# Development ofAlgorithmic Constructions

 02:26:36 9.May 2021

### 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

a:=1;
b:=0;
c:=163;
```liste_max:=100000;

sieving:=proc (stelle, p)
begin
while (stelle<=liste_max) do
erg:=liste[stelle];
while(erg mod p=0) do
// Divison of the stored f(x) by the prime
erg:=erg /p;
end_while;
liste[stelle]:=erg;
stelle:=stelle+p;
end_while;
end_proc;

// Calculation of the values of the polynom for x from 0 to liste_max
for x from 0 to liste_max do
p:=abs (a*x^2+b*x+c);
while (p mod 2=0) p:=p/2;
liste [x]:=p;
end_for;

for x from 0 to liste_max do
p:=liste[x];

if (p>1) then

// Printing the Primes
print (x, p);

// 1. Sieving
sieving (x+p,  p);

t:=(-x-b/a) mod p;      if t=0 then t:=p; end_if;

// 2. Sieving
sieving (t, p);

end_if;

end_for;

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

 A B C D E F G H I J K exponent =log10 (x) <=x number of all primes number of primes p = f(x) number of primes p | f(x) C/x D/x E/x C(n) / C(n-1) D(n) / D(n-1) E(n) / E(n-1) 1 10 11 11 0 1.100000 1.100000 1.100000 2 100 89 89 0 0.890000 0.890000 0.890000 8.090909 8.090909 -nan 3 1.000 828 616 212 0.828000 0.616000 0.828000 9.303370 6.921348 inf 4 10.000 8.014 4.301 3.713 0.801400 0.430100 0.801400 9.678744 6.982143 17.514151 5 100.000 77.749 33.074 44.675 0.777490 0.330740 0.777490 9.701647 7.689839 12.032049 6 1.000.000 762.307 268.954 493.353 0.762307 0.268954 0.762307 9.804718 8.131886 11.043156 7 10.000.000 7.518.017 2.261.339 5.256.678 0.751802 0.226134 0.751802 9.862190 8.407903 10.655004 8 100.000.000 74.401.099 19.531.700 54.869.399 0.744011 0.195317 0.744011 9.896373 8.637228 10.438037 9 1.000.000.000 738.087.517 171.903.430 566.184.087 0.738087 0.171903 0.738087 9.920384 8.801252 10.318758 10 10.000.000.000 7.334.006.609 1.535.355.686 5.798.650.923 0.733401 0.153536 0.733401 9.936501 8.931501 10.241635

 A B C D E F G H I J K exponent =log2 (x) <=x number of all primes number of primes p = f(x) number of primes p | f(x) C/x D/x E/x C(n) / C(n-1) D(n) / D(n-1) E(n) / E(n-1) 1 2 3 3 0 1.500000 1.500000 0.000000 2 4 5 5 0 1.250000 1.250000 0.000000 1.666667 1.666667 -nan 3 8 9 9 0 1.125000 1.125000 0.000000 1.800000 1.800000 -nan 4 16 17 17 0 1.062500 1.062500 0.000000 1.888889 1.888889 -nan 5 32 33 33 0 1.031250 1.031250 0.000000 1.941176 1.941176 -nan 6 64 60 60 0 0.937500 0.937500 0.000000 1.818182 1.818182 -nan 7 128 115 112 3 0.898438 0.875000 0.023438 1.916667 1.866667 inf 8 256 219 200 19 0.855469 0.781250 0.074219 1.904348 1.785714 6.333333 9 512 428 353 75 0.835938 0.689453 0.146484 1.954338 1.765000 3.947368 10 1.024 847 628 219 0.827148 0.613281 0.213867 1.978972 1.779037 2.920000 11 2.048 1.676 1.126 550 0.818359 0.549805 0.268555 1.978749 1.792994 2.511415 12 4.096 3.319 1.996 1.323 0.810303 0.487305 0.322998 1.980310 1.772647 2.405455 13 8.192 6.585 3.628 2.957 0.803833 0.442871 0.360962 1.984031 1.817635 2.235072 14 16.384 13.015 6.628 6.387 0.794373 0.404541 0.389832 1.976462 1.826902 2.159959 15 32.768 25.787 12.245 13.542 0.786957 0.373688 0.413269 1.981329 1.847465 2.120244 16 65.536 51.229 22.631 28.598 0.781693 0.345322 0.436371 1.986621 1.848183 2.111800 17 131.072 101.690 42.243 59.447 0.775833 0.322289 0.453545 1.985008 1.866599 2.078712 18 262.144 201.945 79.183 122.762 0.770359 0.302059 