Inhaltsverzeichnis

Development of
Algorithmic Constructions

01:01:34
Deutsch
29.Mar 2024

Polynom = x^2+17x-41

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) = 41 = 41
f(1) = 23 = 23
f(2) = 3 = 3
f(3) = 19 = 19
f(4) = 43 = 43
f(5) = 69 = 3*23
f(6) = 97 = 97
f(7) = 127 = 127
f(8) = 159 = 3*53
f(9) = 193 = 193
f(10) = 229 = 229
f(11) = 267 = 3*89
f(12) = 307 = 307
f(13) = 349 = 349
f(14) = 393 = 3*131
f(15) = 439 = 439
f(16) = 487 = 487
f(17) = 537 = 3*179
f(18) = 589 = 19*31
f(19) = 643 = 643
f(20) = 699 = 3*233
f(21) = 757 = 757
f(22) = 817 = 19*43
f(23) = 879 = 3*293
f(24) = 943 = 23*41
f(25) = 1009 = 1009
f(26) = 1077 = 3*359
f(27) = 1147 = 31*37
f(28) = 1219 = 23*53
f(29) = 1293 = 3*431
f(30) = 1369 = 37*37
f(31) = 1447 = 1447
f(32) = 1527 = 3*509
f(33) = 1609 = 1609
f(34) = 1693 = 1693
f(35) = 1779 = 3*593
f(36) = 1867 = 1867
f(37) = 1957 = 19*103
f(38) = 2049 = 3*683
f(39) = 2143 = 2143
f(40) = 2239 = 2239
f(41) = 2337 = 3*19*41
f(42) = 2437 = 2437
f(43) = 2539 = 2539
f(44) = 2643 = 3*881
f(45) = 2749 = 2749
f(46) = 2857 = 2857
f(47) = 2967 = 3*23*43
f(48) = 3079 = 3079
f(49) = 3193 = 31*103
f(50) = 3309 = 3*1103
f(51) = 3427 = 23*149
f(52) = 3547 = 3547
f(53) = 3669 = 3*1223
f(54) = 3793 = 3793
f(55) = 3919 = 3919
f(56) = 4047 = 3*19*71
f(57) = 4177 = 4177
f(58) = 4309 = 31*139
f(59) = 4443 = 3*1481
f(60) = 4579 = 19*241
f(61) = 4717 = 53*89
f(62) = 4857 = 3*1619
f(63) = 4999 = 4999
f(64) = 5143 = 37*139
f(65) = 5289 = 3*41*43
f(66) = 5437 = 5437
f(67) = 5587 = 37*151
f(68) = 5739 = 3*1913
f(69) = 5893 = 71*83
f(70) = 6049 = 23*263
f(71) = 6207 = 3*2069
f(72) = 6367 = 6367
f(73) = 6529 = 6529
f(74) = 6693 = 3*23*97
f(75) = 6859 = 19*19*19
f(76) = 7027 = 7027
f(77) = 7197 = 3*2399
f(78) = 7369 = 7369
f(79) = 7543 = 19*397
f(80) = 7719 = 3*31*83
f(81) = 7897 = 53*149
f(82) = 8077 = 41*197
f(83) = 8259 = 3*2753
f(84) = 8443 = 8443
f(85) = 8629 = 8629
f(86) = 8817 = 3*2939
f(87) = 9007 = 9007
f(88) = 9199 = 9199
f(89) = 9393 = 3*31*101
f(90) = 9589 = 43*223
f(91) = 9787 = 9787
f(92) = 9987 = 3*3329
f(93) = 10189 = 23*443
f(94) = 10393 = 19*547
f(95) = 10599 = 3*3533
f(96) = 10807 = 101*107
f(97) = 11017 = 23*479
f(98) = 11229 = 3*19*197
f(99) = 11443 = 11443
f(100) = 11659 = 89*131

