Inhaltsverzeichnis

Development of
Algorithmic Constructions

02:35:38
Deutsch
9.Jul 2020

Polynom = x^2+163

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) = 163 = 163
f(1) = 41 = 41
f(2) = 167 = 167
f(3) = 43 = 43
f(4) = 179 = 179
f(5) = 47 = 47
f(6) = 199 = 199
f(7) = 53 = 53
f(8) = 227 = 227
f(9) = 61 = 61
f(10) = 263 = 263
f(11) = 71 = 71
f(12) = 307 = 307
f(13) = 83 = 83
f(14) = 359 = 359
f(15) = 97 = 97
f(16) = 419 = 419
f(17) = 113 = 113
f(18) = 487 = 487
f(19) = 131 = 131
f(20) = 563 = 563
f(21) = 151 = 151
f(22) = 647 = 647
f(23) = 173 = 173
f(24) = 739 = 739
f(25) = 197 = 197
f(26) = 839 = 839
f(27) = 223 = 223
f(28) = 947 = 947
f(29) = 251 = 251
f(30) = 1063 = 1063
f(31) = 281 = 281
f(32) = 1187 = 1187
f(33) = 313 = 313
f(34) = 1319 = 1319
f(35) = 347 = 347
f(36) = 1459 = 1459
f(37) = 383 = 383
f(38) = 1607 = 1607
f(39) = 421 = 421
f(40) = 1763 = 41*43
f(41) = 461 = 461
f(42) = 1927 = 41*47
f(43) = 503 = 503
f(44) = 2099 = 2099
f(45) = 547 = 547
f(46) = 2279 = 43*53
f(47) = 593 = 593
f(48) = 2467 = 2467
f(49) = 641 = 641
f(50) = 2663 = 2663
f(51) = 691 = 691
f(52) = 2867 = 47*61
f(53) = 743 = 743
f(54) = 3079 = 3079
f(55) = 797 = 797
f(56) = 3299 = 3299
f(57) = 853 = 853
f(58) = 3527 = 3527
f(59) = 911 = 911
f(60) = 3763 = 53*71
f(61) = 971 = 971
f(62) = 4007 = 4007
f(63) = 1033 = 1033
f(64) = 4259 = 4259
f(65) = 1097 = 1097
f(66) = 4519 = 4519
f(67) = 1163 = 1163
f(68) = 4787 = 4787
f(69) = 1231 = 1231
f(70) = 5063 = 61*83
f(71) = 1301 = 1301
f(72) = 5347 = 5347
f(73) = 1373 = 1373
f(74) = 5639 = 5639
f(75) = 1447 = 1447
f(76) = 5939 = 5939
f(77) = 1523 = 1523
f(78) = 6247 = 6247
f(79) = 1601 = 1601
f(80) = 6563 = 6563
f(81) = 1681 = 41*41
f(82) = 6887 = 71*97
f(83) = 1763 = 41*43
f(84) = 7219 = 7219
f(85) = 1847 = 1847
f(86) = 7559 = 7559
f(87) = 1933 = 1933
f(88) = 7907 = 7907
f(89) = 2021 = 43*47
f(90) = 8263 = 8263
f(91) = 2111 = 2111
f(92) = 8627 = 8627
f(93) = 2203 = 2203
f(94) = 8999 = 8999
f(95) = 2297 = 2297
f(96) = 9379 = 83*113
f(97) = 2393 = 2393
f(98) = 9767 = 9767
f(99) = 2491 = 47*53
f(100) = 10163 = 10163

