Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:30:50
Deutsch
28.Mar 2024

Polynom = x^2+188x-997

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) = 997 = 997
f(1) = 101 = 101
f(2) = 617 = 617
f(3) = 53 = 53
f(4) = 229 = 229
f(5) = 1 = 1
f(6) = 167 = 167
f(7) = 23 = 23
f(8) = 571 = 571
f(9) = 97 = 97
f(10) = 983 = 983
f(11) = 149 = 149
f(12) = 1403 = 23*61
f(13) = 101 = 101
f(14) = 1831 = 1831
f(15) = 1 = 1
f(16) = 2267 = 2267
f(17) = 311 = 311
f(18) = 2711 = 2711
f(19) = 367 = 367
f(20) = 3163 = 3163
f(21) = 53 = 53
f(22) = 3623 = 3623
f(23) = 241 = 241
f(24) = 4091 = 4091
f(25) = 541 = 541
f(26) = 4567 = 4567
f(27) = 601 = 601
f(28) = 5051 = 5051
f(29) = 331 = 331
f(30) = 5543 = 23*241
f(31) = 181 = 181
f(32) = 6043 = 6043
f(33) = 787 = 787
f(34) = 6551 = 6551
f(35) = 851 = 23*37
f(36) = 7067 = 37*191
f(37) = 229 = 229
f(38) = 7591 = 7591
f(39) = 491 = 491
f(40) = 8123 = 8123
f(41) = 1049 = 1049
f(42) = 8663 = 8663
f(43) = 1117 = 1117
f(44) = 9211 = 61*151
f(45) = 593 = 593
f(46) = 9767 = 9767
f(47) = 157 = 157
f(48) = 10331 = 10331
f(49) = 1327 = 1327
f(50) = 10903 = 10903
f(51) = 1399 = 1399
f(52) = 11483 = 11483
f(53) = 23 = 23
f(54) = 12071 = 12071
f(55) = 773 = 773
f(56) = 12667 = 53*239
f(57) = 1621 = 1621
f(58) = 13271 = 23*577
f(59) = 1697 = 1697
f(60) = 13883 = 13883
f(61) = 887 = 887
f(62) = 14503 = 14503
f(63) = 463 = 463
f(64) = 15131 = 15131
f(65) = 1931 = 1931
f(66) = 15767 = 15767
f(67) = 2011 = 2011
f(68) = 16411 = 16411
f(69) = 523 = 523
f(70) = 17063 = 113*151
f(71) = 1087 = 1087
f(72) = 17723 = 37*479
f(73) = 2257 = 37*61
f(74) = 18391 = 53*347
f(75) = 2341 = 2341
f(76) = 19067 = 23*829
f(77) = 1213 = 1213
f(78) = 19751 = 19751
f(79) = 157 = 157
f(80) = 20443 = 20443
f(81) = 2599 = 23*113
f(82) = 21143 = 21143
f(83) = 2687 = 2687
f(84) = 21851 = 21851
f(85) = 347 = 347
f(86) = 22567 = 22567
f(87) = 1433 = 1433
f(88) = 23291 = 23291
f(89) = 2957 = 2957
f(90) = 24023 = 24023
f(91) = 3049 = 3049
f(92) = 24763 = 24763
f(93) = 1571 = 1571
f(94) = 25511 = 97*263
f(95) = 809 = 809
f(96) = 26267 = 26267
f(97) = 3331 = 3331
f(98) = 27031 = 27031
f(99) = 3427 = 23*149
f(100) = 27803 = 27803