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+17x-41

f(0)=41
f(1)=23
f(2)=3
f(3)=19
f(4)=43
f(5)=1
f(6)=97
f(7)=127
f(8)=53
f(9)=193
f(10)=229
f(11)=89
f(12)=307
f(13)=349
f(14)=131
f(15)=439
f(16)=487
f(17)=179
f(18)=31
f(19)=643
f(20)=233
f(21)=757
f(22)=1
f(23)=293
f(24)=1
f(25)=1009
f(26)=359
f(27)=37
f(28)=1
f(29)=431
f(30)=1
f(31)=1447
f(32)=509
f(33)=1609
f(34)=1693
f(35)=593
f(36)=1867
f(37)=103
f(38)=683
f(39)=2143
f(40)=2239
f(41)=1
f(42)=2437
f(43)=2539
f(44)=881
f(45)=2749
f(46)=2857
f(47)=1
f(48)=3079
f(49)=1
f(50)=1103
f(51)=149
f(52)=3547
f(53)=1223
f(54)=3793
f(55)=3919
f(56)=71
f(57)=4177
f(58)=139
f(59)=1481
f(60)=241
f(61)=1
f(62)=1619
f(63)=4999
f(64)=1
f(65)=1
f(66)=5437
f(67)=151
f(68)=1913
f(69)=83
f(70)=263
f(71)=2069
f(72)=6367
f(73)=6529
f(74)=1
f(75)=1
f(76)=7027
f(77)=2399
f(78)=7369
f(79)=397
f(80)=1
f(81)=1
f(82)=197
f(83)=2753
f(84)=8443
f(85)=8629
f(86)=2939
f(87)=9007
f(88)=9199
f(89)=101
f(90)=223
f(91)=9787
f(92)=3329
f(93)=443
f(94)=547
f(95)=3533
f(96)=107
f(97)=479
f(98)=1
f(99)=11443

b) Substitution of the polynom
The polynom f(x)=x^2+17x-41 could be written as f(y)= y^2-113.25 with x=y-8.5

