Inhaltsverzeichnis

Development of
Algorithmic Constructions

02:12:47
Deutsch
29.Mar 2024

Polynom = x^2-227

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) = 227 = 227
f(1) = 113 = 113
f(2) = 223 = 223
f(3) = 109 = 109
f(4) = 211 = 211
f(5) = 101 = 101
f(6) = 191 = 191
f(7) = 89 = 89
f(8) = 163 = 163
f(9) = 73 = 73
f(10) = 127 = 127
f(11) = 53 = 53
f(12) = 83 = 83
f(13) = 29 = 29
f(14) = 31 = 31
f(15) = 1 = 1
f(16) = 29 = 29
f(17) = 31 = 31
f(18) = 97 = 97
f(19) = 67 = 67
f(20) = 173 = 173
f(21) = 107 = 107
f(22) = 257 = 257
f(23) = 151 = 151
f(24) = 349 = 349
f(25) = 199 = 199
f(26) = 449 = 449
f(27) = 251 = 251
f(28) = 557 = 557
f(29) = 307 = 307
f(30) = 673 = 673
f(31) = 367 = 367
f(32) = 797 = 797
f(33) = 431 = 431
f(34) = 929 = 929
f(35) = 499 = 499
f(36) = 1069 = 1069
f(37) = 571 = 571
f(38) = 1217 = 1217
f(39) = 647 = 647
f(40) = 1373 = 1373
f(41) = 727 = 727
f(42) = 1537 = 29*53
f(43) = 811 = 811
f(44) = 1709 = 1709
f(45) = 899 = 29*31
f(46) = 1889 = 1889
f(47) = 991 = 991
f(48) = 2077 = 31*67
f(49) = 1087 = 1087
f(50) = 2273 = 2273
f(51) = 1187 = 1187
f(52) = 2477 = 2477
f(53) = 1291 = 1291
f(54) = 2689 = 2689
f(55) = 1399 = 1399
f(56) = 2909 = 2909
f(57) = 1511 = 1511
f(58) = 3137 = 3137
f(59) = 1627 = 1627
f(60) = 3373 = 3373
f(61) = 1747 = 1747
f(62) = 3617 = 3617
f(63) = 1871 = 1871
f(64) = 3869 = 53*73
f(65) = 1999 = 1999
f(66) = 4129 = 4129
f(67) = 2131 = 2131
f(68) = 4397 = 4397
f(69) = 2267 = 2267
f(70) = 4673 = 4673
f(71) = 2407 = 29*83
f(72) = 4957 = 4957
f(73) = 2551 = 2551
f(74) = 5249 = 29*181
f(75) = 2699 = 2699
f(76) = 5549 = 31*179
f(77) = 2851 = 2851
f(78) = 5857 = 5857
f(79) = 3007 = 31*97
f(80) = 6173 = 6173
f(81) = 3167 = 3167
f(82) = 6497 = 73*89
f(83) = 3331 = 3331
f(84) = 6829 = 6829
f(85) = 3499 = 3499
f(86) = 7169 = 67*107
f(87) = 3671 = 3671
f(88) = 7517 = 7517
f(89) = 3847 = 3847
f(90) = 7873 = 7873
f(91) = 4027 = 4027
f(92) = 8237 = 8237
f(93) = 4211 = 4211
f(94) = 8609 = 8609
f(95) = 4399 = 53*83
f(96) = 8989 = 89*101
f(97) = 4591 = 4591
f(98) = 9377 = 9377
f(99) = 4787 = 4787
f(100) = 9773 = 29*337

