Inhaltsverzeichnis

Development of
Algorithmic Constructions

23:01:18
Deutsch
28.Mar 2024

Polynom = x^2+136x-349

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) = 349 = 349
f(1) = 53 = 53
f(2) = 73 = 73
f(3) = 17 = 17
f(4) = 211 = 211
f(5) = 89 = 89
f(6) = 503 = 503
f(7) = 163 = 163
f(8) = 803 = 11*73
f(9) = 239 = 239
f(10) = 1111 = 11*101
f(11) = 317 = 317
f(12) = 1427 = 1427
f(13) = 397 = 397
f(14) = 1751 = 17*103
f(15) = 479 = 479
f(16) = 2083 = 2083
f(17) = 563 = 563
f(18) = 2423 = 2423
f(19) = 649 = 11*59
f(20) = 2771 = 17*163
f(21) = 737 = 11*67
f(22) = 3127 = 53*59
f(23) = 827 = 827
f(24) = 3491 = 3491
f(25) = 919 = 919
f(26) = 3863 = 3863
f(27) = 1013 = 1013
f(28) = 4243 = 4243
f(29) = 1109 = 1109
f(30) = 4631 = 11*421
f(31) = 1207 = 17*71
f(32) = 5027 = 11*457
f(33) = 1307 = 1307
f(34) = 5431 = 5431
f(35) = 1409 = 1409
f(36) = 5843 = 5843
f(37) = 1513 = 17*89
f(38) = 6263 = 6263
f(39) = 1619 = 1619
f(40) = 6691 = 6691
f(41) = 1727 = 11*157
f(42) = 7127 = 7127
f(43) = 1837 = 11*167
f(44) = 7571 = 67*113
f(45) = 1949 = 1949
f(46) = 8023 = 71*113
f(47) = 2063 = 2063
f(48) = 8483 = 17*499
f(49) = 2179 = 2179
f(50) = 8951 = 8951
f(51) = 2297 = 2297
f(52) = 9427 = 11*857
f(53) = 2417 = 2417
f(54) = 9911 = 11*17*53
f(55) = 2539 = 2539
f(56) = 10403 = 101*103
f(57) = 2663 = 2663
f(58) = 10903 = 10903
f(59) = 2789 = 2789
f(60) = 11411 = 11411
f(61) = 2917 = 2917
f(62) = 11927 = 11927
f(63) = 3047 = 11*277
f(64) = 12451 = 12451
f(65) = 3179 = 11*17*17
f(66) = 12983 = 12983
f(67) = 3313 = 3313
f(68) = 13523 = 13523
f(69) = 3449 = 3449
f(70) = 14071 = 14071
f(71) = 3587 = 17*211
f(72) = 14627 = 14627
f(73) = 3727 = 3727
f(74) = 15191 = 11*1381
f(75) = 3869 = 53*73
f(76) = 15763 = 11*1433
f(77) = 4013 = 4013
f(78) = 16343 = 59*277
f(79) = 4159 = 4159
f(80) = 16931 = 16931
f(81) = 4307 = 59*73
f(82) = 17527 = 17*1031
f(83) = 4457 = 4457
f(84) = 18131 = 18131
f(85) = 4609 = 11*419
f(86) = 18743 = 18743
f(87) = 4763 = 11*433
f(88) = 19363 = 17*17*67
f(89) = 4919 = 4919
f(90) = 19991 = 19991
f(91) = 5077 = 5077
f(92) = 20627 = 20627
f(93) = 5237 = 5237
f(94) = 21271 = 89*239
f(95) = 5399 = 5399
f(96) = 21923 = 11*1993
f(97) = 5563 = 5563
f(98) = 22583 = 11*2053
f(99) = 5729 = 17*337
f(100) = 23251 = 23251