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+163

f(0)=163
f(1)=41
f(2)=167
f(3)=43
f(4)=179
f(5)=47
f(6)=199
f(7)=53
f(8)=227
f(9)=61
f(10)=263
f(11)=71
f(12)=307
f(13)=83
f(14)=359
f(15)=97
f(16)=419
f(17)=113
f(18)=487
f(19)=131
f(20)=563
f(21)=151
f(22)=647
f(23)=173
f(24)=739
f(25)=197
f(26)=839
f(27)=223
f(28)=947
f(29)=251
f(30)=1063
f(31)=281
f(32)=1187
f(33)=313
f(34)=1319
f(35)=347
f(36)=1459
f(37)=383
f(38)=1607
f(39)=421
f(40)=1
f(41)=461
f(42)=1
f(43)=503
f(44)=2099
f(45)=547
f(46)=1
f(47)=593
f(48)=2467
f(49)=641
f(50)=2663
f(51)=691
f(52)=1
f(53)=743
f(54)=3079
f(55)=797
f(56)=3299
f(57)=853
f(58)=3527
f(59)=911
f(60)=1
f(61)=971
f(62)=4007
f(63)=1033
f(64)=4259
f(65)=1097
f(66)=4519
f(67)=1163
f(68)=4787
f(69)=1231
f(70)=1
f(71)=1301
f(72)=5347
f(73)=1373
f(74)=5639
f(75)=1447
f(76)=5939
f(77)=1523
f(78)=6247
f(79)=1601
f(80)=6563
f(81)=1
f(82)=1
f(83)=1
f(84)=7219
f(85)=1847
f(86)=7559
f(87)=1933
f(88)=7907
f(89)=1
f(90)=8263
f(91)=2111
f(92)=8627
f(93)=2203
f(94)=8999
f(95)=2297
f(96)=1
f(97)=2393
f(98)=9767
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+163 could be written as f(y)= y^2+163 with x=y+0