0.468300 1.985888 1.874464 2.065066 19 524.288 401.589 148.728 252.861 0.765970 0.283676 0.482294 1.988606 1.878282 2.059766 20 1.048.576 799.035 280.867 518.168 0.762019 0.267856 0.494164 1.989684 1.888461 2.049221 21 2.097.152 1.590.905 531.177 1.059.728 0.758603 0.253285 0.505318 1.991033 1.891205 2.045144 22 4.194.304 3.168.412 1.008.196 2.160.216 0.755408 0.240373 0.515036 1.991578 1.898041 2.038463 23 8.388.608 6.312.489 1.920.027 4.392.462 0.752507 0.228885 0.523622 1.992319 1.904418 2.033344 24 16.777.216 12.580.214 3.663.436 8.916.778 0.749839 0.218358 0.531481 1.992909 1.908013 2.030018 25 33.554.432 25.079.112 7.006.119 18.072.993 0.747416 0.208799 0.538617 1.993536 1.912445 2.026852 26 67.108.864 50.008.474 13.424.634 36.583.840 0.745184 0.200043 0.545142 1.994029 1.916130 2.024227 27 134.217.728 99.746.301 25.767.348 73.978.953 0.743168 0.191982 0.551186 1.994588 1.919408 2.022176 28 268.435.456 198.986.691 49.538.601 149.448.090 0.741283 0.184546 0.556738 1.994928 1.922534 2.020144 29 536.870.912 397.041.137 95.377.255 301.663.882 0.739547 0.177654 0.561893 1.995315 1.925312 2.018519 30 1.073.741.824 792.343.269 183.901.069 608.442.200 0.737927 0.171271 0.566656 1.995620 1.928144 2.016954 31 2.147.483.648 1.581.443.065 355.049.906 1.226.393.159 0.736417 0.165333 0.571084 1.995906 1.930657 2.015628 32 4.294.967.296 3.156.817.961 686.322.881 2.470.495.080 0.735004 0.159797 0.575207 1.996163 1.933032 2.014440 33 8.589.934.592 6.302.254.046 1.328.221.309 4.974.032.737 0.733679 0.154625 0.579054 1.996395 1.935272 2.013375 34 17.179.869.184 12.583.174.610 2.573.130.410 10.010.044.200 0.732437 0.149776 0.582661 1.996615 1.937275 2.012460 35 34.359.738.368 25.126.178.502 4.989.810.499 20.136.368.003 0.731268 0.145223 0.586045 1.996808 1.939198 2.011616 36 68.719.476.736 50.176.715.014 9.685.152.523 40.491.562.491 0.730167 0.140938 0.589230 1.996990 1.940986 2.010867

 A B C D E F G H I exponent =log2 (x) <=x 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 1 2 3 1 2 1 1 0 1 2 4 5 2 3 1 3 0 1 3 8 9 3 6 1 4 1 3 4 16 17 6 11 2 7 2 6 5 32 33 11 22 4 13 4 12 6 64 60 20 40 8 22 8 22 7 128 112 37 75 14 41 15 42 8 256 200 67 133 26 72 27 75 9 512 353 120 233 46 130 47 130 10 1.024 628 208 420 81 232 85 230 11 2.048 1.126 375 751 149 414 146 417 12 4.096 1.996 658 1.338 266 736 258 736 13 8.192 3.628 1.218 2.410 479 1.339 462 1.348 14 16.384 6.628 2.208 4.420 859 2.441 859 2.469 15 32.768 12.245 4.106 8.139 1.598 4.508 1.589 4.550 16 65.536 22.631 7.558 15.073 2.942 8.332 2.952 8.405 17 131.072 42.243 14.036 28.207 5.486 15.633 5.434 15.690 18 262.144 79.183 26.335 52.848 10.148 29.404 10.180 29.451 19 524.288 148.728 49.573 99.155 19.103 55.228 19.135 55.262 20 1.048.576 280.867 93.767 187.100 36.093 104.389 36.119 104.266 21 2.097.152 531.177 177.103 354.074 68.365 197.275 68.245 197.292 22 4.194.304 1.008.196 336.134 672.062 129.309 374.675 129.385 374.827 23 8.388.608 1.920.027 639.910 1.280.117 245.604 714.463 245.701 714.259 24 16.777.216 3.663.436 1.221.523 2.441.913 468.459 1.362.899 468.495 1.363.583 25 33.554.432 7.006.119 2.335.707 4.670.412 895.009 2.608.076 894.643 2.608.391 26 67.108.864 13.424.634 4.475.722 8.948.912 1.714.275 4.999.659 1.712.733 4.997.967 27 134.217.728 25.767.348 8.591.108 17.176.240 3.286.309 9.597.262 3.285.513 9.598.264 28 