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+136x-349

f(0)=349
f(1)=53
f(2)=73
f(3)=17
f(4)=211
f(5)=89
f(6)=503
f(7)=163
f(8)=11
f(9)=239
f(10)=101
f(11)=317
f(12)=1427
f(13)=397
f(14)=103
f(15)=479
f(16)=2083
f(17)=563
f(18)=2423
f(19)=59
f(20)=1
f(21)=67
f(22)=1
f(23)=827
f(24)=3491
f(25)=919
f(26)=3863
f(27)=1013
f(28)=4243
f(29)=1109
f(30)=421
f(31)=71
f(32)=457
f(33)=1307
f(34)=5431
f(35)=1409
f(36)=5843
f(37)=1
f(38)=6263
f(39)=1619
f(40)=6691
f(41)=157
f(42)=7127
f(43)=167
f(44)=113
f(45)=1949
f(46)=1
f(47)=2063
f(48)=499
f(49)=2179
f(50)=8951
f(51)=2297
f(52)=857
f(53)=2417
f(54)=1
f(55)=2539
f(56)=1
f(57)=2663
f(58)=10903
f(59)=2789
f(60)=11411
f(61)=2917
f(62)=11927
f(63)=277
f(64)=12451
f(65)=1
f(66)=12983
f(67)=3313
f(68)=13523
f(69)=3449
f(70)=14071
f(71)=1
f(72)=14627
f(73)=3727
f(74)=1381
f(75)=1
f(76)=1433
f(77)=4013
f(78)=1
f(79)=4159
f(80)=16931
f(81)=1
f(82)=1031
f(83)=4457
f(84)=18131
f(85)=419
f(86)=18743
f(87)=433
f(88)=1
f(89)=4919
f(90)=19991
f(91)=5077
f(92)=20627
f(93)=5237
f(94)=1
f(95)=5399
f(96)=1993
f(97)=5563
f(98)=2053
f(99)=337

b) Substitution of the polynom
The polynom f(x)=x^2+136x-349 could be written as f(y)= y^2-4973 with x=y-68