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+8.5
f'(x)>2x+16

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

41, 23, 3, 19, 43, 1, 97, 127, 53, 193, 229, 89, 307, 349, 131, 439, 487, 179, 31, 643, 233, 757, 1, 293, 1, 1009, 359, 37, 1, 431, 1, 1447, 509, 1609, 1693, 593, 1867, 103, 683, 2143, 2239, 1, 2437, 2539, 881, 2749, 2857, 1, 3079, 1, 1103, 149, 3547, 1223, 3793, 3919, 71, 4177, 139, 1481, 241, 1, 1619, 4999, 1, 1, 5437, 151, 1913, 83, 263, 2069, 6367, 6529, 1, 1, 7027, 2399, 7369, 397, 1, 1, 197, 2753, 8443, 8629, 2939, 9007, 9199, 101, 223, 9787, 3329, 443, 547, 3533, 107, 479, 1, 11443, 1, 1, 12097, 1, 113, 1, 317, 4409, 313, 13693, 4643, 457, 14407, 257, 281, 15139, 1, 823, 15889, 5381, 1, 16657, 5639, 419, 17443, 5903, 17977, 1, 6173, 18793, 19069, 6449, 1033, 463, 1, 20479, 1093, 7019, 577, 941, 1, 601, 727, 331, 23143, 1, 7919, 587, 24379, 8231, 1, 1, 1, 25969, 26293, 467, 26947, 27277, 9203, 27943, 28279, 9539, 1259, 353, 1, 29989, 1319, 1, 31039, 31393, 557, 1, 32467, 1, 1747, 907, 1, 34297, 937, 11681, 35419, 35797, 389, 36559, 36943, 541, 37717, 719, 1, 1, 1, 13229, 40087, 2131, 1, 41299, 1, 1, 599, 42943, 1, 43777, 1, 1, 1453, 1109, 15299, 46327, 1, 15731, 47629, 677, 1, 1579, 49393, 449, 50287, 1, 1, 1201, 983, 17519, 53017, 53479, 17981, 54409, 54877, 971, 55819, 1373, 1, 1, 57727, 19403, 58693, 1, 19889, 1399, 60649, 1, 1163, 62137, 20879, 613, 2053, 21383, 1, 65167, 21893, 1789, 3511, 22409, 1831, 1, 997, 673, 69847, 23459, 3083, 71443, 23993, 571, 1699, 24533, 74143, 3931, 809, 75787, 787, 1, 1889, 78007, 26189, 1493, 79693, 863, 617, 3539, 1, 1, 1933, 1213, 1187, 84859, 1499, 86029, 2341, 709, 4621, 2389, 29663, 887, 90187, 1, 91393, 1, 30869, 1, 929, 31481, 4133, 1, 32099, 5101, 4241, 761, 3187, 5233, 33353, 100693, 947, 829, 102607, 1, 34631, 977, 1, 35279, 1283, 1, 35933, 108457, 5743, 1, 1, 111109, 1, 4889, 1, 1, 1013, 1019, 38609, 1, 117193, 39293, 853, 3847, 39983, 1, 1709, 2141, 883, 123439, 41381, 6571, 1, 42089, 2953, 857, 1861, 129127, 3167, 1, 131293, 132019, 44249, 1051, 1303, 1451, 1, 136417, 1, 3727, 7297, 1, 3259, 1583, 1523, 1, 1, 47969, 144667, 6323, 48731, 821, 147727, 49499, 149269, 1, 50273, 151597, 152377, 2687, 1, 1, 51839, 156307, 1237, 52631, 5119, 3709, 1, 3929, 1669, 1, 7109, 4441, 2897, 1, 4507, 1, 8863, 169219, 56681, 1, 171697, 1, 173359, 4051, 1423, 2477, 2129, 59183, 178393, 7793, 60029, 1, 181789, 2647, 183499, 1, 1, 1, 2633, 62603, 188677, 189547, 63473, 1, 6199, 1, 3659, 194809, 1, 196579, 1, 1787, 8663, 1, 3527, 201937, 8819, 1, 1811, 205549, 68819, 207367, 208279, 1, 210109, 5147, 1, 212869, 1, 3767, 2423, 216577, 72503, 11497, 1, 1, 2281, 222199, 74381, 9743, 225037, 75329, 991, 227893, 76283, 6211, 230767, 77243, 1, 1, 1, 5479, 12451, 79181, 238519, 1723, 1129, 7789, 1, 1531, 1049, 245407, 3571, 247393, 248389, 1, 1, 1, 84131, 1, 254407, 4481, 1721, 257443, 1, 1, 1, 1, 262543, 4973, 1, 11549, 7207, 89231, 268729, 1, 4751, 