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^2-227

f(0)=227
f(1)=113
f(2)=223
f(3)=109
f(4)=211
f(5)=101
f(6)=191
f(7)=89
f(8)=163
f(9)=73
f(10)=127
f(11)=53
f(12)=83
f(13)=29
f(14)=31
f(15)=1
f(16)=1
f(17)=1
f(18)=97
f(19)=67
f(20)=173
f(21)=107
f(22)=257
f(23)=151
f(24)=349
f(25)=199
f(26)=449
f(27)=251
f(28)=557
f(29)=307
f(30)=673
f(31)=367
f(32)=797
f(33)=431
f(34)=929
f(35)=499
f(36)=1069
f(37)=571
f(38)=1217
f(39)=647
f(40)=1373
f(41)=727
f(42)=1
f(43)=811
f(44)=1709
f(45)=1
f(46)=1889
f(47)=991
f(48)=1
f(49)=1087
f(50)=2273
f(51)=1187
f(52)=2477
f(53)=1291
f(54)=2689
f(55)=1399
f(56)=2909
f(57)=1511
f(58)=3137
f(59)=1627
f(60)=3373
f(61)=1747
f(62)=3617
f(63)=1871
f(64)=1
f(65)=1999
f(66)=4129
f(67)=2131
f(68)=4397
f(69)=2267
f(70)=4673
f(71)=1
f(72)=4957
f(73)=2551
f(74)=181
f(75)=2699
f(76)=179
f(77)=2851
f(78)=5857
f(79)=1
f(80)=6173
f(81)=3167
f(82)=1
f(83)=3331
f(84)=6829
f(85)=3499
f(86)=1
f(87)=3671
f(88)=7517
f(89)=3847
f(90)=7873
f(91)=4027
f(92)=8237
f(93)=4211
f(94)=8609
f(95)=1
f(96)=1
f(97)=4591
f(98)=9377
f(99)=4787

b) Substitution of the polynom
The polynom f(x)=x^2-227 could be written as f(y)= y^2-227 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 > 15

