Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:58:27
Deutsch
11.May 2021

Polynom = x^2+1x+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) = 43 = 43
f(2) = 47 = 47
f(3) = 53 = 53
f(4) = 61 = 61
f(5) = 71 = 71
f(6) = 83 = 83
f(7) = 97 = 97
f(8) = 113 = 113
f(9) = 131 = 131
f(10) = 151 = 151
f(11) = 173 = 173
f(12) = 197 = 197
f(13) = 223 = 223
f(14) = 251 = 251
f(15) = 281 = 281
f(16) = 313 = 313
f(17) = 347 = 347
f(18) = 383 = 383
f(19) = 421 = 421
f(20) = 461 = 461
f(21) = 503 = 503
f(22) = 547 = 547
f(23) = 593 = 593
f(24) = 641 = 641
f(25) = 691 = 691
f(26) = 743 = 743
f(27) = 797 = 797
f(28) = 853 = 853
f(29) = 911 = 911
f(30) = 971 = 971
f(31) = 1033 = 1033
f(32) = 1097 = 1097
f(33) = 1163 = 1163
f(34) = 1231 = 1231
f(35) = 1301 = 1301
f(36) = 1373 = 1373
f(37) = 1447 = 1447
f(38) = 1523 = 1523
f(39) = 1601 = 1601
f(40) = 1681 = 41*41
f(41) = 1763 = 41*43
f(42) = 1847 = 1847
f(43) = 1933 = 1933
f(44) = 2021 = 43*47
f(45) = 2111 = 2111
f(46) = 2203 = 2203
f(47) = 2297 = 2297
f(48) = 2393 = 2393
f(49) = 2491 = 47*53
f(50) = 2591 = 2591
f(51) = 2693 = 2693
f(52) = 2797 = 2797
f(53) = 2903 = 2903
f(54) = 3011 = 3011
f(55) = 3121 = 3121
f(56) = 3233 = 53*61
f(57) = 3347 = 3347
f(58) = 3463 = 3463
f(59) = 3581 = 3581
f(60) = 3701 = 3701
f(61) = 3823 = 3823
f(62) = 3947 = 3947
f(63) = 4073 = 4073
f(64) = 4201 = 4201
f(65) = 4331 = 61*71
f(66) = 4463 = 4463
f(67) = 4597 = 4597
f(68) = 4733 = 4733
f(69) = 4871 = 4871
f(70) = 5011 = 5011
f(71) = 5153 = 5153
f(72) = 5297 = 5297
f(73) = 5443 = 5443
f(74) = 5591 = 5591
f(75) = 5741 = 5741
f(76) = 5893 = 71*83
f(77) = 6047 = 6047
f(78) = 6203 = 6203
f(79) = 6361 = 6361
f(80) = 6521 = 6521
f(81) = 6683 = 41*163
f(82) = 6847 = 41*167
f(83) = 7013 = 7013
f(84) = 7181 = 43*167
f(85) = 7351 = 7351
f(86) = 7523 = 7523
f(87) = 7697 = 43*179
f(88) = 7873 = 7873
f(89) = 8051 = 83*97
f(90) = 8231 = 8231
f(91) = 8413 = 47*179
f(92) = 8597 = 8597
f(93) = 8783 = 8783
f(94) = 8971 = 8971
f(95) = 9161 = 9161
f(96) = 9353 = 47*199
f(97) = 9547 = 9547
f(98) = 9743 = 9743
f(99) = 9941 = 9941
f(100) = 10141 = 10141

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+1x+41

f(0)=41
f(1)=43
f(2)=47
f(3)=53
f(4)=61
f(5)=71
f(6)=83
f(7)=97
f(8)=113
f(9)=131
f(10)=151
f(11)=173
f(12)=197
f(13)=223
f(14)=251
f(15)=281
f(16)=313
f(17)=347
f(18)=383
f(19)=421
f(20)=461
f(21)=503
f(22)=547
f(23)=593
f(24)=641
f(25)=691
f(26)=743
f(27)=797
f(28)=853
f(29)=911
f(30)=971
f(31)=1033
f(32)=1097
f(33)=1163
f(34)=1231
f(35)=1301
f(36)=1373
f(37)=1447
f(38)=1523
f(39)=1601
f(40)=1
f(41)=1
f(42)=1847
f(43)=1933
f(44)=1
f(45)=2111
f(46)=2203
f(47)=2297
f(48)=2393
f(49)=1
f(50)=2591
f(51)=2693
f(52)=2797
f(53)=2903
f(54)=3011
f(55)=3121
f(56)=1
f(57)=3347
f(58)=3463
f(59)=3581
f(60)=3701
f(61)=3823
f(62)=3947
f(63)=4073
f(64)=4201
f(65)=1
f(66)=4463
f(67)=4597
f(68)=4733
f(69)=4871
f(70)=5011
f(71)=5153
f(72)=5297
f(73)=5443
f(74)=5591
f(75)=5741
f(76)=1
f(77)=6047
f(78)=6203
f(79)=6361
f(80)=6521
f(81)=163
f(82)=167
f(83)=7013
f(84)=1
f(85)=7351
f(86)=7523
f(87)=179
f(88)=7873
f(89)=1
f(90)=8231
f(91)=1
f(92)=8597
f(93)=8783
f(94)=8971
f(95)=9161
f(96)=199
f(97)=9547
f(98)=9743
f(99)=9941