271849, 8803, 1, 1, 276037, 92363, 278143, 1, 93419, 281317, 9109, 1, 284509, 1, 95549, 1931, 2699, 4201, 15313, 292027, 2383, 12791, 15541, 1, 297457, 1, 99881, 300739, 1, 1, 304039, 305143, 1, 1, 308467, 2789, 310693, 16411, 104309, 3049, 1, 1, 317419, 2819, 1, 6053, 3319, 107693, 324217, 1459, 2531, 3181, 328789, 1549, 3989, 1, 5849, 334549, 1, 112289, 17791, 8273, 113453, 1, 342697, 1381, 1, 15053, 2693, 9421, 349759, 116981, 1, 353317, 118169, 1, 356893, 119363, 6779, 18973, 120563, 3593, 4091, 1, 366517, 8969, 5347, 370159, 3677, 124199, 16253, 375043, 125423, 8779, 1, 126653, 1301, 382429, 1, 384907, 386149, 1, 9479, 12577, 130379, 392389, 10639, 1, 396157, 1, 132893, 1, 401209, 1, 1571, 1, 135431, 1, 3821, 136709, 411409, 4637, 137993, 3881, 5867, 4493, 10223, 420439, 1, 3331, 424339, 2677, 1, 1777, 1, 430879, 1, 1741, 434827, 436147, 145823, 438793, 4273, 1, 442777, 444109, 4013, 1951, 5399, 1, 450799, 1, 151163, 1, 1, 1, 1, 4073, 1, 20129, 2357, 155231, 1, 1667, 1, 15199, 472543, 3853, 475297, 2663, 8387, 5387, 4957, 160739, 25453, 484999, 1, 487789, 21269, 3803, 13297, 493393, 1, 13411, 1, 1, 500443, 16189, 167759, 1, 506119, 169181, 508969, 26863, 170609, 7229, 16603, 4001, 1483, 1847, 173483, 22691, 1, 174929, 9929, 22943, 1, 530599, 1, 177839, 6011, 536443, 9437, 1, 1, 180773, 543793, 14737, 5879, 548227, 1, 1, 12853, 554167, 8053, 1, 558643, 1, 24419, 1, 1759, 1, 567649, 1, 570667, 572179, 191231, 10853, 576727, 192749, 1, 581293, 1, 584347, 585877, 1901, 1, 25673, 197339, 1, 31321, 8647, 598189, 8447, 5417, 602839, 4759, 1, 19597, 609067, 2287, 612193, 613759, 1, 616897, 1, 1, 621619, 3229, 1, 1, 627943, 1, 1, 27509, 211433, 635893, 637489, 213029, 640687, 20719, 214631, 645499, 1, 1, 650329, 651943, 1663, 34483, 21187, 1, 17839, 1999, 9613, 17971, 666559, 222731, 29123, 671467, 2521, 674749, 2161, 226013, 1, 681337, 227663, 684643, 1, 5333, 4567, 13043, 7451, 694609, 4673, 2803, 2207, 30491, 234323, 704647, 1, 1, 709693, 37441, 237689, 10067, 716449, 1, 719839, 19501, 241079, 724939, 1, 2269, 1, 731767, 244493, 735193, 1, 246209, 32189, 742069, 13049, 24049, 1, 249659, 39511, 2861, 251393, 1, 8513, 253133, 1, 6007, 254879, 766387, 768139, 1, 3917, 773407, 258389, 1, 778693, 11311, 14759, 41263, 7079, 1, 1, 7127, 18439, 794659, 1, 19469, 1, 1, 1, 3517, 269063, 1877, 1, 1, 814393, 1, 1, 8117, 1, 3307, 825259, 9293, 1, 830719, 8243, 278123, 1, 19489, 3943, 15881, 10163, 14831, 4733, 849049, 283631, 1, 854587, 9209, 23197, 860143, 287333, 1, 865717, 289193, 869443, 1, 1, 1, 1, 5527, 46351, 6737, 294809, 2887, 46747, 296693, 891967, 893857, 298583, 4651, 1, 1, 903337, 22079, 13147, 6101, 910957, 3137, 1, 48247, 7121, 10343, 1, 16217, 926293, 25087, 310049, 1, 934009, 311981, 2411, 2243, 5923, 943699, 9181, 1, 1, 1, 1, 1, 957349, 13903, 50593, 6379, 1, 6491, 1, 323699, 973069, 1, 325673, 978997, 1, 1, 24023, 986929, 329639, 52153, 32029, 8963, 1, 2767, 1, 1, 43691, 335633, 1, 1010917, 3343, 1,