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

227, 113, 223, 109, 211, 101, 191, 89, 163, 73, 127, 53, 83, 29, 31, 1, 1, 1, 97, 67, 173, 107, 257, 151, 349, 199, 449, 251, 557, 307, 673, 367, 797, 431, 929, 499, 1069, 571, 1217, 647, 1373, 727, 1, 811, 1709, 1, 1889, 991, 1, 1087, 2273, 1187, 2477, 1291, 2689, 1399, 2909, 1511, 3137, 1627, 3373, 1747, 3617, 1871, 1, 1999, 4129, 2131, 4397, 2267, 4673, 1, 4957, 2551, 181, 2699, 179, 2851, 5857, 1, 6173, 3167, 1, 3331, 6829, 3499, 1, 3671, 7517, 3847, 7873, 4027, 8237, 4211, 8609, 1, 1, 4591, 9377, 4787, 337, 4987, 10177, 1, 10589, 5399, 1, 1, 11437, 5827, 383, 6047, 1, 6271, 1, 1, 13229, 1, 13697, 6967, 14173, 7207, 14657, 7451, 15149, 7699, 15649, 7951, 1, 283, 16673, 8467, 593, 8731, 17729, 8999, 18269, 1, 607, 9547, 19373, 317, 19937, 10111, 20509, 10399, 21089, 10691, 409, 10987, 22273, 11287, 22877, 1, 1, 1, 24109, 12211, 853, 12527, 25373, 443, 26017, 13171, 26669, 13499, 27329, 13831, 27997, 457, 541, 1, 947, 14851, 1, 15199, 1, 15551, 379, 15907, 32173, 16267, 491, 16631, 33629, 1, 34369, 599, 35117, 17747, 1237, 18127, 36637, 1, 37409, 18899, 38189, 1, 38977, 19687, 1283, 1, 40577, 661, 41389, 20899, 42209, 1, 43037, 21727, 601, 22147, 461, 22571, 45569, 1, 1601, 23431, 47297, 823, 719, 1, 49057, 467, 1, 1, 50849, 1, 709, 26107, 52673, 857, 53597, 27031, 1759, 1, 55469, 1, 56417, 28447, 57373, 28927, 58337, 29411, 1, 1031, 60289, 30391, 2113, 1, 62273, 31387, 63277, 31891, 1213, 1, 65309, 32911, 66337, 33427, 757, 1, 2207, 34471, 547, 1129, 70529, 35531, 71597, 36067, 72673, 36607, 73757, 1, 1, 37699, 1433, 1319, 1, 1, 78173, 39367, 1, 1, 80429, 40499, 81569, 613, 1, 41647, 83873, 42227, 85037, 1381, 86209, 43399, 2819, 43991, 877, 44587, 839, 619, 90977, 1579, 92189, 46399, 3221, 887, 937, 1, 95873, 48247, 97117, 48871, 98369, 49499, 1487, 50131, 1, 50767, 1231, 51407, 103457, 52051, 1, 1, 1, 1721, 107357, 1019, 997, 54667, 3793, 55331, 683, 1931, 1, 56671, 1009, 57347, 1021, 58027, 1091, 58711, 1423, 59399, 119489, 60091, 120877, 1, 122273, 61487, 123677, 62191, 1867, 2029, 1733, 63611, 4127, 64327, 2441, 2243, 130817, 739, 4561, 66499, 773, 67231, 135197, 67967, 1409, 1, 138157, 1, 1913, 70199, 1, 70951, 142657, 71707, 144173, 72467, 2749, 1093, 147229, 73999, 4799, 74771, 787, 2437, 5237, 1, 153437, 2659, 155009, 77899, 156589, 78691, 1, 1, 159773, 80287, 161377, 977, 162989, 81899, 1697, 1, 166237, 827, 167873, 84347, 1, 1607, 171169, 85999, 172829, 2801, 1, 3023, 5683, 1321, 6133, 1, 179549, 90199, 859, 1097, 182957, 1259, 1223, 92767, 186397, 1, 1051, 1783, 1049, 1, 2153, 883, 193373, 97127, 195137, 98011, 1, 98899, 198689, 1367, 1, 941, 3019, 1, 2293, 907, 1, 103399, 207709, 104311, 1, 105227, 211373, 1, 213217, 107071, 1, 107999, 4093, 1123, 218797, 1, 220673, 110807, 222557, 111751, 224449, 1, 1499, 3919, 228257, 3697, 7937, 1, 7487, 116531, 234029, 117499, 2843, 118471, 1, 119447, 239873, 120427, 1, 1, 243809, 122399, 245789, 1, 1951, 1, 2473, 125387, 3449, 1303, 253789, 127399, 8821, 128411, 257837, 4463, 1, 130447, 261917, 4241, 2467, 132499, 266029, 1993, 2111, 2539, 3701, 135607, 272257, 