1, 409, 17029, 17291, 3511, 1, 18089, 1669, 601, 1, 19181, 1, 19739, 20021, 1, 349, 20879, 21169, 1951, 1, 22051, 22349, 1, 389, 4651, 23561, 23869, 24179, 1289, 1, 25121, 25439, 25759, 2371, 5281, 26731, 27059, 449, 1459, 1, 1, 28729, 709, 29411, 541, 971, 30449, 1621, 31151, 6301, 1, 1, 32579, 32941, 6661, 3061, 34039, 1811, 34781, 1, 35531, 1, 3299, 36671, 7411, 37441, 1, 38219, 38611, 1, 1, 39799, 659, 3691, 1, 41411, 1, 1, 42641, 1, 1499, 43889, 751, 1091, 821, 2399, 1, 46439, 4261, 9461, 47741, 48179, 48619, 691, 9901, 1, 499, 50849, 1, 941, 479, 1699, 53129, 53591, 569, 54521, 4999, 1789, 55931, 1, 5171, 1399, 57839, 58321, 619, 1, 59779, 5479, 60761, 12251, 61751, 5659, 1, 3329, 1, 1, 1, 2251, 5981, 1, 66821, 67339, 1, 1, 13781, 69431, 69959, 70489, 1, 1301, 72091, 1231, 3851, 6701, 14851, 571, 1, 1069, 859, 15401, 641, 1, 1, 79241, 1451, 1, 2791, 81509, 1039, 1, 83231, 1, 84389, 2741, 1, 1, 86729, 3011, 87911, 1, 89101, 4721, 8209, 90901, 18301, 92111, 8429, 93329, 93941, 18911, 5009, 95789, 1, 8821, 19531, 3389, 98909, 9049, 919, 20161, 1, 102079, 1741, 2521, 1, 104651, 3631, 1049, 881, 1129, 1, 719, 109229, 109891, 22111, 10111, 2729, 112559, 1, 1, 769, 115259, 1, 839, 809, 117991, 10789, 1511, 1, 24151, 1, 122149, 122849, 123551, 24851, 1, 125669, 11489, 6689, 25561, 128521, 1, 129959, 130681, 1, 1861, 132859, 1, 12211, 27011, 1, 1, 12479, 4759, 27751, 139501, 140249, 1, 1, 2591, 143261, 2441, 144779, 1, 1009, 2411, 1, 148609, 149381, 1, 13721, 1, 4919, 153271, 2801, 154841, 8191, 156419, 157211, 31601, 1, 1319, 5531, 161201, 32401, 1, 1249, 164429, 1109, 33211, 166871, 2749, 15319, 5839, 34031, 8999, 15619, 5569, 4231, 1, 175141, 175979, 176819, 1, 1879, 179351, 2281, 1, 1801, 36551, 183611, 6361, 1, 186191, 1, 187921, 2659, 2131, 17321, 38281, 3259, 1, 6691, 10259, 39161, 17881, 1, 198469, 1, 1, 201151, 202049, 202949, 10729, 1321, 205661, 1, 207479, 208391, 1021, 1, 1061, 11161, 212981, 1, 214831, 1, 19699, 1, 43711, 219491, 1, 1, 222311, 44651, 224201, 225149, 226099, 20641, 1471, 228961, 12101, 1, 231841, 1, 7541, 1229, 5749, 236681, 1, 3361, 12611, 240589, 21961, 1, 1019, 244529, 245519, 246511, 1, 1, 249499, 250499, 251501, 4591, 3209, 254519, 1, 256541, 51511, 1, 23599, 260609, 261631, 1, 23971, 8539, 265739, 9199, 2819, 268841, 269879, 2239, 1, 54601, 3469, 1, 1, 6761, 1, 279311, 1549, 281429, 1, 56711, 1181, 2621, 1, 15149, 57781, 1, 1, 1, 293221, 5351, 3319, 296479, 1, 1429, 59951, 300851, 301949, 303049, 304151, 61051, 27851, 1, 1, 1, 5651, 1151, 4409, 314159, 315281, 63281, 1, 1, 16831, 5261, 1571, 2671, 324329, 10499, 326611, 65551, 328901, 1, 30109, 1, 66701, 334661, 30529, 8219, 338141, 1, 3371, 17981, 11821, 31271, 69031, 346331, 347509, 31699, 349871, 1, 18539, 353429, 354619, 355811, 6491, 358201, 1279, 1709, 1, 1, 1, 365419, 1601, 367841, 2381, 1, 1381, 372709, 2089, 1, 4229, 1, 6421, 380071, 76261, 382541, 1, 1, 9421, 4079, 1, 389999, 391249, 1, 1291, 3911, 396269, 1, 1, 1, 401321, 36599, 1609, 2719, 81281, 13151, 408959, 410239, 1, 82561, 1, 415379, 37879, 417961, 1, 420551, 10289, 22271, 424451, 7741, 7001, 428369, 429679, 39181, 86461, 2179, 434939, 22961, 1, 2141, 1, 4051, 2099, 3391, 8101, 446891, 1, 449569, 1871, 3119, 453601, 1, 2389, 457651, 91801, 41851, 1, 1489, 5879, 1579, 467171, 4639, 42719, 471281, 94531, 1, 3929, 1, 478171, 95911, 480941, 15559, 483719, 44101, 97301, 25679, 489299, 1439, 1, 1, 12071, 3331, 497729, 499141, 1, 501971, 503389, 