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+188x-997

f(0)=997
f(1)=101
f(2)=617
f(3)=53
f(4)=229
f(5)=1
f(6)=167
f(7)=23
f(8)=571
f(9)=97
f(10)=983
f(11)=149
f(12)=61
f(13)=1
f(14)=1831
f(15)=1
f(16)=2267
f(17)=311
f(18)=2711
f(19)=367
f(20)=3163
f(21)=1
f(22)=3623
f(23)=241
f(24)=4091
f(25)=541
f(26)=4567
f(27)=601
f(28)=5051
f(29)=331
f(30)=1
f(31)=181
f(32)=6043
f(33)=787
f(34)=6551
f(35)=37
f(36)=191
f(37)=1
f(38)=7591
f(39)=491
f(40)=8123
f(41)=1049
f(42)=8663
f(43)=1117
f(44)=151
f(45)=593
f(46)=9767
f(47)=157
f(48)=10331
f(49)=1327
f(50)=10903
f(51)=1399
f(52)=11483
f(53)=1
f(54)=12071
f(55)=773
f(56)=239
f(57)=1621
f(58)=577
f(59)=1697
f(60)=13883
f(61)=887
f(62)=14503
f(63)=463
f(64)=15131
f(65)=1931
f(66)=15767
f(67)=2011
f(68)=16411
f(69)=523
f(70)=113
f(71)=1087
f(72)=479
f(73)=1
f(74)=347
f(75)=2341
f(76)=829
f(77)=1213
f(78)=19751
f(79)=1
f(80)=20443
f(81)=1
f(82)=21143
f(83)=2687
f(84)=21851
f(85)=1
f(86)=22567
f(87)=1433
f(88)=23291
f(89)=2957
f(90)=24023
f(91)=3049
f(92)=24763
f(93)=1571
f(94)=263
f(95)=809
f(96)=26267
f(97)=3331
f(98)=27031
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+188x-997 could be written as f(y)= y^2-9833 with x=y-94