136651, 274349, 137699, 276449, 1559, 278557, 1, 280673, 140867, 282797, 141931, 1, 4931, 1117, 144071, 9973, 1, 291373, 1, 2693, 147311, 9539, 148399, 297889, 149491, 2753, 150587, 302273, 151687, 304477, 152791, 306689, 2297, 2887, 1, 311137, 156127, 313373, 157247, 315617, 158371, 1, 159499, 2833, 1, 322397, 1949, 324673, 162907, 1, 164051, 1, 1, 331549, 166351, 333857, 1, 1, 1, 338497, 169831, 5087, 1, 343169, 172171, 345517, 173347, 347873, 174527, 350237, 1, 6653, 176899, 12241, 178091, 357377, 179287, 3709, 1787, 362177, 5861, 1, 182899, 11839, 184111, 1, 185327, 371873, 1847, 374317, 1, 376769, 188999, 4261, 1, 381697, 191467, 384173, 1801, 1, 193951, 389149, 1, 1, 1301, 394157, 1, 4457, 198967, 399197, 200231, 12959, 201499, 404269, 1, 406817, 204047, 2287, 205327, 411937, 1, 2543, 1637, 417089, 3947, 5749, 210487, 2333, 1, 5119, 213091, 14741, 214399, 1, 1979, 432737, 217027, 2671, 2251, 438017, 2053, 440669, 7129, 6073, 1, 14387, 223667, 1, 1, 2609, 1, 1, 227699, 5503, 2027, 8669, 2039, 15937, 1, 464897, 8039, 467629, 234499, 1, 235871, 473117, 3541, 1, 238627, 478637, 240011, 481409, 1, 15619, 2503, 486977, 7877, 9241, 245587, 4877, 1, 495389, 2791, 498209, 249811, 501037, 8663, 1, 1, 1, 254071, 509569, 255499, 512429, 256931, 7691, 2903, 1831, 3559, 1493, 261251, 523949, 1, 2099, 8521, 4951, 265607, 17183, 5039, 3547, 268531, 538529, 3253, 541469, 271471, 18773, 3739, 547373, 9463, 550337, 275911, 553309, 1, 556289, 278891, 559277, 1, 562273, 281887, 4451, 5347, 568289, 284899, 8527, 286411, 1, 3469, 5297, 9337, 580417, 290971, 583469, 1, 5381, 10139, 1, 295567, 1, 1, 4691, 298651, 1777, 1, 601949, 301751, 4007, 303307, 2423, 304867, 611297, 306431, 11593, 4597, 1, 309571, 620717, 10037, 5521, 312727, 1, 1, 1, 315899, 21841, 317491, 3517, 11003, 1, 320687, 642977, 3191, 646189, 323899, 12253, 1997, 7333, 327127, 2069, 1, 9029, 3271, 662369, 331999, 665629, 333631, 8059, 1873, 21683, 1693, 675457, 1, 678749, 11731, 682049, 341851, 23633, 1, 1, 1907, 691997, 346831, 695329, 3257, 698669, 6607, 702017, 351847, 3343, 353527, 8539, 355211, 712109, 2063, 715489, 358591, 3407, 360287, 722273, 11677, 10831, 363691, 1, 3767, 732509, 12659, 735937, 6959, 739373, 370547, 742817, 372271, 1, 373999, 749729, 5147, 753197, 3463, 756673, 379207, 1, 380951, 763649, 3511, 2003, 3593, 770657, 3041, 774173, 387967, 25087, 1, 781229, 1, 27061, 393271, 788317, 395047, 1, 4091, 2591, 398611, 3583, 1, 802589, 4519, 806177, 3181, 1, 4889, 813377, 3607, 8089, 3623, 820609, 411211, 3631, 413027, 827873, 1, 1, 13441, 8269, 14431, 27059, 420331, 842497, 6301, 1, 424007, 849857, 425851, 1901, 5153, 857249, 429551, 860957, 2857, 864673, 433267, 868397, 435131, 872129, 436999, 2311, 1, 879617, 1, 12101, 15263, 8291, 8387, 1, 446399, 894689, 14461, 898477, 2357, 2939, 452087, 9341, 453991, 909889, 455899, 913709, 6833, 12569, 1, 921373, 2417, 925217, 1, 929069, 8783, 2281, 467431, 936797, 469367, 1, 2633, 10613, 16319, 948449, 15329, 952349, 1, 1, 2647, 960173, 481067, 964097, 483031, 1, 484999, 1, 486971, 10061, 488947, 979873, 490927, 1, 492911, 987809, 494899, 18713, 496891, 5563, 17203,