b) Substitution of the polynom
The polynom f(x)=x^2+1x+41 could be written as f(y)= y^2+40.75 with x=y-0.5

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+0.5
f'(x)>2x

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, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1, 1, 1847, 1933, 1, 2111, 2203, 2297, 2393, 1, 2591, 2693, 2797, 2903, 3011, 3121, 1, 3347, 3463, 3581, 3701, 3823, 3947, 4073, 4201, 1, 4463, 4597, 4733, 4871, 5011, 5153, 5297, 5443, 5591, 5741, 1, 6047, 6203, 6361, 6521, 163, 167, 7013, 1, 7351, 7523, 179, 7873, 1, 8231, 1, 8597, 8783, 8971, 9161, 199, 9547, 9743, 9941, 10141, 10343, 1, 10753, 1, 11171, 11383, 11597, 11813, 227, 12251, 12473, 12697, 12923, 13151, 13381, 13613, 1, 14083, 14321, 14561, 1, 367, 373, 15541, 15791, 263, 379, 16553, 16811, 397, 17333, 17597, 17863, 18131, 18401, 1, 18947, 409, 19501, 1, 20063, 20347, 439, 20921, 21211, 21503, 307, 22093, 22391, 22691, 22993, 23297, 23603, 23911, 457, 24533, 24847, 25163, 1, 25801, 1, 499, 653, 661, 27431, 27763, 28097, 28433, 28771, 677, 29453, 359, 701, 30491, 30841, 31193, 31547, 523, 32261, 32621, 32983, 33347, 33713, 1, 733, 1, 577, 35573, 35951, 773, 36713, 37097, 37483, 37871, 38261, 38653, 39047, 39443, 39841, 40241, 419, 41047, 41453, 1021, 1031, 42683, 607, 821, 1, 44351, 44773, 45197, 1061, 46051, 877, 1091, 1, 673, 48221, 48661, 49103, 49547, 49993, 50441, 50891, 51343, 51797, 52253, 52711, 53171, 53633, 1151, 54563, 487, 55501, 1, 1201, 56923, 941, 57881, 58363, 709, 59333, 1, 1471, 1483, 61297, 1013, 1, 62791, 1, 1, 64303, 64811, 787, 1531, 66347, 66863, 1567, 67901, 1291, 68947, 69473, 70001, 1, 1, 71597, 1361, 72671, 1, 563, 74297, 74843, 75391, 75941, 76493, 77047, 1093, 1663, 78721, 79283, 79847, 829, 1723, 81551, 2003, 2017, 83273, 1181, 84431, 1, 85597, 86183, 86771, 1, 1, 88547, 919, 2087, 1481, 1, 2129, 92153, 92761, 93371, 93983, 94597, 95213, 1571, 96451, 97073, 647, 98323, 1867, 99581, 100213, 100847, 101483, 102121, 102761, 1951, 104047, 104693, 105341, 1277, 2269, 2617, 2633, 1, 967, 2339, 110597, 111263, 1, 112601, 1, 113947, 1381, 115301, 115981, 116663, 2729, 118033, 1, 2777, 120103, 1069, 121493, 1721, 122891, 123593, 124297, 125003, 125711, 126421, 127133, 739, 128563, 129281, 1831, 2143, 131447, 132173, 1, 133631, 134363, 2549, 3313, 3331, 2251, 138053, 138797, 2969, 2647, 141041, 141793, 142547, 3049, 144061, 1493, 1, 146347, 1123, 147881, 3457, 149423, 150197, 3511, 1, 152531, 153313, 154097, 154883, 155671, 1613, 157253, 158047, 158843, 1, 160441, 1, 1237, 162853, 163661, 164471, 839, 1019, 2011, 4091, 4111, 169373, 170197, 171023, 171851, 172681, 1039, 174347, 1049, 176021, 1, 2141, 2927, 179393, 180241, 3853, 181943, 3449, 4271, 184511, 185371, 1, 1, 187963, 188831, 189701, 190573, 191447, 192323, 193201, 1, 194963, 1297, 1741, 197621, 1109, 199403, 200297, 201193, 1129, 4951, 4973, 204797, 205703, 1, 207521, 208433, 209347, 210263, 947, 1877, 213023, 213947, 1423, 215801, 216731, 1, 4651, 219533, 220471, 221411, 5171, 4751, 4231, 5237, 226141, 227093, 2351, 229003, 229961, 4357, 1, 232847, 3833, 234781, 235751, 236723, 1, 238673, 1, 240631, 1, 1, 243583, 1229, 245561, 1, 247547, 248543, 249541, 