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+94
f'(x)>2x+187

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

997, 101, 617, 53, 229, 1, 167, 23, 571, 97, 983, 149, 61, 1, 1831, 1, 2267, 311, 2711, 367, 3163, 1, 3623, 241, 4091, 541, 4567, 601, 5051, 331, 1, 181, 6043, 787, 6551, 37, 191, 1, 7591, 491, 8123, 1049, 8663, 1117, 151, 593, 9767, 157, 10331, 1327, 10903, 1399, 11483, 1, 12071, 773, 239, 1621, 577, 1697, 13883, 887, 14503, 463, 15131, 1931, 15767, 2011, 16411, 523, 113, 1087, 479, 1, 347, 2341, 829, 1213, 19751, 1, 20443, 1, 21143, 2687, 21851, 1, 22567, 1433, 23291, 2957, 24023, 3049, 24763, 1571, 263, 809, 26267, 3331, 27031, 1, 27803, 881, 283, 1811, 1277, 1, 1, 3821, 30971, 1, 859, 503, 32603, 4127, 1, 4231, 34267, 271, 35111, 2221, 35963, 4549, 1601, 4657, 37691, 2383, 38567, 1, 39451, 4987, 40343, 5099, 41243, 1303, 691, 2663, 43067, 5441, 43991, 5557, 269, 2837, 45863, 1, 46811, 257, 1291, 163, 48731, 769, 2161, 3137, 50683, 6397, 317, 6521, 52667, 3323, 281, 1693, 1, 6899, 1051, 7027, 56731, 1789, 947, 3643, 2557, 7417, 59863, 7549, 60923, 1, 61991, 977, 63067, 7951, 64151, 8087, 1231, 1, 66343, 1, 1823, 8501, 68567, 8641, 69691, 4391, 70823, 1, 71963, 9067, 647, 1, 3229, 2339, 75431, 4751, 76603, 9649, 77783, 1, 1, 4973, 80167, 631, 81371, 10247, 307, 10399, 1, 1319, 3697, 1, 1, 10861, 87511, 1, 2399, 1, 90023, 2833, 91291, 11491, 92567, 1, 93851, 2953, 95143, 5987, 96443, 1, 409, 12301, 1, 1, 100391, 1579, 101723, 12799, 4481, 12967, 397, 821, 105767, 6653, 719, 13477, 108503, 13649, 109883, 6911, 111271, 3499, 1847, 383, 3083, 14347, 5021, 3631, 116903, 7351, 1, 1, 1, 15061, 2287, 7621, 122663, 1, 124123, 15607, 125591, 15791, 449, 1997, 128551, 8081, 130043, 16349, 131543, 1, 133051, 8363, 2539, 4229, 1, 17107, 137623, 17299, 439, 4373, 3803, 1, 1259, 17881, 143831, 18077, 1499, 9137, 146983, 2309, 1471, 18671, 6529, 1, 151771, 1, 941, 419, 155003, 19477, 156631, 19681, 1567, 1, 159911, 5023, 161563, 1, 163223, 20507, 911, 5179, 166567, 10463, 1, 919, 169943, 1, 4639, 10781, 7537, 1361, 175067, 21991, 176791, 1, 1069, 2803, 180263, 11321, 182011, 22861, 1217, 23081, 185531, 1, 187303, 5881, 8221, 23747, 190871, 23971, 192667, 1, 194471, 12211, 1, 1, 499, 24877, 199931, 12553, 201767, 3167, 5503, 1, 205463, 25799, 1373, 1627, 3947, 1, 211067, 26501, 3491, 26737, 9341, 13487, 216743, 6803, 218651, 1, 1, 27691, 1, 6983, 224423, 14087, 4271, 1, 228311, 28661, 563, 1, 1, 1, 234203, 29399, 1, 1289, 238171, 1, 6491, 15073, 242171, 1, 4003, 30649, 246203, 15451, 248231, 7789, 250267, 1, 1321, 31667, 2251, 1, 256423, 16091, 258491, 32441, 11329, 1, 262651, 16481, 264743, 4153, 1, 33487, 599, 33751, 271067, 1063, 1, 1, 275323, 34549, 7499, 1, 12157, 1, 281767, 8839, 569, 1549, 286103, 35899, 288283, 9043, 290471, 18223, 292667, 36721, 1979, 36997, 297083, 18637, 1307, 2347, 301531, 37831, 303767, 1657, 306011, 4799, 308263, 1, 587, 1, 1873, 39241, 315067, 19763, 317351, 1, 1, 40099, 3319, 40387, 324251, 10169, 326567, 20483, 328891, 41257, 14401, 41549, 333563, 20921, 3463, 1, 338267, 1, 6427, 42727, 5623, 2689, 607, 21661, 