2789, 46021, 101531, 1, 8369, 511939, 513371, 5419, 1, 17851, 519119, 5849, 9491, 1, 524899, 16979, 27779, 1, 530711, 1, 18401, 1, 1, 1, 2711, 2011, 28549, 1, 1, 546859, 1, 9319, 110261, 552791, 1, 29251, 557261, 111751, 19319, 9209, 563249, 51341, 1, 567761, 569269, 2731, 18461, 114761, 4139, 19891, 7321, 579881, 1, 582931, 1619, 585989, 53411, 117811, 2579, 592129, 4271, 1, 1, 1, 1, 9859, 602951, 1, 1831, 607619, 609179, 610741, 1, 32309, 55949, 1, 618581, 4001, 1949, 8779, 3491, 7039, 2129, 1, 631229, 57529, 2341, 1, 637601, 58109, 10861, 1, 6779, 645611, 647219, 2309, 2039, 130411, 1, 655289, 1, 3121, 6949, 16141, 663409, 665039, 666671, 1, 21611, 8501, 3761, 3229, 135301, 1889, 21929, 681449, 16661, 136951, 62401, 2539, 6829, 36389, 12601, 694721, 3041, 24071, 11471, 140281, 703081, 1, 37181, 2029, 141961, 1, 1, 714869, 24709, 143651, 719951, 23279, 3461, 12289, 1, 4889, 1, 23609, 733591, 1, 1759, 1, 10429, 1, 148781, 745631, 25771, 68099, 3931, 3671, 1, 39791, 3019, 12451, 13841, 763001, 2459, 26431, 1, 154001, 1, 1, 1, 777041, 1, 6451, 782339, 784109, 1, 14321, 41549, 791209, 792989, 4391, 159311, 27529, 72739, 19559, 1, 1, 1, 9091, 1, 812701, 162901, 2339, 28211, 1, 821741, 8669, 1, 1, 14051, 26801, 1, 2081, 11779, 838139, 4019, 5431, 1, 1, 77029, 849161, 5869, 852851, 1, 856549, 45179, 15641, 6581, 2221, 865829, 1, 2851, 1, 5861, 46061, 21391, 175781, 80071, 882659, 3701, 886421, 1, 1931, 892079, 47051, 895861, 1, 29021, 1999, 903449, 31219, 1, 82651, 47951, 912979, 914891, 183361, 15061, 1, 83869, 1, 6389, 928331, 4451, 932189, 934121, 187211, 937991, 1, 8641, 1, 189151, 1, 949649, 86509, 953551, 1, 1, 30949, 33151, 963341, 17551, 50909, 969239, 13679, 88471, 1, 10979, 979109, 23929, 1, 10369, 1, 989029, 1, 993011, 1, 997001, 5581, 1000999,

6. Sequence of the polynom (only primes)

5, 11, 19, 29, 41, 71, 89, 109, 131, 31, 181, 239, 271, 61, 379, 419, 461, 101, 599, 59, 701, 151, 811, 79, 929, 991, 211, 1259, 281, 1481, 1559, 149, 1721, 1979, 2069, 2161, 2351, 2549, 241, 2861, 2969, 3079, 3191, 661, 311, 3539, 3659, 199, 139, 4159, 4289, 4421, 911, 4691, 439, 4969, 269, 1051, 491, 179, 5851, 1201, 229, 1361, 6971, 7309, 7481, 1531, 191, 8009, 431, 761, 8741, 8929, 829, 9311, 1901, 521, 10099, 10301, 10711, 359, 1031, 2311, 631, 421, 401, 2531, 1171, 13109, 13339, 331, 251, 739, 14519, 509, 3001, 1409, 15749, 16001, 3251, 409, 17029, 17291, 3511, 18089, 1669, 601, 19181, 19739, 20021, 349, 20879, 21169, 1951, 22051, 22349, 389, 4651, 23561, 23869, 24179, 1289, 25121, 25439, 25759, 2371, 5281, 26731, 27059, 449, 1459, 28729, 709, 29411, 541, 971, 30449, 1621, 31151, 6301, 32579, 32941, 6661, 3061, 34039, 1811, 34781, 35531, 3299, 36671, 7411, 37441, 38219, 38611, 39799, 659, 3691, 41411, 42641, 1499, 43889, 751, 1091, 821, 2399, 46439, 4261, 9461, 47741, 48179, 48619, 691, 9901, 499, 50849, 941, 479, 1699, 53129, 53591, 569, 54521, 4999, 1789, 55931, 5171, 1399, 57839, 58321, 619, 59779, 5479, 60761, 12251, 61751, 5659, 3329, 2251, 5981, 66821, 67339, 13781, 69431, 69959, 70489, 1301, 72091, 1231, 3851, 6701, 14851, 571, 1069, 859, 15401, 641, 79241, 1451, 2791, 81509, 1039, 83231, 84389, 2741, 86729, 3011, 87911, 89101, 4721, 8209, 90901, 18301, 92111, 8429, 93329, 93941, 18911, 5009, 95789, 8821, 19531, 3389, 98909, 9049, 919, 20161, 102079, 1741, 2521, 104651, 3631, 1049, 881, 1129, 719, 109229, 109891, 22111, 10111, 2729, 112559, 769, 115259, 839, 809, 117991, 10789, 1511, 