6. Sequence of the polynom (only primes)

41, 23, 3, 19, 43, 97, 127, 53, 193, 229, 89, 307, 349, 131, 439, 487, 179, 31, 643, 233, 757, 293, 1009, 359, 37, 431, 1447, 509, 1609, 1693, 593, 1867, 103, 683, 2143, 2239, 2437, 2539, 881, 2749, 2857, 3079, 1103, 149, 3547, 1223, 3793, 3919, 71, 4177, 139, 1481, 241, 1619, 4999, 5437, 151, 1913, 83, 263, 2069, 6367, 6529, 7027, 2399, 7369, 397, 197, 2753, 8443, 8629, 2939, 9007, 9199, 101, 223, 9787, 3329, 443, 547, 3533, 107, 479, 11443, 12097, 113, 317, 4409, 313, 13693, 4643, 457, 14407, 257, 281, 15139, 823, 15889, 5381, 16657, 5639, 419, 17443, 5903, 17977, 6173, 18793, 19069, 6449, 1033, 463, 20479, 1093, 7019, 577, 941, 601, 727, 331, 23143, 7919, 587, 24379, 8231, 25969, 26293, 467, 26947, 27277, 9203, 27943, 28279, 9539, 1259, 353, 29989, 1319, 31039, 31393, 557, 32467, 1747, 907, 34297, 937, 11681, 35419, 35797, 389, 36559, 36943, 541, 37717, 719, 13229, 40087, 2131, 41299, 599, 42943, 43777, 1453, 1109, 15299, 46327, 15731, 47629, 677, 1579, 49393, 449, 50287, 1201, 983, 17519, 53017, 53479, 17981, 54409, 54877, 971, 55819, 1373, 57727, 19403, 58693, 19889, 1399, 60649, 1163, 62137, 20879, 613, 2053, 21383, 65167, 21893, 1789, 3511, 22409, 1831, 997, 673, 69847, 23459, 3083, 71443, 23993, 571, 1699, 24533, 74143, 3931, 809, 75787, 787, 1889, 78007, 26189, 1493, 79693, 863, 617, 3539, 1933, 1213, 1187, 84859, 1499, 86029, 2341, 709, 4621, 2389, 29663, 887, 90187, 91393, 30869, 929, 31481, 4133, 32099, 5101, 4241, 761, 3187, 5233, 33353, 100693, 947, 829, 102607, 34631, 977, 35279, 1283, 35933, 108457, 5743, 111109, 4889, 1013, 1019, 38609, 117193, 39293, 853, 3847, 39983, 1709, 2141, 883, 123439, 41381, 6571, 42089, 2953, 857, 1861, 129127, 3167, 131293, 132019, 44249, 1051, 1303, 1451, 136417, 3727, 7297, 3259, 1583, 1523, 47969, 144667, 6323, 48731, 821, 147727, 49499, 149269, 50273, 151597, 152377, 2687, 51839, 156307, 1237, 52631, 5119, 3709, 3929, 1669, 7109, 4441, 2897, 4507, 8863, 169219, 56681, 171697, 173359, 4051, 1423, 2477, 2129, 59183, 178393, 7793, 60029, 181789, 2647, 183499, 2633, 62603, 188677, 189547, 63473, 6199, 3659, 194809, 196579, 1787, 8663, 3527, 201937, 8819, 1811, 205549, 68819, 207367, 208279, 210109, 5147, 212869, 3767, 2423, 216577, 72503, 11497, 2281, 222199, 74381, 9743, 225037, 75329, 991, 227893, 76283, 6211, 230767, 77243, 5479, 12451, 79181, 238519, 1723, 1129, 7789, 1531, 1049, 245407, 3571, 247393, 248389, 84131, 254407, 4481, 1721, 257443, 262543, 4973, 11549, 7207, 89231, 268729, 4751, 271849, 8803, 276037, 92363, 278143, 93419, 281317, 9109, 284509, 95549, 1931, 2699, 4201, 15313, 292027, 