347771, 1, 3467, 1, 1301, 22111, 354983, 11131, 1483, 1949, 613, 45131, 9791, 1, 1, 22871, 367163, 46049, 3271, 1, 1, 23333, 1, 1, 377051, 47287, 1, 47599, 6263, 1, 1609, 24113, 16829, 48541, 389591, 48857, 2053, 1, 701, 12373, 397211, 49811, 2683, 50131, 402331, 12613, 2579, 1, 407483, 1381, 11083, 51421, 412667, 25873, 415271, 1, 417883, 1, 420503, 52727, 18397, 1, 1, 26693, 428411, 53717, 431063, 54049, 433723, 27191, 2411, 13679, 733, 55051, 441751, 1, 444443, 13931, 19441, 28031, 449851, 56401, 8539, 2467, 455291, 28541, 12379, 1, 1, 1, 463511, 58111, 466267, 7307, 469031, 29401, 471803, 59149, 474583, 59497, 9007, 1, 480167, 1, 482971, 1, 21121, 60899, 488603, 15313, 1, 30803, 494267, 61961, 497111, 1, 1901, 31337, 8243, 7879, 505691, 63391, 508567, 1723, 1, 4007, 514343, 32237, 517243, 2819, 520151, 1, 3209, 32783, 1, 1, 1, 1, 531863, 1, 534811, 16759, 4759, 33703, 3581, 67777, 543703, 2963, 546683, 34261, 9011, 2153, 24029, 1, 555671, 1, 558683, 8753, 561703, 35201, 15263, 1913, 567767, 71161, 2017, 35771, 573863, 17981, 823, 72307, 1, 1, 6011, 18269, 2179, 1597, 589243, 73849, 592343, 1, 595451, 37313, 3307, 9377, 6203, 75407, 604823, 1, 11471, 2381, 611111, 1, 614267, 3347, 1, 77377, 620603, 1, 1, 19543, 1637, 78571, 630167, 1, 6271, 19843, 12011, 39887, 2677, 80177, 643031, 1, 646267, 40493, 649511, 5087, 1, 81799, 656023, 82207, 2879, 1, 662567, 41513, 2141, 83437, 1, 1, 672443, 42131, 675751, 21169, 679067, 85091, 18443, 2311, 685723, 21481, 689063, 1877, 11351, 1, 2887, 87181, 1, 43801, 702503, 11003, 705883, 1, 709271, 1, 4783, 5581, 1709, 1, 719483, 90149, 722903, 1, 726331, 45503, 31729, 22859, 2113, 91867, 2801, 4013, 740123, 1, 743591, 1, 1, 93601, 2671, 1, 754043, 47237, 757543, 1, 761051, 1, 929, 95791, 2423, 1, 771623, 48337, 775163, 97117, 33857, 97561, 782267, 49003, 14827, 1, 4133, 98899, 792983, 99347, 796571, 1, 4909, 50123, 4813, 1, 807383, 101149, 953, 1, 814631, 12757, 15439, 4457, 821911, 102967, 1, 1, 829223, 51941, 2713, 1, 836567, 104801, 1, 52631, 2749, 26431, 5399, 106187, 851351, 4637, 855067, 1, 1, 53791, 37501, 1, 3061, 108517, 5839, 54493, 873767, 6841, 877531, 2971, 23819, 2083, 885083, 13859, 888871, 55673, 892667, 111821, 38977, 112297, 900283, 1, 904103, 1, 5783, 113731, 14947, 1, 915611, 1, 9479, 57587, 8171, 115657, 927191, 116141, 931067, 58313, 934951, 14639, 938843, 5113, 9719, 1, 1019, 7411, 1, 1, 954491, 119557, 958423, 120049, 962363, 60271, 2633, 30259, 970267, 121531, 15971, 122027, 978203, 30631, 982183, 61511, 1, 1, 1039, 124021, 1, 2707, 9883, 3907, 1, 125527, 1006231, 126031, 3931, 15817, 1, 63521, 4447, 127549, 1, 3461, 27743, 64283, 3251, 1, 1034651, 129587, 3833, 130099, 45341, 32653, 1046951, 65563, 5807, 131641, 1055191, 132157, 1059323, 66337, 1063463, 16649, 1067611, 133711, 1, 1, 1, 4211, 1, 1, 4219, 1, 1088471, 5927, 1092667, 1, 1096871, 1, 29759, 3727, 6781, 138427, 1109531, 34739, 1093, 1, 1118011, 1, 1122263, 140549, 1126523, 3067, 10007, 1, 7517, 142151, 49537, 1, 1143643, 17903, 1, 71881, 1109, 1, 1156567, 1, 19031, 1, 1165223, 1, 1169563, 1, 4871, 147011, 51229, 1, 31963, 74051,