1259, 251543, 3557, 253553, 3067, 255571, 1, 1489, 258613, 259631, 260651, 261673, 262697, 263723, 1, 265781, 1063, 6229, 268883, 5743, 270961, 272003, 273047, 274093, 1, 276191, 5231, 1, 279353, 280411, 1627, 1, 6917, 1, 285731, 286801, 287873, 288947, 290023, 291101, 292181, 293263, 294347, 295433, 4861, 1, 1, 1, 300893, 301991, 303091, 304193, 305297, 5023, 307511, 308621, 2741, 7229, 311963, 313081, 7307, 6709, 4457, 1399, 318701, 319831, 6829, 322097, 323233, 1, 1997, 1439, 327797, 1, 1, 331241, 332393, 1187, 1699, 6337, 337021, 338183, 1, 2039, 341681, 2053, 6491, 4159, 346373, 3583, 348731, 349913, 351097, 2333, 353471, 354661, 355853, 357047, 358243, 359441, 8387, 361843, 363047, 1, 365461, 6011, 367883, 369097, 7879, 1, 372751, 1, 1, 8009, 1, 9241, 1, 381347, 382583, 383821, 1, 386303, 1, 388793, 2179, 391291, 392543, 393797, 2207, 396311, 1, 398833, 7549, 5653, 402631, 403901, 1, 406447, 407723, 7717, 410281, 411563, 1319, 9631, 5851, 3181, 9721, 419297, 420593, 421891, 423191, 424493, 1619, 427103, 428411, 1, 10513, 5209, 433663, 434981, 9283, 1, 1669, 1, 3371, 1, 7283, 445597, 446933, 3967, 5417, 4649, 452297, 453643, 454991, 7481, 1, 459047, 460403, 1, 2677, 464483, 2089, 467213, 468581, 8867, 1, 472697, 474073, 11057, 476831, 478213, 9049, 2417, 482371, 483761, 11833, 11867, 487943, 2459, 490741, 1, 10501, 2861, 6991, 497771, 499183, 10651, 502013, 503431, 504851, 1459, 507697, 509123, 510551, 7211, 1, 514847, 516283, 517721, 519161, 1, 522047, 523493, 524941, 526391, 527843, 8677, 530753, 12377, 533671, 535133, 12479, 3301, 1, 541001, 8893, 13267, 1, 1, 6607, 549863, 551347, 552833, 3671, 10487, 557303, 1, 560293, 11953, 3373, 564793, 3391, 6841, 12113, 570821, 572333, 573847, 1, 576881, 578401, 1889, 581447, 582973, 1, 3881, 587563, 2347, 6089, 1, 593711, 595253, 13879, 1949, 599891, 1, 4603, 604547, 14783, 14821, 609221, 610783, 5419, 6329, 1607, 617051, 8713, 620197, 10193, 623351, 2753, 11821, 628097, 629683, 631271, 4831, 13499, 10427, 1, 5657, 3253, 13669, 644047, 3607, 647261, 648871, 650483, 3643, 653713, 655331, 656951, 658573, 660197, 661823, 1, 15467, 1, 668347, 15581, 16381, 16421, 674903, 1, 8171, 1, 1, 683143, 2437, 686453, 688111, 689771, 691433, 693097, 694763, 1, 1, 8431, 701447, 703123, 704801, 706481, 708163, 709847, 15139, 13457, 714911, 10093, 1, 15319, 11831, 723391, 13681, 726797, 4211, 1, 731921, 1999, 10357, 1, 738781, 18061, 1, 743947, 745673, 747401, 7723, 750863, 1, 754333, 2027, 757811, 759553, 761297, 763043, 1, 766541, 4441, 770047, 3461, 773561, 7993, 1, 778847, 780613, 782381, 2069, 785923, 787697, 789473, 791251, 1, 1, 4003, 798383, 800171, 1, 4931, 15199, 4057, 809141, 810941, 1, 19867, 816353, 1, 3677, 9901, 6287, 825413, 827231, 13591, 830873, 7369, 834523, 836351, 3187, 5563, 1, 1, 1, 847361, 849203, 2719, 852893, 2153, 3257, 6553, 860297, 12143, 864011, 865871, 867733, 869597, 871463, 873331, 875201, 877073, 18701, 880823, 882701, 1, 886463, 1, 21713, 20747, 894011, 895903, 20879, 899693, 901591, 17047, 1, 907297, 909203, 911111, 2917, 2237, 17299, 918763, 1, 922601, 924523, 9551, 1, 930301, 932231, 1, 936097, 938033, 939971, 941911, 15473, 4801, 947743, 949691, 951641, 953593, 9851, 2609, 959461, 15761, 1, 5393, 23593, 1, 971251, 5437, 22679, 13763, 1, 22817, 983113, 985097, 987083, 989071, 2657, 993053, 5051, 997043, 14071,