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
f'(x)>2x-1 with x > 13

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

163, 41, 167, 43, 179, 47, 199, 53, 227, 61, 263, 71, 307, 83, 359, 97, 419, 113, 487, 131, 563, 151, 647, 173, 739, 197, 839, 223, 947, 251, 1063, 281, 1187, 313, 1319, 347, 1459, 383, 1607, 421, 1, 461, 1, 503, 2099, 547, 1, 593, 2467, 641, 2663, 691, 1, 743, 3079, 797, 3299, 853, 3527, 911, 1, 971, 4007, 1033, 4259, 1097, 4519, 1163, 4787, 1231, 1, 1301, 5347, 1373, 5639, 1447, 5939, 1523, 6247, 1601, 6563, 1, 1, 1, 7219, 1847, 7559, 1933, 7907, 1, 8263, 2111, 8627, 2203, 8999, 2297, 1, 2393, 9767, 1, 10163, 2591, 10567, 2693, 10979, 2797, 11399, 2903, 11827, 3011, 12263, 3121, 1, 1, 13159, 3347, 13619, 3463, 14087, 3581, 14563, 3701, 367, 3823, 379, 3947, 373, 4073, 16547, 4201, 1, 1, 409, 4463, 18119, 4597, 397, 4733, 19207, 4871, 19763, 5011, 20327, 5153, 20899, 5297, 457, 5443, 22067, 5591, 1, 5741, 439, 1, 23879, 6047, 24499, 6203, 25127, 6361, 25763, 6521, 26407, 1, 27059, 1, 523, 7013, 28387, 1, 29063, 7351, 1, 7523, 499, 1, 31139, 7873, 31847, 1, 32563, 8231, 33287, 1, 34019, 8597, 34759, 8783, 35507, 8971, 36263, 9161, 607, 1, 37799, 9547, 1, 9743, 39367, 9941, 40163, 10141, 577, 10343, 1019, 1, 1039, 10753, 43427, 1, 44263, 11171, 1049, 11383, 45959, 11597, 46819, 11813, 1109, 1, 48563, 12251, 1, 12473, 709, 12697, 51239, 12923, 52147, 13151, 1129, 13381, 53987, 13613, 54919, 1, 673, 14083, 56807, 14321, 1229, 14561, 58727, 1, 59699, 1, 60679, 1, 61667, 15541, 1, 15791, 63667, 1, 64679, 1, 65699, 16553, 1259, 16811, 67763, 1, 829, 17333, 69859, 17597, 70919, 17863, 71987, 18131, 73063, 18401, 1399, 1, 75239, 18947, 787, 1, 77447, 19501, 1, 1, 79687, 20063, 80819, 20347, 1999, 1, 2027, 20921, 84263, 21211, 85427, 21503, 86599, 1, 1439, 22093, 2069, 22391, 90163, 22691, 91367, 22993, 2153, 23297, 967, 23603, 95027, 23911, 96263, 1, 1, 24533, 1619, 24847, 100019, 25163, 101287, 1, 102563, 25801, 919, 1, 2237, 1, 653, 1, 107747, 661, 109063, 27431, 1, 27763, 2377, 28097, 677, 28433, 114407, 28771, 115763, 1, 117127, 29453, 1669, 1, 1, 701, 121267, 30491, 122663, 30841, 124067, 31193, 1, 31547, 1123, 1, 128327, 32261, 129763, 32621, 733, 32983, 2503, 33347, 1889, 33713, 3307, 1, 3343, 1, 138547, 1, 1069, 1, 141539, 35573, 2699, 35951, 144563, 773, 1, 36713, 3433, 37097, 149159, 37483, 150707, 37871, 3541, 38261, 1, 38653, 155399, 39047, 156979, 39443, 158567, 39841, 160163, 40241, 1949, 1, 821, 41047, 164999, 41453, 166627, 1021, 168263, 1031, 1297, 42683, 171559, 1, 173219, 1, 1, 1, 1, 44351, 178247, 44773, 179939, 45197, 181639, 1061, 1, 46051, 185063, 877, 1237, 1091, 188519, 1, 3119, 1, 192007, 48221, 193763, 48661, 195527, 49103, 197299, 49547, 1, 49993, 200867, 50441, 4943, 50891, 4987, 51343, 206279, 51797, 208099, 52253, 209927, 52711, 1, 53171, 941, 53633, 215459, 1151, 217319, 54563, 219187, 1, 1, 55501, 222947, 1, 1489, 1201, 5273, 56923, 228647, 1, 230563, 57881, 232487, 58363, 4423, 1, 3329, 59333, 238307, 1, 240263, 1471, 242227, 1483, 244199, 61297, 1423, 1013, 248167, 1, 2579, 62791, 1, 1, 254179, 1, 256199, 64303, 3637, 64811, 260263, 1, 5581, 1531, 264359, 