24151, 122149, 122849, 123551, 24851, 125669, 11489, 6689, 25561, 128521, 129959, 130681, 1861, 132859, 12211, 27011, 12479, 4759, 27751, 139501, 140249, 2591, 143261, 2441, 144779, 1009, 2411, 148609, 149381, 13721, 4919, 153271, 2801, 154841, 8191, 156419, 157211, 31601, 1319, 5531, 161201, 32401, 1249, 164429, 1109, 33211, 166871, 2749, 15319, 5839, 34031, 8999, 15619, 5569, 4231, 175141, 175979, 176819, 1879, 179351, 2281, 1801, 36551, 183611, 6361, 186191, 187921, 2659, 2131, 17321, 38281, 3259, 6691, 10259, 39161, 17881, 198469, 201151, 202049, 202949, 10729, 1321, 205661, 207479, 208391, 1021, 1061, 11161, 212981, 214831, 19699, 43711, 219491, 222311, 44651, 224201, 225149, 226099, 20641, 1471, 228961, 12101, 231841, 7541, 1229, 5749, 236681, 3361, 12611, 240589, 21961, 1019, 244529, 245519, 246511, 249499, 250499, 251501, 4591, 3209, 254519, 256541, 51511, 23599, 260609, 261631, 23971, 8539, 265739, 9199, 2819, 268841, 269879, 2239, 54601, 3469, 6761, 279311, 1549, 281429, 56711, 1181, 2621, 15149, 57781, 293221, 5351, 3319, 296479, 1429, 59951, 300851, 301949, 303049, 304151, 61051, 27851, 5651, 1151, 4409, 314159, 315281, 63281, 16831, 5261, 1571, 2671, 324329, 10499, 326611, 65551, 328901, 30109, 66701, 334661, 30529, 8219, 338141, 3371, 17981, 11821, 31271, 69031, 346331, 347509, 31699, 349871, 18539, 353429, 354619, 355811, 6491, 358201, 1279, 1709, 365419, 1601, 367841, 2381, 1381, 372709, 2089, 4229, 6421, 380071, 76261, 382541, 9421, 4079, 389999, 391249, 1291, 3911, 396269, 401321, 36599, 1609, 2719, 81281, 13151, 408959, 410239, 82561, 415379, 37879, 417961, 420551, 10289, 22271, 424451, 7741, 7001, 428369, 429679, 39181, 86461, 2179, 434939, 22961, 2141, 4051, 2099, 3391, 8101, 446891, 449569, 1871, 3119, 453601, 2389, 457651, 91801, 41851, 1489, 5879, 1579, 467171, 4639, 42719, 471281, 94531, 3929, 478171, 95911, 480941, 15559, 483719, 44101, 97301, 25679, 489299, 1439, 12071, 3331, 497729, 499141, 501971, 503389, 2789, 46021, 101531, 8369, 511939, 513371, 5419, 17851, 519119, 5849, 9491, 524899, 16979, 27779, 530711, 18401, 2711, 2011, 28549, 546859, 9319, 110261, 552791, 29251, 557261, 111751, 19319, 9209, 563249, 51341, 567761, 569269, 2731, 18461, 114761, 4139, 19891, 7321, 579881, 582931, 1619, 585989, 53411, 117811, 2579, 592129, 4271, 9859, 602951, 1831, 607619, 609179, 610741, 32309, 55949, 618581, 4001, 1949, 8779, 3491, 7039, 2129, 631229, 57529, 2341, 637601, 58109, 10861, 6779, 645611, 647219, 2309, 2039, 130411, 655289, 3121, 6949, 16141, 663409, 665039, 666671, 21611, 8501, 3761, 3229, 135301, 1889, 21929, 681449, 16661, 136951, 62401, 2539, 6829, 36389, 12601, 694721, 3041, 24071, 11471, 140281, 703081, 37181, 2029, 141961, 714869, 24709, 143651, 719951, 23279, 3461, 12289, 4889, 23609, 733591, 1759, 10429, 148781, 745631, 25771, 68099, 3931, 3671, 39791, 3019, 12451, 13841, 763001, 2459, 26431, 154001, 777041, 6451, 782339, 784109, 14321, 41549, 791209, 792989, 4391, 159311, 27529, 72739, 19559, 9091, 812701, 162901, 2339, 28211, 821741, 8669, 14051, 26801, 2081, 11779, 838139, 4019, 5431, 77029, 849161, 5869, 852851, 856549, 45179, 15641, 6581, 2221, 865829, 2851, 5861, 46061, 21391, 175781, 80071, 882659, 3701, 886421, 1931, 892079, 47051, 895861, 29021, 1999, 903449, 31219, 82651, 47951, 912979, 914891, 183361, 15061, 83869, 6389, 928331, 4451, 932189, 934121, 187211, 937991, 8641, 189151, 949649, 86509, 953551, 30949, 33151, 963341, 17551, 50909, 969239, 13679, 88471, 10979, 979109, 23929, 10369, 989029, 993011, 997001, 5581, 1000999,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+3x+1 and