6. Sequence of the polynom (only primes)

997, 101, 617, 53, 229, 167, 23, 571, 97, 983, 149, 61, 1831, 2267, 311, 2711, 367, 3163, 3623, 241, 4091, 541, 4567, 601, 5051, 331, 181, 6043, 787, 6551, 37, 191, 7591, 491, 8123, 1049, 8663, 1117, 151, 593, 9767, 157, 10331, 1327, 10903, 1399, 11483, 12071, 773, 239, 1621, 577, 1697, 13883, 887, 14503, 463, 15131, 1931, 15767, 2011, 16411, 523, 113, 1087, 479, 347, 2341, 829, 1213, 19751, 20443, 21143, 2687, 21851, 22567, 1433, 23291, 2957, 24023, 3049, 24763, 1571, 263, 809, 26267, 3331, 27031, 27803, 881, 283, 1811, 1277, 3821, 30971, 859, 503, 32603, 4127, 4231, 34267, 271, 35111, 2221, 35963, 4549, 1601, 4657, 37691, 2383, 38567, 39451, 4987, 40343, 5099, 41243, 1303, 691, 2663, 43067, 5441, 43991, 5557, 269, 2837, 45863, 46811, 257, 1291, 163, 48731, 769, 2161, 3137, 50683, 6397, 317, 6521, 52667, 3323, 281, 1693, 6899, 1051, 7027, 56731, 1789, 947, 3643, 2557, 7417, 59863, 7549, 60923, 61991, 977, 63067, 7951, 64151, 8087, 1231, 66343, 1823, 8501, 68567, 8641, 69691, 4391, 70823, 71963, 9067, 647, 3229, 2339, 75431, 4751, 76603, 9649, 77783, 4973, 80167, 631, 81371, 10247, 307, 10399, 1319, 3697, 10861, 87511, 2399, 90023, 2833, 91291, 11491, 92567, 93851, 2953, 95143, 5987, 96443, 409, 12301, 100391, 1579, 101723, 12799, 4481, 12967, 397, 821, 105767, 6653, 719, 13477, 108503, 13649, 109883, 6911, 111271, 3499, 1847, 383, 3083, 14347, 5021, 3631, 116903, 7351, 15061, 2287, 7621, 122663, 124123, 15607, 125591, 15791, 449, 1997, 128551, 8081, 130043, 16349, 131543, 133051, 8363, 2539, 4229, 17107, 137623, 17299, 439, 4373, 3803, 1259, 17881, 143831, 18077, 1499, 9137, 146983, 2309, 1471, 18671, 6529, 151771, 941, 419, 155003, 19477, 156631, 19681, 1567, 159911, 5023, 161563, 163223, 20507, 911, 5179, 166567, 10463, 919, 169943, 4639, 10781, 7537, 1361, 175067, 21991, 176791, 1069, 2803, 180263, 11321, 182011, 22861, 1217, 23081, 185531, 187303, 5881, 8221, 23747, 190871, 23971, 192667, 194471, 12211, 499, 24877, 199931, 12553, 201767, 3167, 5503, 205463, 25799, 1373, 1627, 3947, 211067, 26501, 3491, 26737, 9341, 13487, 216743, 6803, 218651, 27691, 6983, 224423, 14087, 4271, 228311, 28661, 563, 234203, 29399, 1289, 238171, 6491, 15073, 242171, 4003, 30649, 246203, 15451, 248231, 7789, 250267, 1321, 31667, 2251, 256423, 16091, 258491, 32441, 11329, 262651, 16481, 264743, 4153, 33487, 599, 33751, 271067, 1063, 275323, 34549, 7499, 12157, 281767, 8839, 569, 1549, 286103, 35899, 288283, 9043, 290471, 18223, 292667, 36721, 1979, 36997, 297083, 18637, 1307, 2347, 301531, 37831, 303767, 1657, 306011, 4799, 308263, 587, 1873, 39241, 315067, 19763, 317351, 40099, 3319, 40387, 324251, 10169, 326567, 20483, 328891, 41257, 14401, 41549, 333563, 20921, 3463, 338267, 6427, 42727, 5623, 2689, 607, 21661, 347771, 3467, 1301, 22111, 354983, 11131, 1483, 1949, 613, 45131, 9791, 22871, 367163, 46049, 3271, 23333, 377051, 47287, 47599, 6263, 1609, 24113, 16829, 48541, 389591, 48857, 2053, 701, 12373, 397211, 49811, 2683, 50131, 402331, 12613, 2579, 407483, 1381, 11083, 51421, 412667, 25873, 415271, 417883, 420503, 52727, 18397, 26693, 428411, 53717, 431063, 54049, 433723, 27191, 2411, 13679, 733, 55051, 441751, 444443, 13931, 19441, 28031, 449851, 56401, 8539, 2467, 455291, 28541, 12379, 463511, 58111, 466267, 7307, 469031, 29401, 471803, 59149, 474583, 59497, 9007, 480167, 482971, 21121, 60899, 488603, 15313, 30803, 494267, 61961, 497111, 1901, 31337, 8243, 7879, 505691, 63391, 508567, 1723, 4007, 514343, 32237, 517243, 2819, 520151, 3209, 32783, 531863, 534811, 16759, 4759, 33703, 3581, 67777, 543703, 2963, 546683, 34261, 9011, 2153, 24029, 555671, 558683, 8753, 561703, 35201, 15263, 1913, 567767, 71161, 2017, 35771, 573863, 17981, 823, 72307, 6011, 18269, 2179, 1597, 589243, 73849, 592343, 595451, 37313, 3307, 9377, 6203, 75407, 604823, 11471, 2381, 611111, 614267, 3347, 77377, 620603, 19543, 1637, 78571, 630167, 6271, 19843, 12011, 39887, 2677, 80177, 643031, 646267, 40493, 649511, 5087, 81799, 656023, 82207, 2879, 662567, 41513, 2141, 83437, 672443, 42131, 675751, 21169, 679067, 85091, 18443, 2311, 685723, 21481, 689063, 1877, 11351, 2887, 87181, 43801, 702503, 11003, 705883, 709271, 4783, 5581, 1709, 719483, 90149, 722903, 726331, 45503, 31729, 22859, 2113, 91867, 2801, 4013, 740123, 743591, 93601, 2671, 754043, 47237, 757543, 761051, 929, 95791, 2423, 771623, 48337, 775163, 97117, 33857, 97561, 782267, 49003, 14827, 4133, 98899, 792983, 99347, 796571, 4909, 50123, 4813, 807383, 101149, 953, 814631, 12757, 15439, 4457, 821911, 102967, 829223, 51941, 2713, 836567, 104801, 52631, 2749, 26431, 5399, 106187, 851351, 4637, 855067, 53791, 37501, 3061, 108517, 5839, 54493, 873767, 6841, 877531, 2971, 23819, 2083, 885083, 13859, 888871, 55673, 892667, 111821, 38977, 112297, 900283, 904103, 5783, 113731, 14947, 915611, 9479, 57587, 8171, 115657, 927191, 116141, 931067, 58313, 934951, 14639, 938843, 5113, 9719, 1019, 7411, 954491, 119557, 958423, 120049, 962363, 60271, 2633, 30259, 970267, 121531, 15971, 122027, 978203, 30631, 982183, 61511, 1039, 124021, 2707, 9883, 3907, 125527, 1006231, 126031, 3931, 15817, 63521, 4447, 127549, 3461, 27743, 64283, 3251, 1034651, 129587, 3833, 130099, 45341, 32653, 1046951, 65563, 5807, 131641, 1055191, 132157, 1059323, 66337, 1063463, 16649, 1067611, 133711, 4211, 4219, 1088471, 5927, 1092667, 1096871, 29759, 3727, 6781, 138427, 1109531, 34739, 1093, 1118011, 1122263, 140549, 1126523, 3067, 10007, 7517, 142151, 49537, 1143643, 17903, 71881, 1109, 1156567, 19031, 1165223, 1169563, 4871, 147011, 51229, 31963, 74051,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+188x-997 and
the reducible primes which appear as divisor for the first time
p | x^2+188x-997 and p < x^2+188x-997