6. Sequence of the polynom (only primes)

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797, 2903, 3011, 3121, 3347, 3463, 3581, 3701, 3823, 3947, 4073, 4201, 4463, 4597, 4733, 4871, 5011, 5153, 5297, 5443, 5591, 5741, 6047, 6203, 6361, 6521, 163, 167, 7013, 7351, 7523, 179, 7873, 8231, 8597, 8783, 8971, 9161, 199, 9547, 9743, 9941, 10141, 10343, 10753, 11171, 11383, 11597, 11813, 227, 12251, 12473, 12697, 12923, 13151, 13381, 13613, 14083, 14321, 14561, 367, 373, 15541, 15791, 263, 379, 16553, 16811, 397, 17333, 17597, 17863, 18131, 18401, 18947, 409, 19501, 20063, 20347, 439, 20921, 21211, 21503, 307, 22093, 22391, 22691, 22993, 23297, 23603, 23911, 457, 24533, 24847, 25163, 25801, 499, 653, 661, 27431, 27763, 28097, 28433, 28771, 677, 29453, 359, 701, 30491, 30841, 31193, 31547, 523, 32261, 32621, 32983, 33347, 33713, 733, 577, 35573, 35951, 773, 36713, 37097, 37483, 37871, 38261, 38653, 39047, 39443, 39841, 40241, 419, 41047, 41453, 1021, 1031, 42683, 607, 821, 44351, 44773, 45197, 1061, 46051, 877, 1091, 673, 48221, 48661, 49103, 49547, 49993, 50441, 50891, 51343, 51797, 52253, 52711, 53171, 53633, 1151, 54563, 487, 55501, 1201, 56923, 941, 57881, 58363, 709, 59333, 1471, 1483, 61297, 1013, 62791, 64303, 64811, 787, 1531, 66347, 66863, 1567, 67901, 1291, 68947, 69473, 70001, 71597, 1361, 72671, 563, 74297, 74843, 75391, 75941, 76493, 77047, 1093, 1663, 78721, 79283, 79847, 829, 1723, 81551, 2003, 2017, 83273, 1181, 84431, 85597, 86183, 86771, 88547, 919, 2087, 1481, 2129, 92153, 92761, 93371, 93983, 94597, 95213, 1571, 96451, 97073, 647, 98323, 1867, 99581, 100213, 100847, 101483, 102121, 102761, 1951, 104047, 104693, 105341, 1277, 2269, 2617, 2633, 967, 2339, 110597, 111263, 112601, 113947, 1381, 115301, 115981, 116663, 2729, 118033, 2777, 120103, 1069, 121493, 1721, 122891, 123593, 124297, 125003, 125711, 126421, 127133, 739, 128563, 129281, 1831, 2143, 131447, 132173, 133631, 134363, 2549, 3313, 3331, 2251, 138053, 138797, 2969, 2647, 141041, 141793, 142547, 3049, 144061, 1493, 146347, 1123, 147881, 3457, 149423, 150197, 3511, 152531, 153313, 154097, 154883, 155671, 1613, 157253, 158047, 158843, 160441, 1237, 162853, 163661, 164471, 839, 1019, 2011, 4091, 4111, 169373, 170197, 171023, 171851, 172681, 1039, 174347, 1049, 176021, 2141, 2927, 179393, 180241, 3853, 181943, 3449, 4271, 184511, 185371, 187963, 188831, 189701, 190573, 191447, 192323, 193201, 194963, 1297, 1741, 197621, 1109, 199403, 200297, 201193, 1129, 4951, 4973, 204797, 205703, 207521, 208433, 209347, 210263, 947, 1877, 213023, 213947, 1423, 215801, 216731, 4651, 219533, 220471, 221411, 5171, 4751, 4231, 5237, 226141, 227093, 2351, 229003, 229961, 4357, 232847, 3833, 234781, 235751, 236723, 238673, 240631, 243583, 1229, 245561, 247547, 248543, 249541, 1259, 251543, 3557, 253553, 3067, 255571, 1489, 258613, 259631, 260651, 261673, 262697, 263723, 265781, 1063, 6229, 268883, 5743, 270961, 272003, 273047, 274093, 276191, 5231, 279353, 280411, 1627, 6917, 285731, 286801, 287873, 