268.435.456 49.538.601 16.514.746 33.023.855 6.313.508 18.457.975 6.313.158 18.453.960 29 536.870.912 95.377.255 31.791.118 63.586.137 12.142.866 35.547.087 12.145.159 35.542.143 30 1.073.741.824 183.901.069 61.297.693 122.603.376 23.405.241 68.548.797 23.402.012 68.545.019 31 2.147.483.648 355.049.906 118.349.505 236.700.401 45.154.900 132.381.076 45.147.800 132.366.130 32 4.294.967.296 686.322.881 228.783.782 457.539.099 87.231.356 255.933.929 87.226.319 255.931.277 33 8.589.934.592 1.328.221.309 442.752.180 885.469.129 168.712.195 495.390.401 168.714.314 495.404.399 34 17.179.869.184 2.573.130.410 857.730.022 1.715.400.388 326.695.765 959.878.823 326.671.815 959.884.007 35 34.359.738.368 4.989.810.499 1.663.294.885 3.326.515.614 633.205.369 1.861.700.812 633.178.147 1.861.726.171 36 68.719.476.736 9.685.152.523 3.228.407.876 6.456.744.647 1.228.482.279 3.614.077.630 1.228.455.988 3.614.136.626

 A B C D E F G H I exponent =log2 (x) <=x 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 1 2 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 3 8 0 0 0 0 0 0 0 4 16 0 0 0 0 0 0 0 5 32 0 0 0 0 0 0 0 6 64 0 0 0 0 0 0 0 7 128 3 3 0 0 1 1 1 8 256 19 15 4 6 4 5 4 9 512 75 45 30 15 15 27 18 10 1.024 219 123 96 54 50 61 54 11 2.048 550 302 248 155 125 146 124 12 4.096 1.323 713 610 362 308 352 301 13 8.192 2.957 1.560 1.397 772 687 789 709 14 16.384 6.387 3.350 3.037 1.664 1.511 1.683 1.529 15 32.768 13.542 7.119 6.423 3.500 3.234 3.486 3.322 16 65.536 28.598 14.912 13.686 7.417 6.857 7.354 6.970 17 131.072 59.447 30.801 28.646 15.356 14.356 15.307 14.428 18 262.144 122.762 63.210 59.552 31.578 29.678 31.722 29.784 19 524.288 252.861 129.976 122.885 64.821 61.640 64.845 61.555 20 1.048.576 518.168 265.698 252.470 132.628 126.560 132.419 126.561 21 2.097.152 1.059.728 542.480 517.248 270.509 259.773 270.438 259.008 22 4.194.304 2.160.216 1.104.402 1.055.814 550.285 529.792 550.564 529.575 23 8.388.608 4.392.462 2.242.715 2.149.747 1.117.276 1.078.479 1.117.930 1.078.777 24 16.777.216 8.916.778 4.546.552 4.370.226 2.265.982 2.192.913 2.266.351 2.191.532 25 33.554.432 18.072.993 9.202.404 8.870.589 4.587.275 4.450.247 4.588.575 4.446.896 26 67.108.864 36.583.840 18.607.276 17.976.564 9.279.229 9.013.881 9.281.234 9.009.496 27 134.217.728 73.978.953 37.598.184 36.380.769 18.753.807 18.237.925 18.749.488 18.237.733 28 268.435.456 149.448.090 75.892.597 73.555.493 37.855.559 36.873.677 37.848.302 36.870.552 29 536.870.912 301.663.882 153.075.036 148.588.846 76.362.535 74.469.203 76.358.796 74.473.348 30 1.073.741.824 608.442.200 308.516.532 299.925.668 153.924.247 150.293.806 153.924.388 150.299.759 31 2.147.483.648 1.226.393.159 621.455.404 604.937.755 310.081.642 303.115.104 310.073.976 303.122.437 32 4.294.967.296 2.470.495.080 1.251.148.873 1.219.346.207 624.316.861 610.935.383 624.327.284 610.915.552 33 8.589.934.592 4.974.032.737 2.517.684.431 2.456.348.306 1.256.420.711 1.230.601.798 1.256.434.285 1.230.575.943 34 17.179.869.184 10.010.044.200 5.064.292.741 4.945.751.459 2.527.435.270 2.477.567.072 2.527.474.794 2.477.567.064 35 34.359.738.368 20.136.368.003 10.182.801.550 9.953.566.453 5.082.291.048 4.985.848.419 5.082.390.840 4.985.837.696 36 68.719.476.736 40.491.562.491 20.467.539.099 20.024.023.392 10.216.145.612 10.029.550.221 10.216.431.718 10.029.434.940