2383, 12791, 15541, 297457, 99881, 300739, 304039, 305143, 308467, 2789, 310693, 16411, 104309, 3049, 317419, 2819, 6053, 3319, 107693, 324217, 1459, 2531, 3181, 328789, 1549, 3989, 5849, 334549, 112289, 17791, 8273, 113453, 342697, 1381, 15053, 2693, 9421, 349759, 116981, 353317, 118169, 356893, 119363, 6779, 18973, 120563, 3593, 4091, 366517, 8969, 5347, 370159, 3677, 124199, 16253, 375043, 125423, 8779, 126653, 1301, 382429, 384907, 386149, 9479, 12577, 130379, 392389, 10639, 396157, 132893, 401209, 1571, 135431, 3821, 136709, 411409, 4637, 137993, 3881, 5867, 4493, 10223, 420439, 3331, 424339, 2677, 1777, 430879, 1741, 434827, 436147, 145823, 438793, 4273, 442777, 444109, 4013, 1951, 5399, 450799, 151163, 4073, 20129, 2357, 155231, 1667, 15199, 472543, 3853, 475297, 2663, 8387, 5387, 4957, 160739, 25453, 484999, 487789, 21269, 3803, 13297, 493393, 13411, 500443, 16189, 167759, 506119, 169181, 508969, 26863, 170609, 7229, 16603, 4001, 1483, 1847, 173483, 22691, 174929, 9929, 22943, 530599, 177839, 6011, 536443, 9437, 180773, 543793, 14737, 5879, 548227, 12853, 554167, 8053, 558643, 24419, 1759, 567649, 570667, 572179, 191231, 10853, 576727, 192749, 581293, 584347, 585877, 1901, 25673, 197339, 31321, 8647, 598189, 8447, 5417, 602839, 4759, 19597, 609067, 2287, 612193, 613759, 616897, 621619, 3229, 627943, 27509, 211433, 635893, 637489, 213029, 640687, 20719, 214631, 645499, 650329, 651943, 1663, 34483, 21187, 17839, 1999, 9613, 17971, 666559, 222731, 29123, 671467, 2521, 674749, 2161, 226013, 681337, 227663, 684643, 5333, 4567, 13043, 7451, 694609, 4673, 2803, 2207, 30491, 234323, 704647, 709693, 37441, 237689, 10067, 716449, 719839, 19501, 241079, 724939, 2269, 731767, 244493, 735193, 246209, 32189, 742069, 13049, 24049, 249659, 39511, 2861, 251393, 8513, 253133, 6007, 254879, 766387, 768139, 3917, 773407, 258389, 778693, 11311, 14759, 41263, 7079, 7127, 18439, 794659, 19469, 3517, 269063, 1877, 814393, 8117, 3307, 825259, 9293, 830719, 8243, 278123, 19489, 3943, 15881, 10163, 14831, 4733, 849049, 283631, 854587, 9209, 23197, 860143, 287333, 865717, 289193, 869443, 5527, 46351, 6737, 294809, 2887, 46747, 296693, 891967, 893857, 298583, 4651, 903337, 22079, 13147, 6101, 910957, 3137, 48247, 7121, 10343, 16217, 926293, 25087, 310049, 934009, 311981, 2411, 2243, 5923, 943699, 9181, 957349, 13903, 50593, 6379, 6491, 323699, 973069, 325673, 978997, 24023, 986929, 329639, 52153, 32029, 8963, 2767, 43691, 335633, 1010917, 3343,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+17x-41 and
the reducible primes which appear as divisor for the first time
p | x^2+17x-41 and p < x^2+17x-41