288947, 290023, 291101, 292181, 293263, 294347, 295433, 4861, 300893, 301991, 303091, 304193, 305297, 5023, 307511, 308621, 2741, 7229, 311963, 313081, 7307, 6709, 4457, 1399, 318701, 319831, 6829, 322097, 323233, 1997, 1439, 327797, 331241, 332393, 1187, 1699, 6337, 337021, 338183, 2039, 341681, 2053, 6491, 4159, 346373, 3583, 348731, 349913, 351097, 2333, 353471, 354661, 355853, 357047, 358243, 359441, 8387, 361843, 363047, 365461, 6011, 367883, 369097, 7879, 372751, 8009, 9241, 381347, 382583, 383821, 386303, 388793, 2179, 391291, 392543, 393797, 2207, 396311, 398833, 7549, 5653, 402631, 403901, 406447, 407723, 7717, 410281, 411563, 1319, 9631, 5851, 3181, 9721, 419297, 420593, 421891, 423191, 424493, 1619, 427103, 428411, 10513, 5209, 433663, 434981, 9283, 1669, 3371, 7283, 445597, 446933, 3967, 5417, 4649, 452297, 453643, 454991, 7481, 459047, 460403, 2677, 464483, 2089, 467213, 468581, 8867, 472697, 474073, 11057, 476831, 478213, 9049, 2417, 482371, 483761, 11833, 11867, 487943, 2459, 490741, 10501, 2861, 6991, 497771, 499183, 10651, 502013, 503431, 504851, 1459, 507697, 509123, 510551, 7211, 514847, 516283, 517721, 519161, 522047, 523493, 524941, 526391, 527843, 8677, 530753, 12377, 533671, 535133, 12479, 3301, 541001, 8893, 13267, 6607, 549863, 551347, 552833, 3671, 10487, 557303, 560293, 11953, 3373, 564793, 3391, 6841, 12113, 570821, 572333, 573847, 576881, 578401, 1889, 581447, 582973, 3881, 587563, 2347, 6089, 593711, 595253, 13879, 1949, 599891, 4603, 604547, 14783, 14821, 609221, 610783, 5419, 6329, 1607, 617051, 8713, 620197, 10193, 623351, 2753, 11821, 628097, 629683, 631271, 4831, 13499, 10427, 5657, 3253, 13669, 644047, 3607, 647261, 648871, 650483, 3643, 653713, 655331, 656951, 658573, 660197, 661823, 15467, 668347, 15581, 16381, 16421, 674903, 8171, 683143, 2437, 686453, 688111, 689771, 691433, 693097, 694763, 8431, 701447, 703123, 704801, 706481, 708163, 709847, 15139, 13457, 714911, 10093, 15319, 11831, 723391, 13681, 726797, 4211, 731921, 1999, 10357, 738781, 18061, 743947, 745673, 747401, 7723, 750863, 754333, 2027, 757811, 759553, 761297, 763043, 766541, 4441, 770047, 3461, 773561, 7993, 778847, 780613, 782381, 2069, 785923, 787697, 789473, 791251, 4003, 798383, 800171, 4931, 15199, 4057, 809141, 810941, 19867, 816353, 3677, 9901, 6287, 825413, 827231, 13591, 830873, 7369, 834523, 836351, 3187, 5563, 847361, 849203, 2719, 852893, 2153, 3257, 6553, 860297, 12143, 864011, 865871, 867733, 869597, 871463, 873331, 875201, 877073, 18701, 880823, 882701, 886463, 21713, 20747, 894011, 895903, 20879, 899693, 901591, 17047, 907297, 909203, 911111, 2917, 2237, 17299, 918763, 922601, 924523, 9551, 930301, 932231, 936097, 938033, 939971, 941911, 15473, 4801, 947743, 949691, 951641, 953593, 9851, 2609, 959461, 15761, 5393, 23593, 971251, 5437, 22679, 13763, 22817, 983113, 985097, 987083, 989071, 2657, 993053, 5051, 997043, 14071,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+1x+41 and
the reducible primes which appear as divisor for the first time
p | x^2+1x+41 and p < x^2+1x+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)
110110111.1000000.0000001.100000
2100913880.9100000.0300000.8800008.272727inf8.000000