66347, 1, 66863, 268487, 1567, 270563, 67901, 5801, 1291, 274739, 68947, 276839, 69473, 278947, 70001, 281063, 1, 6907, 1, 6959, 71597, 1093, 1361, 289607, 72671, 4783, 1, 1699, 1, 296099, 74297, 1, 74843, 2659, 75391, 302663, 75941, 304867, 76493, 307079, 77047, 7193, 1, 5107, 1663, 313763, 78721, 7349, 79283, 318259, 79847, 1627, 1, 3889, 1723, 325063, 81551, 327347, 2003, 329639, 2017, 6263, 83273, 334247, 1181, 336563, 84431, 2999, 1, 341219, 85597, 343559, 86183, 345907, 86771, 6571, 1, 1, 1, 4253, 88547, 355379, 1, 357767, 2087, 360163, 1481, 1, 1, 364979, 2129, 7817, 92153, 369827, 92761, 372263, 93371, 1, 93983, 9199, 94597, 1, 95213, 382087, 1571, 1, 96451, 387047, 97073, 389539, 1, 1277, 98323, 5557, 1867, 397063, 99581, 399587, 100213, 1, 100847, 3089, 101483, 407207, 102121, 1, 102761, 1, 1951, 414899, 104047, 417479, 104693, 9769, 105341, 5953, 1, 2609, 2269, 427879, 2617, 430499, 2633, 1, 1, 435763, 1, 7187, 2339, 4547, 110597, 2657, 111263, 446387, 1, 2689, 112601, 451747, 1, 3469, 113947, 457139, 1381, 459847, 115301, 7583, 115981, 8779, 116663, 468019, 2729, 1, 118033, 473507, 1, 1, 2777, 479027, 120103, 4967, 1, 1, 121493, 11887, 1721, 10429, 122891, 492967, 123593, 1, 124297, 498599, 125003, 501427, 125711, 10729, 126421, 2833, 127133, 509959, 1, 512819, 128563, 515687, 129281, 2897, 1831, 521447, 2143, 2089, 131447, 1, 132173, 12329, 1, 533063, 133631, 1493, 134363, 1, 2549, 541859, 3313, 544807, 3331, 547763, 2251, 1, 138053, 553699, 138797, 556679, 2969, 559667, 2647, 562663, 141041, 565667, 141793, 568679, 142547, 571699, 3049, 574727, 144061, 6961, 1, 580807, 1, 583859, 146347, 586919, 1, 1, 147881, 8353, 3457, 596147, 149423, 5303, 150197, 3989, 3511, 14767, 1, 14843, 152531, 2437, 153313, 10079, 154097, 617959, 154883, 11719, 155671, 3137, 1613, 8837, 157253, 13417, 158047, 1, 158843, 636967, 1, 640163, 160441, 1, 1, 13757, 1, 649799, 162853, 5779, 163661, 656263, 164471, 2347, 1, 15413, 1, 666019, 2011, 669287, 4091, 15641, 4111, 1, 169373, 1, 170197, 682439, 171023, 685747, 171851, 689063, 172681, 692387, 1, 695719, 174347, 699059, 1, 702407, 176021, 705763, 1, 4099, 2141, 712499, 2927, 715879, 179393, 719267, 180241, 722663, 3853, 1, 181943, 1741, 3449, 1997, 4271, 7591, 184511, 18043, 185371, 18127, 1, 746659, 1, 750119, 187963, 753587, 188831, 757063, 189701, 2039, 190573, 2719, 191447, 767539, 192323, 771047, 193201, 1, 1, 2053, 194963, 781619, 1, 785159, 1, 1, 197621, 792263, 1, 795827, 199403, 15083, 200297, 802979, 201193, 1, 1, 1, 4951, 813767, 4973, 817379, 204797, 1, 205703, 15559, 1, 1, 207521, 11717, 208433, 835559, 209347, 3697, 210263, 842887, 1, 846563, 1877, 850247, 213023, 13999, 213947, 10333, 1, 861347, 215801, 2179, 216731, 868787, 1, 12289, 4651, 6689, 219533, 880007, 220471, 883763, 221411, 21647, 5171, 21739, 4751, 895079, 4231, 898867, 5237, 2207, 226141, 1, 227093, 910279, 2351, 1, 229003, 917927, 229961, 4679, 4357, 925607, 1, 929459, 232847, 933319, 3833, 937187, 234781, 1, 235751, 3019, 236723, 948839, 1, 952739, 238673, 5869, 1, 1, 240631, 20521, 1, 968419, 1, 22613, 243583, 976307, 1, 1, 245561, 1, 1, 988199, 247547, 992179, 248543, 996167, 249541,