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)
11010911.0000000.9000001.0000000.0000000.0000000.000000
21007844340.7800000.4400000.7800007.8000004.88888934.000000
31.0007512604910.7510000.2600000.7510009.6282055.90909114.441176
410.0007.3561.8805.4760.7356000.1880000.7356009.7949407.23076911.152749
5100.00073.00514.71758.2880.7300500.1471700.7300509.9245517.82819110.644266
61.000.000723.761119.964603.7970.7237610.1199640.7237619.9138568.15138910.358856
710.000.0007.192.2131.013.6586.178.5550.7192210.1013660.7192219.9372768.44968510.232835
8100.000.00071.568.3868.782.97962.785.4070.7156840.0878300.7156849.9508158.66463810.161827
91.000.000.000713.039.22077.502.001635.537.2190.7130390.0775020.7130399.9630488.82411310.122372
1010.000.000.0007.109.478.757693.605.9006.415.872.8570.7109480.0693610.7109489.9706708.94952210.095197


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
245501.2500001.2500000.0000001.6666671.666667-nan
388711.0000000.8750000.1250001.6000001.400000inf
416161331.0000000.8125000.1875002.0000001.8571433.000000
532261790.8125000.5312500.2812501.6250001.3076923.000000
6645232200.8125000.5000000.3125002.0000001.8823532.222222
71289952470.7734380.4062500.3671881.9038461.6250002.350000
8256197921050.7695310.3593750.4101561.9898991.7692312.234043
95123911462450.7636720.2851560.4785161.9847721.5869572.333333
101.0247722675050.7539060.2607420.4931641.9744251.8287672.061224
112.0481.5304901.0400.7470700.2392580.5078121.9818651.8352062.059406
124.0963.0398642.1750.7419430.2109380.5310061.9862741.7632652.091346
138.1926.0481.5774.4710.7382810.1925050.5457761.9901281.8252312.055632
1416.38412.0262.9209.1060.7340090.1782230.5557861.9884261.8516172.036681
1532.76824.0255.38518.6400.7331850.1643370.5688481.9977551.8441782.047002
1665.53647.90510.05037.8550.7309720.1533510.5776211.9939651.8662952.030848
17131.07295.56118.75376.8080.7290730.1430740.5859991.9948021.8659702.029006
18262.144190.54135.216155.3250.7268560.1343380.5925181.9939201.8778862.022250
19524.288380.34466.278314.0660.7254490.1264150.5990331.9961271.8820422.021993
201.048.576758.808125.219633.5890.7236560.1194180.6042381.9950571.8893002.017375
212.097.1521.514.348237.5601.276.7880.7220970.1132770.6088201.9956931.8971562.015167
224.194.3043.023.248451.8852.571.3630.7207980.1077380.6130611.9964021.9021932.013931
238.388.6086.035.662860.7915.174.8710.7195070.1026140.6168931.9964161.9048892.012501
2416.777.21612.051.5661.642.44610.409.1200.7183290.0978970.6204321.9967271.9080662.011474
2533.554.43224.066.0963.146.24120.919.8550.7172260.0937650.6234601.9969271.9155832.009762
2667.108.86448.065.0956.033.23942.031.8560.7162260.0899020.6263231.9972121.9176022.009185
27134.217.72896.007.57011.590.73684.416.8340.7153120.0863580.6289541.9974491.9211472.008401
28268.435.456191.792.39422.300.503169.491.8910.7144820.0830760.6314061.9976801.9239942.007797
29536.870.912383.162.04642.973.752340.188.2940.7136950.0800450.6336501.9977961.9270312.007107
301.073.741.824765.548.11882.915.152682.632.9660.7129720.0772210.6357511.9979751.9294372.006633
312.147.483.6481.529.639.812160.205.2671.369.434.5450.7122940.0746010.6376931.9980981.9321592.006107
324.294.967.2963.056.570.809309.842.8232.746.727.9860.7116630.0721410.6395221.9982291.9340362.005739
338.589.934.5926.108.065.581599.970.1235.508.095.4580.7110720.0698460.6412271.9983391.9363692.005330
3417.179.869.18412.206.616.1941.162.902.49411.043.713.7000.7105190.0676900.6428291.9984421.9382672.004997