the reducible primes which appear as divisor for the first time
p | x^2+3x+1 and p < x^2+3x+1

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)
1109090.9000000.0000000.900000
21008030500.8000000.3000000.5000008.888889inf5.555555
310007694563130.7690000.4560000.3130009.61250015.2000006.260000
4100007539535121880.7539000.5351000.2188009.80364111.7346496.990416
51000007414557070170750.7414500.5707000.1707509.83485910.6652977.803931
610000007330615935771394840.7330610.5935770.1394849.88685710.4008598.168901
7100000007269658608989211797660.7269660.6089890.1179779.91685310.2596498.458074
81000000007224897762028899102200780.7224900.6202890.1022019.93842910.1855508.662801
91000000000719036536628877070901594660.7190370.6288770.0901599.95220410.1384538.821798
1010000000000716297292663560439418069289850.7162970.6356040.0806939.96190410.1069748.950020


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)
121010.5000000.0000000.500000
243030.7500000.0000000.7500003.000000-nan3.000000
387070.8750000.0000000.8750002.333333-nan2.333333
416142120.8750000.1250000.7500002.000000inf1.714286
532277200.8437500.2187500.6250001.9285713.5000001.666667
6645114370.7968750.2187500.5781251.8888892.0000001.850000
712810246560.7968750.3593750.4375002.0000003.2857141.513514
8256200951050.7812500.3710940.4101561.9607842.0652171.875000
95123962091870.7734380.4082030.3652341.9800002.2000001.780952
1010247874663210.7685550.4550780.3134771.9873742.2296651.716578
112048156810075610.7656250.4916990.2739261.9923762.1609441.747663
124096312121259960.7619630.5187990.2431641.9904342.1102281.775401
1381926186433018560.7551270.5285640.2265621.9820572.0376471.863454
141638412325896733580.7522580.5473020.2049561.9924022.0709011.809267
1532768244811825262290.7471010.5570070.1900941.9862882.0354631.854973
16655364874137112116290.7437290.5662840.1774441.9909732.0333111.866913
171310729700275223217790.7400670.5739060.1661611.9901522.0269181.872818
18262144193389152497408920.7377200.5817300.1559911.9936602.0272661.877589
19524288385401308348770530.7350940.5881270.1469671.9928802.0219941.884305
2010485767684836227461457370.7328820.5938970.1389861.9939832.0196211.891386
212097152153274812566322761160.7308710.5992090.1316621.9945112.0178891.894618
224194304305784725323255255220.7290480.6037530.1252941.9950102.0151681.903265
2383886086101716510038810013280.7273810.6080140.1193681.9954292.0141131.905397
2416777216121776781026518419124940.7258460.6118530.1139941.9957792.0126281.909958
2533554432243093532064981836595350.7244750.6154130.1090631.9962222.0116361.913488
2667108864485324884151149170209970.7231900.6185690.1046211.9964532.0102591.918549
271342177289690544783420010134854370.7220020.6215270.1004741.9967132.0095641.920730
28268435456193515509167568579259469300.7209010.6242420.0966601.9969522.0087341.924070
29536870912386482508336484899499976090.7198800.6267520.0931281.9971662.0080431.926918
301073741824771959328675502032964572960.7189430.6291100.0898331.9973982.0075251.929238
312147483648154201731613556559251863613910.7180580.6312760.0867811.9975372.0068861.932061
324294967296308049942727200372143604622130.7172350.6333080.0839271.9977072.0064361.934211
338589934592615435043254563682496979821830.7164610.6352050.0812561.9978422.0059901.936353