6. Sequence of the polynom (only primes)

163, 41, 167, 43, 179, 47, 199, 53, 227, 61, 263, 71, 307, 83, 359, 97, 419, 113, 487, 131, 563, 151, 647, 173, 739, 197, 839, 223, 947, 251, 1063, 281, 1187, 313, 1319, 347, 1459, 383, 1607, 421, 461, 503, 2099, 547, 593, 2467, 641, 2663, 691, 743, 3079, 797, 3299, 853, 3527, 911, 971, 4007, 1033, 4259, 1097, 4519, 1163, 4787, 1231, 1301, 5347, 1373, 5639, 1447, 5939, 1523, 6247, 1601, 6563, 7219, 1847, 7559, 1933, 7907, 8263, 2111, 8627, 2203, 8999, 2297, 2393, 9767, 10163, 2591, 10567, 2693, 10979, 2797, 11399, 2903, 11827, 3011, 12263, 3121, 13159, 3347, 13619, 3463, 14087, 3581, 14563, 3701, 367, 3823, 379, 3947, 373, 4073, 16547, 4201, 409, 4463, 18119, 4597, 397, 4733, 19207, 4871, 19763, 5011, 20327, 5153, 20899, 5297, 457, 5443, 22067, 5591, 5741, 439, 23879, 6047, 24499, 6203, 25127, 6361, 25763, 6521, 26407, 27059, 523, 7013, 28387, 29063, 7351, 7523, 499, 31139, 7873, 31847, 32563, 8231, 33287, 34019, 8597, 34759, 8783, 35507, 8971, 36263, 9161, 607, 37799, 9547, 9743, 39367, 9941, 40163, 10141, 577, 10343, 1019, 1039, 10753, 43427, 44263, 11171, 1049, 11383, 45959, 11597, 46819, 11813, 1109, 48563, 12251, 12473, 709, 12697, 51239, 12923, 52147, 13151, 1129, 13381, 53987, 13613, 54919, 673, 14083, 56807, 14321, 1229, 14561, 58727, 59699, 60679, 61667, 15541, 15791, 63667, 64679, 65699, 16553, 1259, 16811, 67763, 829, 17333, 69859, 17597, 70919, 17863, 71987, 18131, 73063, 18401, 1399, 75239, 18947, 787, 77447, 19501, 79687, 20063, 80819, 20347, 1999, 2027, 20921, 84263, 21211, 85427, 21503, 86599, 1439, 22093, 2069, 22391, 90163, 22691, 91367, 22993, 2153, 23297, 967, 23603, 95027, 23911, 96263, 24533, 1619, 24847, 100019, 25163, 101287, 102563, 25801, 919, 2237, 653, 107747, 661, 109063, 27431, 27763, 2377, 28097, 677, 28433, 114407, 28771, 115763, 117127, 29453, 1669, 701, 121267, 30491, 122663, 30841, 124067, 31193, 31547, 1123, 128327, 32261, 129763, 32621, 733, 32983, 2503, 33347, 1889, 33713, 3307, 3343, 138547, 1069, 141539, 35573, 2699, 35951, 144563, 773, 36713, 3433, 37097, 149159, 37483, 150707, 37871, 3541, 38261, 38653, 155399, 39047, 156979, 39443, 158567, 39841, 160163, 40241, 1949, 821, 41047, 164999, 41453, 166627, 1021, 168263, 1031, 1297, 42683, 171559, 173219, 44351, 178247, 44773, 179939, 45197, 181639, 1061, 46051, 185063, 877, 1237, 1091, 188519, 3119, 192007, 48221, 193763, 48661, 195527, 49103, 197299, 49547, 49993, 200867, 50441, 4943, 50891, 4987, 51343, 206279, 51797, 208099, 52253, 209927, 52711, 53171, 941, 53633, 215459, 1151, 217319, 54563, 219187, 55501, 222947, 1489, 1201, 5273, 56923, 228647, 230563, 57881, 232487, 58363, 4423, 3329, 59333, 238307, 240263, 1471, 242227, 1483, 244199, 61297, 1423, 1013, 248167, 2579, 62791, 254179, 256199, 64303, 3637, 64811, 260263, 5581, 1531, 264359, 66347, 66863, 268487, 1567, 270563, 67901, 5801, 1291, 274739, 68947, 276839, 69473, 278947, 70001, 281063, 6907, 