6. Sequence of the polynom (only primes)

227, 113, 223, 109, 211, 101, 191, 89, 163, 73, 127, 53, 83, 29, 31, 97, 67, 173, 107, 257, 151, 349, 199, 449, 251, 557, 307, 673, 367, 797, 431, 929, 499, 1069, 571, 1217, 647, 1373, 727, 811, 1709, 1889, 991, 1087, 2273, 1187, 2477, 1291, 2689, 1399, 2909, 1511, 3137, 1627, 3373, 1747, 3617, 1871, 1999, 4129, 2131, 4397, 2267, 4673, 4957, 2551, 181, 2699, 179, 2851, 5857, 6173, 3167, 3331, 6829, 3499, 3671, 7517, 3847, 7873, 4027, 8237, 4211, 8609, 4591, 9377, 4787, 337, 4987, 10177, 10589, 5399, 11437, 5827, 383, 6047, 6271, 13229, 13697, 6967, 14173, 7207, 14657, 7451, 15149, 7699, 15649, 7951, 283, 16673, 8467, 593, 8731, 17729, 8999, 18269, 607, 9547, 19373, 317, 19937, 10111, 20509, 10399, 21089, 10691, 409, 10987, 22273, 11287, 22877, 24109, 12211, 853, 12527, 25373, 443, 26017, 13171, 26669, 13499, 27329, 13831, 27997, 457, 541, 947, 14851, 15199, 15551, 379, 15907, 32173, 16267, 491, 16631, 33629, 34369, 599, 35117, 17747, 1237, 18127, 36637, 37409, 18899, 38189, 38977, 19687, 1283, 40577, 661, 41389, 20899, 42209, 43037, 21727, 601, 22147, 461, 22571, 45569, 1601, 23431, 47297, 823, 719, 49057, 467, 50849, 709, 26107, 52673, 857, 53597, 27031, 1759, 55469, 56417, 28447, 57373, 28927, 58337, 29411, 1031, 60289, 30391, 2113, 62273, 31387, 63277, 31891, 1213, 65309, 32911, 66337, 33427, 757, 2207, 34471, 547, 1129, 70529, 35531, 71597, 36067, 72673, 36607, 73757, 37699, 1433, 1319, 78173, 39367, 80429, 40499, 81569, 613, 41647, 83873, 42227, 85037, 1381, 86209, 43399, 2819, 43991, 877, 44587, 839, 619, 90977, 1579, 92189, 46399, 3221, 887, 937, 95873, 48247, 97117, 48871, 98369, 49499, 1487, 50131, 50767, 1231, 51407, 103457, 52051, 1721, 107357, 1019, 997, 54667, 3793, 55331, 683, 1931, 56671, 1009, 57347, 1021, 58027, 1091, 58711, 1423, 59399, 119489, 60091, 120877, 122273, 61487, 123677, 62191, 1867, 2029, 1733, 63611, 4127, 64327, 2441, 2243, 130817, 739, 4561, 66499, 773, 67231, 135197, 67967, 1409, 138157, 1913, 70199, 70951, 142657, 71707, 144173, 72467, 2749, 1093, 147229, 73999, 4799, 74771, 787, 2437, 5237, 153437, 2659, 155009, 77899, 156589, 78691, 159773, 80287, 161377, 977, 162989, 81899, 1697, 166237, 827, 167873, 84347, 1607, 171169, 85999, 172829, 2801, 3023, 5683, 1321, 6133, 179549, 90199, 859, 1097, 182957, 1259, 1223, 92767, 186397, 1051, 1783, 1049, 2153, 883, 193373, 97127, 195137, 98011, 98899, 198689, 1367, 941, 3019, 2293, 907, 103399, 207709, 104311, 105227, 211373, 213217, 107071, 107999, 4093, 1123, 218797, 220673, 110807, 222557, 111751, 224449, 1499, 3919, 228257, 3697, 7937, 7487, 116531, 234029, 117499, 2843, 118471, 119447, 239873, 120427, 243809, 122399, 245789, 1951, 2473, 125387, 3449, 1303, 253789, 127399, 8821, 128411, 257837, 4463, 130447, 261917, 4241, 2467, 132499, 266029, 1993, 2111, 2539, 3701, 135607, 272257, 136651, 274349, 137699, 276449, 1559, 278557, 280673, 140867, 282797, 141931, 4931, 1117, 144071, 9973, 291373, 2693, 147311, 9539, 148399, 297889, 149491, 2753, 150587, 302273, 151687, 304477, 152791, 306689, 2297, 2887, 311137, 156127, 313373, 157247, 315617, 158371, 159499, 2833, 322397, 1949, 324673, 162907, 164051, 331549, 166351, 333857, 338497, 169831, 5087, 343169, 172171, 345517, 173347, 347873, 174527, 350237, 6653, 176899, 12241, 178091, 357377, 179287, 3709, 1787, 362177, 5861, 182899, 11839, 184111, 185327, 371873, 1847, 374317, 376769, 188999, 4261, 381697, 191467, 384173, 1801, 193951, 389149, 1301, 394157, 4457, 198967, 399197, 200231, 12959, 201499, 404269, 406817, 204047, 2287, 205327, 411937, 2543, 1637, 417089, 3947, 5749, 210487, 2333, 5119, 213091, 14741, 214399, 1979, 432737, 217027, 2671, 2251, 438017, 2053, 440669, 7129, 6073, 14387, 223667, 2609, 227699, 5503, 2027, 8669, 2039, 15937, 464897, 8039, 467629, 234499, 235871, 473117, 3541, 238627, 478637, 240011, 481409, 15619, 2503, 486977, 7877, 9241, 245587, 4877, 495389, 2791, 498209, 249811, 501037, 8663, 254071, 509569, 255499, 512429, 256931, 7691, 2903, 1831, 3559, 1493, 261251, 523949, 2099, 8521, 4951, 265607, 17183, 5039, 3547, 268531, 538529, 3253, 541469, 271471, 18773, 3739, 547373, 9463, 550337, 275911, 553309, 556289, 278891, 559277, 562273, 281887, 4451, 5347, 568289, 284899, 8527, 286411, 3469, 5297, 9337, 580417, 290971, 583469, 5381, 10139, 295567, 4691, 298651, 1777, 601949, 301751, 4007, 303307, 2423, 304867, 611297, 306431, 11593, 4597, 309571, 620717, 10037, 5521, 312727, 315899, 21841, 317491, 3517, 11003, 320687, 642977, 3191, 646189, 323899, 12253, 1997, 7333, 327127, 2069, 9029, 3271, 662369, 331999, 665629, 333631, 8059, 1873, 21683, 1693, 675457, 678749, 11731, 682049, 341851, 23633, 1907, 691997, 346831, 695329, 3257, 698669, 6607, 702017, 351847, 3343, 353527, 8539, 355211, 712109, 2063, 715489, 358591, 3407, 360287, 722273, 11677, 10831, 363691, 3767, 732509, 12659, 735937, 6959, 739373, 370547, 742817, 372271, 373999, 749729, 5147, 753197, 3463, 756673, 379207, 380951, 763649, 3511, 2003, 3593, 770657, 3041, 774173, 387967, 25087, 781229, 27061, 393271, 788317, 395047, 4091, 2591, 398611, 3583, 802589, 4519, 806177, 3181, 4889, 813377, 3607, 8089, 3623, 820609, 411211, 3631, 413027, 827873, 13441, 8269, 14431, 27059, 420331, 842497, 6301, 424007, 849857, 425851, 1901, 5153, 857249, 429551, 860957, 2857, 864673, 433267, 868397, 435131, 872129, 436999, 2311, 879617, 12101, 15263, 8291, 8387, 446399, 894689, 14461, 898477, 2357, 2939, 452087, 9341, 453991, 909889, 455899, 913709, 6833, 12569, 921373, 2417, 925217, 929069, 8783, 2281, 467431, 936797, 469367, 2633, 10613, 16319, 948449, 15329, 952349, 2647, 960173, 481067, 964097, 483031, 484999, 486971, 10061, 488947, 979873, 490927, 492911, 987809, 494899, 18713, 496891, 5563, 17203,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x-227 and
the reducible primes which appear as divisor for the first time
p | x^2x-227 and p < x^2x-227