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 8 5 3 1 0.625 0.375
4 16 14 8 6 0.875 0.5 0.375
5 32 28 15 13 0.875 0.46875 0.40625
6 64 58 27 31 0.90625 0.421875 0.484375
7 128 112 48 64 0.875 0.375 0.5
8 256 222 86 136 0.8671875 0.3359375 0.53125
9 512 432 153 279 0.84375 0.29882813 0.54492188
10 1024 835 277 558 0.81542969 0.27050781 0.54492188
11 2048 1641 499 1142 0.80126953 0.24365234 0.55761719
12 4096 3203 932 2271 0.78198242 0.22753906 0.55444336
13 8192 6320 1704 4616 0.77148438 0.20800781 0.56347656
14 16384 12519 3102 9417 0.76409912 0.18933105 0.57476807
15 32768 24815 5835 18980 0.7572937 0.17807007 0.57922363
16 65536 49220 10854 38366 0.7510376 0.1656189 0.5854187
17 131072 97865 20331 77534 0.7466507 0.15511322 0.59153748
18 262144 194844 38148 156696 0.74327087 0.14552307 0.5977478
19 524288 388059 72049 316010 0.7401638 0.13742256 0.60274124
20 1048576 773256 136296 636960 0.73743439 0.12998199 0.60745239
21 2097152 1541037 258101 1282936 0.7348237 0.12307215 0.61175156
22 4194304 3073304 491121 2582183 0.73273277 0.11709237 0.6156404
23 8388608 6131246 935466 5195780 0.73090148 0.11151624 0.61938524
24 16777216 12233195 1786864 10446331 0.72915524 0.10650539 0.62264985


8. Check for existing Integer Sequences by OEIS

Found in Database : 997, 101, 617, 53, 229, 1, 167, 23, 571, 97, 983, 149, 61, 1, 1831, 1, 2267, 311, 2711, 367,
Found in Database : 997, 101, 617, 53, 229, 167, 23, 571, 97, 983, 149, 61, 1831, 2267, 311, 2711, 367, 3163, 3623, 241, 4091, 541, 4567, 601, 5051, 331, 181, 6043, 787, 6551, 37, 191, 7591, 491,
Found in Database : 23, 37, 53, 61, 97, 101, 113, 149,