310008732905830.8730000.2900000.5830009.59340796.6666646.625000
4100008338418841500.8338000.4188000.4150009.55097414.4413807.118353
51000008056748581319860.8056700.4858100.3198609.66262911.6000487.707470
610000007862915252092610820.7862910.5252090.2610829.75946710.8109968.162383
7100000007724383551618522081980.7724380.5516190.2208209.82382210.5028388.457871
81000000007621857557085921191326540.7621860.5708590.1913279.86727010.3488048.664374
910000000007543212785855145361688067420.7543210.5855150.1688079.89681710.2567248.822965
10100000000007480541977596986517415106768030.7480540.5969870.1510689.91691810.1959308.949150


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)
123031.5000000.0000001.500000
245051.2500000.0000001.2500001.666667-nan1.666667
389091.1250000.0000001.1250001.800000-nan1.800000
416170171.0625000.0000001.0625001.888889-nan1.888889
532330331.0312500.0000001.0312501.941176-nan1.941176
664600600.9375000.0000000.9375001.818182-nan1.818182
712811581070.8984380.0625000.8359381.916667inf1.783333
8256230421880.8984380.1640620.7343752.0000005.2500001.757009
95124571223350.8925780.2382810.6542971.9869562.9047621.781915
1010248953005950.8740230.2929690.5810551.9584242.4590161.776119
112048176371310500.8608400.3481450.5126951.9698322.3766671.764706
1240963479157919000.8493650.3854980.4638671.9733412.2145861.809524
1381926864338634780.8378910.4133300.4245611.9729812.1443951.830526
141638413559717963800.8275760.4381710.3894041.9753792.1202011.834388
15327682684815125117230.8193360.4615780.3577581.9800872.1068391.837461
16655365315631354218020.8110960.4784240.3326721.9798872.0729921.859763
1713107210525764400408570.8030470.4913330.3117141.9801532.0539641.874002
18262144208890132294765960.7968520.5046620.2921911.9845712.0542551.874734
195242884147262703171444090.7910270.5155890.2754381.9853802.0433051.885333
2010485768242695515492727200.7860840.5259980.2600861.9875032.0403791.888525
212097152163871011217825169280.7813980.5349070.2464901.9880772.0338751.895453
224194304325994722767569831910.7772320.5428210.2344111.9893372.0295891.901988
2383886086487372461302318743490.7733550.5499150.2234401.9900242.0261391.906394
241677721612916716933677235799440.7698960.5565150.2133811.9910552.0240031.909967
2533554432257257631887168168540820.7666880.5624200.2042681.9916642.0212211.914578
26671088645125568638112318131433680.7637690.5679180.1958511.9923872.0195511.917597
2713421772810215013576900823252493120.7610780.5729560.1881221.9929522.0177421.921069
28268435456203630641155054863485757780.7585830.5776240.1809591.9934452.0162971.923846
29536870912406013475312408070936054050.7562590.5819050.1743541.9938722.0148231.926998
3010737418248097135016291158461805976550.7541040.5859100.1681951.9943022.0137631.929351
312147483648161508990312661813583489085450.7520850.5896120.1624731.9946442.0126361.931966
324294967296322207721325472162926748609210.7501980.5930700.1571281.9949832.0117311.934206
3385899345926428940628512220655413067340740.7484270.5963030.1521241.9952782.0109041.936301
3417179869184128292853751029650961725327757580.7467630.5993360.1474271.9955522.0101711.938249
3534359738368256047753922069085291949139224730.7451970.6021830.1430141.9958072.0095021.940133