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
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
1 2 3 2 1 1.5 1 0.5
2 4 5 3 2 1.25 0.75 0.5
3 8 9 5 4 1.125 0.625 0.5
4 16 15 8 7 0.9375 0.5 0.4375
5 32 30 16 14 0.9375 0.5 0.4375
6 64 58 29 29 0.90625 0.453125 0.453125
7 128 108 50 58 0.84375 0.390625 0.453125
8 256 214 91 123 0.8359375 0.35546875 0.48046875
9 512 427 150 277 0.83398438 0.29296875 0.54101563
10 1024 853 275 578 0.83300781 0.26855469 0.56445313
11 2048 1674 494 1180 0.81738281 0.24121094 0.57617188
12 4096 3296 872 2424 0.8046875 0.21289063 0.59179688
13 8192 6526 1553 4973 0.79663086 0.1895752 0.60705566
14 16384 12912 2884 10028 0.78808594 0.17602539 0.61206055
15 32768 25602 5319 20283 0.78131104 0.162323 0.61898804
16 65536 50747 9945 40802 0.77433777 0.15174866 0.62258911
17 131072 100743 18618 82125 0.76860809 0.14204407 0.62656403
18 262144 200381 34716 165665 0.76439285 0.13243103 0.63196182
19 524288 398673 65520 333153 0.7604084 0.12496948 0.63543892
20 1048576 793365 124137 669228 0.75661182 0.11838627 0.63822556
21 2097152 1580519 234829 1345690 0.75365019 0.11197519 0.641675
22 4194304 3148761 446969 2701792 0.75072312 0.10656571 0.64415741
23 8388608 6276245 851542 5424703 0.74818671 0.10151172 0.64667499
24 16777216 12512309 1625480 10886829 0.74579173 0.09688616 0.64890558


8. Check for existing Integer Sequences by OEIS

Found in Database : 227, 113, 223, 109, 211, 101, 191, 89, 163, 73, 127, 53, 83, 29, 31, 1, 1, 1, 97, 67,
Found in Database : 227, 113, 223, 109, 211, 101, 191, 89, 163, 73, 127, 53, 83, 29, 31, 97, 67, 173, 107, 257, 151, 349, 199, 449, 251, 557, 307, 673, 367, 797, 431, 929, 499, 1069, 571, 1217, 647,
Found in Database : 29, 31, 53, 67, 73, 83, 89, 97, 101, 107, 109, 113, 127,