6959, 71597, 1093, 1361, 289607, 72671, 4783, 1699, 296099, 74297, 74843, 2659, 75391, 302663, 75941, 304867, 76493, 307079, 77047, 7193, 5107, 1663, 313763, 78721, 7349, 79283, 318259, 79847, 1627, 3889, 1723, 325063, 81551, 327347, 2003, 329639, 2017, 6263, 83273, 334247, 1181, 336563, 84431, 2999, 341219, 85597, 343559, 86183, 345907, 86771, 6571, 4253, 88547, 355379, 357767, 2087, 360163, 1481, 364979, 2129, 7817, 92153, 369827, 92761, 372263, 93371, 93983, 9199, 94597, 95213, 382087, 1571, 96451, 387047, 97073, 389539, 1277, 98323, 5557, 1867, 397063, 99581, 399587, 100213, 100847, 3089, 101483, 407207, 102121, 102761, 1951, 414899, 104047, 417479, 104693, 9769, 105341, 5953, 2609, 2269, 427879, 2617, 430499, 2633, 435763, 7187, 2339, 4547, 110597, 2657, 111263, 446387, 2689, 112601, 451747, 3469, 113947, 457139, 1381, 459847, 115301, 7583, 115981, 8779, 116663, 468019, 2729, 118033, 473507, 2777, 479027, 120103, 4967, 121493, 11887, 1721, 10429, 122891, 492967, 123593, 124297, 498599, 125003, 501427, 125711, 10729, 126421, 2833, 127133, 509959, 512819, 128563, 515687, 129281, 2897, 1831, 521447, 2143, 2089, 131447, 132173, 12329, 533063, 133631, 1493, 134363, 2549, 541859, 3313, 544807, 3331, 547763, 2251, 138053, 553699, 138797, 556679, 2969, 559667, 2647, 562663, 141041, 565667, 141793, 568679, 142547, 571699, 3049, 574727, 144061, 6961, 580807, 583859, 146347, 586919, 147881, 8353, 3457, 596147, 149423, 5303, 150197, 3989, 3511, 14767, 14843, 152531, 2437, 153313, 10079, 154097, 617959, 154883, 11719, 155671, 3137, 1613, 8837, 157253, 13417, 158047, 158843, 636967, 640163, 160441, 13757, 649799, 162853, 5779, 163661, 656263, 164471, 2347, 15413, 666019, 2011, 669287, 4091, 15641, 4111, 169373, 170197, 682439, 171023, 685747, 171851, 689063, 172681, 692387, 695719, 174347, 699059, 702407, 176021, 705763, 4099, 2141, 712499, 2927, 715879, 179393, 719267, 180241, 722663, 3853, 181943, 1741, 3449, 1997, 4271, 7591, 184511, 18043, 185371, 18127, 746659, 750119, 187963, 753587, 188831, 757063, 189701, 2039, 190573, 2719, 191447, 767539, 192323, 771047, 193201, 2053, 194963, 781619, 785159, 197621, 792263, 795827, 199403, 15083, 200297, 802979, 201193, 4951, 813767, 4973, 817379, 204797, 205703, 15559, 207521, 11717, 208433, 835559, 209347, 3697, 210263, 842887, 846563, 1877, 850247, 213023, 13999, 213947, 10333, 861347, 215801, 2179, 216731, 868787, 12289, 4651, 6689, 219533, 880007, 220471, 883763, 221411, 21647, 5171, 21739, 4751, 895079, 4231, 898867, 5237, 2207, 226141, 227093, 910279, 2351, 229003, 917927, 229961, 4679, 4357, 925607, 929459, 232847, 933319, 3833, 937187, 234781, 235751, 3019, 236723, 948839, 952739, 238673, 5869, 240631, 20521, 968419, 22613, 243583, 976307, 245561, 988199, 247547, 992179, 248543, 996167, 249541,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+163 and
the reducible primes which appear as divisor for the first time
p | x^2x+163 and p < x^2x+163