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
123021101
245221301
387422302
416131023424
532171424535
6643229288610
7128524921415914
82569289224211829
9512146143236333344
101.024267264264656672
112.0484904872121124126119
124.0968648612215224223202
138.1921.5771.5742408398398373
1416.3842.9202.9172751746735688
1532.7685.3855.38221.3741.3501.3761.285
1665.53610.05010.04722.5242.5112.5252.490
17131.07218.75318.75024.7344.6934.6924.634
18262.14435.21635.21328.8568.8658.7748.721
19524.28866.27866.275216.63216.66816.54116.437
201.048.576125.219125.216231.43631.46931.18831.126
212.097.152237.560237.557259.66259.40559.29659.197
224.194.304451.885451.8822113.184113.084112.825112.792
238.388.608860.791860.7882215.896214.864214.935215.096
2416.777.2161.642.4461.642.4432411.190410.226410.596410.434
2533.554.4323.146.2413.146.2382786.911785.965786.620786.745
2667.108.8646.033.2396.033.23621.509.1351.507.3581.508.6371.508.109
27134.217.72811.590.73611.590.73322.897.9122.897.0052.898.2112.897.608
28268.435.45622.300.50322.300.50025.575.9245.573.9665.576.1915.574.422
29536.870.91242.973.75242.973.749210.745.49210.741.05310.743.18410.744.023
301.073.741.82482.915.15282.915.149220.731.93020.729.47820.725.93320.727.811
312.147.483.648160.205.267160.205.264240.054.78940.051.26440.048.76040.050.454
324.294.967.296309.842.823309.842.820277.458.72877.466.66577.456.79977.460.631
338.589.934.592599.970.123599.970.1202149.995.941150.002.834149.977.562149.993.786
3417.179.869.1841.162.902.4941.162.902.4912290.736.909290.734.846290.705.195290.725.544


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
381010010
4163031110
5329092232
664203176545
71284793814101112
8256105327326262726
95122458116461666058
101.024505175330118130134123
112.0481.040383657251259274256
124.0962.1758541.321525554536560
138.1924.4711.7682.7031.0971.1281.1241.122
1416.3849.1063.7465.3602.2872.2662.2792.274
1532.76818.6407.86410.7764.6234.6734.7014.643
1665.53637.85516.21021.6459.4069.5129.4619.476
17131.07276.80833.29443.51419.05719.22819.14119.382
18262.144155.32568.12387.20238.66038.83039.03838.797
19524.288314.066139.156174.91078.27978.55578.81578.417
201.048.576633.589283.243350.346158.208158.442158.811158.128
212.097.1521.276.788575.172701.616319.013319.578319.675318.522
224.194.3042.571.3631.165.5941.405.769642.568643.046643.432642.317
238.388.6085.174.8712.360.1332.814.7381.293.2891.293.4761.294.5321.293.574
2416.777.21610.409.1204.770.7645.638.3562.601.1672.601.0552.603.1552.603.743
2533.554.43220.919.8559.631.64311.288.2125.228.3365.229.2025.231.2555.231.062
2667.108.86442.031.85619.432.53222.599.32410.504.15110.507.44510.511.58410.508.676
27134.217.72884.416.83439.171.32545.245.50921.099.47821.106.45721.107.58821.103.311
28268.435.456169.491.89178.905.05490.586.83742.373.44242.377.78542.369.42842.371.236
29536.870.912340.188.294158.850.567181.337.72785.050.74085.057.15185.046.18185.034.222
301.073.741.824682.632.966319.634.031362.998.935170.656.538170.666.313170.655.542170.654.573
312.147.483.6481.369.434.545642.878.355726.556.190342.353.115342.387.262342.348.262342.345.906
324.294.967.2962.746.727.9861.292.549.3871.454.178.599686.660.930686.697.378686.689.610686.680.068
338.589.934.5925.508.095.4582.597.736.1542.910.359.3041.377.008.2171.377.028.4811.377.029.9681.377.028.792
3417.179.869.18411.043.713.7005.219.228.0085.824.485.6922.760.931.8772.760.899.0912.760.962.3052.760.920.427


8. Check for existing Integer Sequences by OEIS

Found in Database : 41, 23, 3, 19, 43, 1, 97, 127, 53, 193, 229, 89, 307, 349, 131, 439, 487, 179, 31, 643,
Found in Database : 41, 23, 3, 19, 43, 97, 127, 53, 193, 229, 89, 307, 349, 131, 439, 487, 179, 31, 643, 233, 757, 293, 1009, 359, 37, 431, 1447, 509, 1609, 1693, 593, 1867, 103, 683, 2143,
Found in Database : 3, 19, 23, 31, 37, 41, 43, 53, 71, 83, 89, 97, 101, 103, 107, 113, 127, 131, 139, 149,