8. Check for existing Integer Sequences by OEIS

Found in Database : 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421,
Found in Database : 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601,
Found in Database : 41, 43, 47, 53, 61, 71, 83, 97, 113, 131,

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
1231211010000000
2452311210000000
3893632220000000
4161761154440000000
53233112298880000000
664602040151415160000000
71281073572272527288440314
825618864124464947464225175111511
951233511022582858583122665624303632
10102459519739814914714615330016213875748071
1120481050340710267262258263713382331169184183177
124096190064012604804794624791579828751380394402403
138192347811672311860879859880338617851601840833866847
141638463802144423615981590158916037179378333961758177718131831
15327681172339177806294229012952292815125789172343772375637723825
16655362180272421456054865439543454433135416228151267807784778457855
171310724085713566272911014810280101801024964400330943130616030160421625616072
1826214476596255025109419103191871913519171132294679586433633010330633317733044
1952428814440947988964213609336122361193607527031713855713176067528675756769267522
2010485762727209086818185268365680816824568029551549282056269493137616138138138127137668
2120971525169281722543446741293091294001293851288341121782572844548938280135280508280795280344
224194304983191327542655649245604246242245701245644227675611614211115335568146569311569392569907
238388608187434962489812494514684594688074684954685884613023235141222616111152370115451311526631153477
24167772163579944119364123863038950098955558946438947379336772475338145833912331872233555623348152334529
2533554432685408222853544568728171427517141661712733171290818871681959739692742854715498471778147182914720111
2667108864131433684382214876115432863093286972328551332845743811231819367233187450859528026952710595272689529919
2713421772825249312841721416832098631350863132666313158630938076900823390484603785236319224324192240671922637819226054
2826843545648575778161934033238237512142866121463341214515912141419155054863786701797638468438762259387593913876512038768093
295368709129360540531204228624011772340524123399411234020122339874131240807015839501215401305878096198781001407810447678107256
3010737418241805976556020246312039519245154900451497534514780045145202629115846318769485310346361157275414157278833157271969157289630
312147483648348908545116310266232598279872313568722970687226319872211641266181358641183998624997360316533312316561920316534447316551679
324294967296674860921224964481449896440168712195168717851168714314168716561254721629212892127541258003538636792762636836275636771671636815584
33858993459213067340744355937068711403683266957653266790983266718153266873965122206554259125100425309555501280528078128059513212805186551280564689
34171798691842532775758844277390168849836863320536963319494863317814763319729410296509617520650630050900033172574135423257415086325741095172574113814
353435973836849139224731638006939327591553412284822821228476950122845599012285072512069085291910458107269102327456505172646501517276849451727645045172673420