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)
110111101.1000001.1000001.100000
2100898900.8900000.8900000.8900008.0909098.090909-nan
31.0008286162120.8280000.6160000.8280009.3033706.921348inf
410.0008.0144.3013.7130.8014000.4301000.8014009.6787446.98214317.514151
5100.00077.74933.07444.6750.7774900.3307400.7774909.7016477.68983912.032049
61.000.000762.307268.954493.3530.7623070.2689540.7623079.8047188.13188611.043156
710.000.0007.518.0172.261.3395.256.6780.7518020.2261340.7518029.8621908.40790310.655004
8100.000.00074.401.09919.531.70054.869.3990.7440110.1953170.7440119.8963738.63722810.438037
91.000.000.000738.087.517171.903.430566.184.0870.7380870.1719030.7380879.9203848.80125210.318758
1010.000.000.0007.334.006.6091.535.355.6865.798.650.9230.7334010.1535360.7334019.9365018.93150110.241635


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)
123301.5000001.5000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
389901.1250001.1250000.0000001.8000001.800000-nan
416171701.0625001.0625000.0000001.8888891.888889-nan
532333301.0312501.0312500.0000001.9411761.941176-nan
664606000.9375000.9375000.0000001.8181821.818182-nan
712811511230.8984380.8750000.0234381.9166671.866667inf
8256219200190.8554690.7812500.0742191.9043481.7857146.333333
9512428353750.8359380.6894530.1464841.9543381.7650003.947368
101.0248476282190.8271480.6132810.2138671.9789721.7790372.920000
112.0481.6761.1265500.8183590.5498050.2685551.9787491.7929942.511415
124.0963.3191.9961.3230.8103030.4873050.3229981.9803101.7726472.405455
138.1926.5853.6282.9570.8038330.4428710.3609621.9840311.8176352.235072
1416.38413.0156.6286.3870.7943730.4045410.3898321.9764621.8269022.159959
1532.76825.78712.24513.5420.7869570.3736880.4132691.9813291.8474652.120244
1665.53651.22922.63128.5980.7816930.3453220.4363711.9866211.8481832.111800
17131.072101.69042.24359.4470.7758330.3222890.4535451.9850081.8665992.078712
18262.144201.94579.183122.7620.7703590.3020590.4683001.9858881.8744642.065066
19524.288401.589148.728252.8610.7659700.2836760.4822941.9886061.8782822.059766
201.048.576799.035280.867518.1680.7620190.2678560.4941641.9896841.8884612.049221
212.097.1521.590.905531.1771.059.7280.7586030.2532850.5053181.9910331.8912052.045144
224.194.3043.168.4121.008.1962.160.2160.7554080.2403730.5150361.9915781.8980412.038463
238.388.6086.312.4891.920.0274.392.4620.7525070.2288850.5236221.9923191.9044182.033344
2416.777.21612.580.2143.663.4368.916.7780.7498390.2183580.5314811.9929091.9080132.030018
2533.554.43225.079.1127.006.11918.072.9930.7474160.2087990.5386171.9935361.9124452.026852
2667.108.86450.008.47413.424.63436.583.8400.7451840.2000430.5451421.9940291.9161302.024227
27134.217.72899.746.30125.767.34873.978.9530.7431680.1919820.5511861.9945881.9194082.022176
28268.435.456198.986.69149.538.601149.448.0900.7412830.1845460.5567381.9949281.9225342.020144
29536.870.912397.041.13795.377.255301.663.8820.7395470.1776540.5618931.9953151.9253122.018519
301.073.741.824792.343.269183.901.069608.442.2000.7379270.1712710.5666561.9956201.9281442.016954
312.147.483.6481.581.443.065355.049.9061.226.393.1590.7364170.1653330.5710841.9959061.9306572.015628
324.294.967.2963.156.817.961686.322.8812.470.495.0800.7350040.1597970.5752071.9961631.9330322.014440
338.589.934.5926.302.254.0461.328.221.3094.974.032.7370.7336790.1546250.5790541.9963951.9352722.013375
3417.179.869.18412.583.174.6102.573.130.41010.010.044.2000.7324370.1497760.5826611.9966151.9372752.012460
3534.359.738.36825.126.178.5024.989.810.49920.136.368.0030.7312680.1452230.5860451.9968081.9391982.011616
3668.719.476.73650.176.715.0149.685.152.52340.491.562.4910.7301670.1409380.5892301.9969901.9409862.010867


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
123121101
245231301
389361413
416176112726
532331122413412
664602040822822
7128112377514411542
82562006713326722775
95123531202334613047130
101.0246282084208123285230
112.0481.126375751149414146417
124.0961.9966581.338266736258736
138.1923.6281.2182.4104791.3394621.348
1416.3846.6282.2084.4208592.4418592.469
1532.76812.2454.1068.1391.5984.5081.5894.550
1665.53622.6317.55815.0732.9428.3322.9528.405
17131.07242.24314.03628.2075.48615.6335.43415.690
18262.14479.18326.33552.84810.14829.40410.18029.451
19524.288148.72849.57399.15519.10355.22819.13555.262
201.048.576280.86793.767187.10036.093104.38936.119104.266
212.097.152531.177177.103354.07468.365197.27568.245197.292
224.194.3041.008.196336.134672.062129.309374.675129.385374.827
238.388.6081.920.027639.9101.280.117245.604714.463245.701714.259
2416.777.2163.663.4361.221.5232.441.913468.4591.362.899468.4951.363.583
2533.554.4327.006.1192.335.7074.670.412895.0092.608.076894.6432.608.391
2667.108.86413.424.6344.475.7228.948.9121.714.2754.999.6591.712.7334.997.967
27134.217.72825.767.3488.591.10817.176.2403.286.3099.597.2623.285.5139.598.264
28268.435.45649.538.60116.514.74633.023.8556.313.50818.457.9756.313.15818.453.960
29536.870.91295.377.25531.791.11863.586.13712.142.86635.547.08712.145.15935.542.143
301.073.741.824183.901.06961.297.693122.603.37623.405.24168.548.79723.402.01268.545.019
312.147.483.648355.049.906118.349.505236.700.40145.154.900132.381.07645.147.800132.366.130
324.294.967.296686.322.881228.783.782457.539.09987.231.356255.933.92987.226.319255.931.277
338.589.934.5921.328.221.309442.752.180885.469.129168.712.195495.390.401168.714.314495.404.399
3417.179.869.1842.573.130.410857.730.0221.715.400.388326.695.765959.878.823326.671.815959.884.007
3534.359.738.3684.989.810.4991.663.294.8853.326.515.614633.205.3691.861.700.812633.178.1471.861.726.171
3668.719.476.7369.685.152.5233.228.407.8766.456.744.6471.228.482.2793.614.077.6301.228.455.9883.614.136.626