3417179869184122962376631094334163613528960270.7157350.6369860.0787491.9979752.0056091.938296
3534359738368245691208642194428889326248319710.7150560.6386630.0763931.9981012.0052641.940158


8. Check for existing Integer Sequences by OEIS

Found in Database : 1, 5, 11, 19, 29, 41, 1, 71, 89, 109, 131, 31, 181, 1, 239, 271, 61, 1, 379, 419,
Found in Database : 5, 11, 19, 29, 41, 71, 89, 109, 131, 31, 181, 239, 271, 61, 379, 419, 461, 101, 599, 59, 701, 151, 811, 79, 929, 991, 211, 1259, 281, 1481, 1559, 149,
Found in Database : 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149,

ABCDEFGHIJKLMNOP
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 number 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
1210101000000000
2431202100000000
3872522120000000
416125723342200011
5322081236567520223
66437112681081114952345
71285618381315141446232310131013
825610533722731222595534216322621
951218760127464746482091179255564652
10102432110022182808178466240226104124123115
1120485611883731371481401361007531476240244262261
124096996337659242261261232212511041021512519548546
138192185662612304564864694454330224720831042109810941096
14163843358112122378308738657908967469442732202224822562261
1532768622920604169155315811597149818252953287204584453745544577
1665536116293840778928772895297328843711219385177279235919793469334
1713107221779723214547542754895483538075223391243609918757188501886218754
1826214440892136352725710136102201030310233152497791447335338131380463828738033
195242887705325708513451914119222194991919130834815936514898376851772057739676896
201048576145737485449719336383363673667336314622746321282301464155484155474156180155608
21209715227611692189183927688356890069447689341256632646877609755314090313279315102314161
224194304525522175090350432130979131350131538131655253232513012701231055633418631782634148632977
23838860810013283339026674262497972505572504382505365100388261712124832671274858127494712754201275163
24167772161912494638091127440347737547842347806647863010265184526103750041472565482256869825657752565229
25335544323659535122022124393149143419142249152239157472064981810573049100767695163253516382051622605160485
2667108864702099723414494679548175472617551411755577175555341511491212301722028131910377643103780251037819010377633
271342177281348543744980988987339337097733700003373648337081283420010426198634080014720857128208539082085225320856721
28268435456259469308650127172968036486066648509964891716486594167568579855251218204345841896566418919864188882141891206
295368709124999760916668406333292031249647312499863125029801249829333648489917159692416488797584122619841240928412139384116795
30107374182496457296321553836430191324112275241135862411761224113823675502032344211141331290891168873783168881978168882449168863822
31214748364818636139162124339124237052465908124658819646592772465896111355655925690293653665362272338904565338914861338936040338900459
32429496729636046221312015820424030400990117543901092899011490390120478272003721413840980381335939176679998048679983145680048073680007948
3385899345926979821832326673894653147941744954731744805771744994311745067025456368249277474176026816264891364049636136405187613641468761364119861
3417179869184135289602745097340490192262333823589633820564533822866533822582110943341636556181837753815232592735806560273582174227359042982735809036
3534359738368262483197187493965717498923146562109256561913596562100166562196712194428889311146758934107975299595486047693548610754654861026445486031010