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
380000000
4160000000
5320000000
6640000000
71283300111
8256191546454
951275453015152718
101.0242191239654506154
112.048550302248155125146124
124.0961.323713610362308352301
138.1922.9571.5601.397772687789709
1416.3846.3873.3503.0371.6641.5111.6831.529
1532.76813.5427.1196.4233.5003.2343.4863.322
1665.53628.59814.91213.6867.4176.8577.3546.970
17131.07259.44730.80128.64615.35614.35615.30714.428
18262.144122.76263.21059.55231.57829.67831.72229.784
19524.288252.861129.976122.88564.82161.64064.84561.555
201.048.576518.168265.698252.470132.628126.560132.419126.561
212.097.1521.059.728542.480517.248270.509259.773270.438259.008
224.194.3042.160.2161.104.4021.055.814550.285529.792550.564529.575
238.388.6084.392.4622.242.7152.149.7471.117.2761.078.4791.117.9301.078.777
2416.777.2168.916.7784.546.5524.370.2262.265.9822.192.9132.266.3512.191.532
2533.554.43218.072.9939.202.4048.870.5894.587.2754.450.2474.588.5754.446.896
2667.108.86436.583.84018.607.27617.976.5649.279.2299.013.8819.281.2349.009.496
27134.217.72873.978.95337.598.18436.380.76918.753.80718.237.92518.749.48818.237.733
28268.435.456149.448.09075.892.59773.555.49337.855.55936.873.67737.848.30236.870.552
29536.870.912301.663.882153.075.036148.588.84676.362.53574.469.20376.358.79674.473.348
301.073.741.824608.442.200308.516.532299.925.668153.924.247150.293.806153.924.388150.299.759
312.147.483.6481.226.393.159621.455.404604.937.755310.081.642303.115.104310.073.976303.122.437
324.294.967.2962.470.495.0801.251.148.8731.219.346.207624.316.861610.935.383624.327.284610.915.552
338.589.934.5924.974.032.7372.517.684.4312.456.348.3061.256.420.7111.230.601.7981.256.434.2851.230.575.943
3417.179.869.18410.010.044.2005.064.292.7414.945.751.4592.527.435.2702.477.567.0722.527.474.7942.477.567.064
3534.359.738.36820.136.368.00310.182.801.5509.953.566.4535.082.291.0484.985.848.4195.082.390.8404.985.837.696
3668.719.476.73640.491.562.49120.467.539.09920.024.023.39210.216.145.61210.029.550.22110.216.431.71810.029.434.940


8. Check for existing Integer Sequences by OEIS

Not in Database : 163, 41, 167, 43, 179, 47, 199, 53, 227, 61, 263, 71, 307, 83, 359, 97, 419, 113, 487, 131,
Not in Database : 163, 41, 167, 43, 179, 47, 199, 53, 227, 61, 263, 71, 307, 83, 359, 97, 419, 113, 487, 131, 563, 151, 647, 173, 739, 197, 839, 223, 947, 251, 1063, 281, 1187, 313, 1319, 347, 1459, 383, 1607, 421,
Found in Database : 41, 43, 47, 53, 61, 71, 83, 97, 113, 131,