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+68
f'(x)>2x+135

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

349, 53, 73, 17, 211, 89, 503, 163, 11, 239, 101, 317, 1427, 397, 103, 479, 2083, 563, 2423, 59, 1, 67, 1, 827, 3491, 919, 3863, 1013, 4243, 1109, 421, 71, 457, 1307, 5431, 1409, 5843, 1, 6263, 1619, 6691, 157, 7127, 167, 113, 1949, 1, 2063, 499, 2179, 8951, 2297, 857, 2417, 1, 2539, 1, 2663, 10903, 2789, 11411, 2917, 11927, 277, 12451, 1, 12983, 3313, 13523, 3449, 14071, 1, 14627, 3727, 1381, 1, 1433, 4013, 1, 4159, 16931, 1, 1031, 4457, 18131, 419, 18743, 433, 1, 4919, 19991, 5077, 20627, 5237, 1, 5399, 1993, 5563, 2053, 337, 23251, 5897, 1, 6067, 24611, 367, 25303, 1, 26003, 599, 26711, 1, 27427, 6947, 28151, 7129, 1699, 1, 2693, 7499, 251, 7687, 1831, 7877, 31891, 8069, 1, 8263, 631, 769, 34231, 787, 35027, 521, 35831, 9059, 36643, 1, 37463, 557, 1, 9677, 3557, 9887, 39971, 10099, 40823, 10313, 571, 10529, 2503, 977, 43427, 997, 607, 1, 2659, 1, 46103, 1, 887, 11867, 4357, 12097, 4441, 12329, 49783, 739, 50723, 12799, 1, 13037, 52627, 1, 53591, 1229, 54563, 13763, 829, 14009, 56531, 269, 57527, 1, 313, 14759, 5413, 15013, 853, 15269, 3623, 15527, 62627, 15787, 63671, 1459, 1097, 1483, 1, 281, 66851, 991, 67927, 17117, 69011, 17389, 6373, 1039, 6473, 17939, 1, 18217, 727, 1, 74551, 1, 75683, 1733, 4519, 1759, 757, 1, 1181, 19927, 4723, 20219, 81463, 1, 683, 20809, 7621, 21107, 85027, 21407, 1627, 1277, 87443, 22013, 88663, 2029, 89891, 1, 91127, 22937, 1301, 347, 373, 23563, 1, 23879, 8741, 24197, 1, 24517, 98711, 1, 100003, 25163, 1, 359, 102611, 2347, 1009, 2377, 105251, 26479, 2011, 26813, 107923, 1597, 967, 27487, 1, 27827, 10181, 1657, 113363, 28513, 114743, 28859, 116131, 29207, 1, 2687, 118931, 2719, 7079, 1, 121763, 1, 123191, 30977, 7331, 31337, 1, 31699, 11593, 32063, 128983, 32429, 1787, 32797, 131927, 1951, 1499, 3049, 431, 3083, 136403, 2017, 137911, 34667, 2081, 1, 2389, 1, 12953, 1, 13093, 1, 8563, 36587, 881, 1, 148691, 37369, 8839, 3433, 1, 3469, 2161, 38557, 155027, 1, 156631, 39359, 158243, 2339, 14533, 40169, 1, 40577, 163127, 2411, 164771, 41399, 1021, 41813, 168083, 1, 169751, 3877, 171427, 43067, 1, 1, 2609, 43913, 176503, 439, 953, 1, 1487, 45197, 487, 443, 183383, 1, 185123, 46499, 186871, 1, 3559, 1, 190391, 47819, 3257, 1, 193943, 1, 195731, 49157, 17957, 1, 18121, 1, 2833, 50513, 11939, 50969, 204791, 51427, 206627, 1, 12263, 4759, 210323, 52813, 212183, 53279, 214051, 1, 215927, 54217, 19801, 3217, 19973, 55163, 221603, 55639, 1, 3301, 225427, 56597, 2251, 5189, 229283, 5233, 231223, 58049, 541, 58537, 13831, 1, 237091, 1123, 1, 60013, 1289, 60509, 243031, 61007, 4153, 61507, 247031, 1051, 1, 5683, 251063, 1, 3467, 63527, 255127, 64037, 257171, 3797, 1, 65063, 23753, 65579, 1, 1, 265427, 66617, 267511, 67139, 15859, 1, 271703, 6199, 2711, 6247, 16231, 69247, 278051, 69779, 280183, 70313, 2741, 70849, 2351, 71387, 1, 4231, 593, 72469, 290963, 73013, 5531, 4327, 1, 6737, 3343, 617, 299731, 75209, 301943, 1, 304163, 1, 1, 1303, 28057, 1, 1, 77999, 1, 1, 2791, 1493, 317651, 79697, 319927, 7297, 1, 7349, 324503, 4789, 1163, 1, 329111, 82567, 2111, 1, 30341, 1, 30553, 84313, 338423, 1, 1, 85487, 343127, 86077, 20323, 7879, 643, 7933, 4933, 1, 20743, 1669, 355027, 89057, 357431, 89659, 32713, 90263, 32933, 1, 364691, 5381, 367127, 1297, 2213, 92699, 372023, 1, 374483, 8539, 6389, 94547, 7159, 1613, 381911, 95789, 384403, 1439, 2069, 97039, 35401, 1, 1, 98297, 23203, 98929, 396983, 99563, 399523, 9109, 402071, 1, 1693, 101477, 407191, 6007, 691, 102763, 412343, 103409, 1, 6121, 37957, 104707, 7927, 105359, 5791, 106013, 425363, 106669, 427991, 1, 1, 9817, 433271, 108649, 435923, 109313, 25799, 1549, 441251, 110647, 40357, 111317, 3691, 2113, 449303, 112663, 2879, 1, 454711, 1, 7753, 10427, 6481, 1, 4583, 1, 1, 1, 468371, 117437, 1877, 118127, 1, 118819, 2549, 119513, 5387, 120209, 482231, 120907, 1, 121607, 487831, 11119, 490643, 1, 493463, 123719, 496291, 124427, 499127, 1, 501971, 1, 45893, 1889, 46153, 7487, 510551, 127997, 513427, 128717, 1, 129439, 3109, 11833, 9851, 1, 1, 131617, 5227, 132347, 530851, 1823, 1847, 133813, 1, 1, 1, 2293, 3329, 136027, 5297, 136769, 1, 8089, 551543, 12569, 554531, 12637, 1553, 8221, 560531, 1979, 563543, 141263, 566563, 142019, 1, 2131, 52057, 143537, 33863, 144299, 1, 145063, 1667, 145829, 34403, 13327, 587927, 13397, 2801, 148139, 594103, 148913, 1, 149689, 600311, 1, 4987, 151247, 823, 152029, 609683, 1, 612823, 1, 615971, 154387, 5479, 14107, 5507, 1, 2617, 2657, 36979, 157559, 10709, 158357, 8699, 159157, 3413, 1, 58313, 1, 8831, 161569, 647891, 2287, 651127, 3079, 654371, 877, 1, 14983, 2633, 1861, 664151, 9791, 667427, 167267, 670711, 168089, 863, 168913, 1, 1, 680611, 1, 40231, 1697, 12967, 1, 690583, 15733, 40819, 15809, 697271, 174737, 700627, 1, 703991, 176419, 2099, 1721, 64613, 10477, 64921, 2671, 717527, 179807, 720931, 10627, 12277, 181513, 727763, 1, 731191, 16657, 734627, 184087, 738071, 184949, 1, 185813, 1, 186679, 68041, 187547, 4021, 188417, 2383, 2593, 8527, 1, 1, 191039, 2447, 1, 10837, 1, 1, 193679, 776483, 3671, 7723, 11497, 783571, 196337, 1, 1, 71881, 2957, 5059, 2803, 1801, 199909, 47143, 200807, 805027, 1, 937, 1, 47779, 1, 1, 1, 819491, 205327, 823127, 206237, 75161, 3511, 6863, 12239, 1, 3943, 1, 1, 841427, 12401, 845111, 19249, 848803, 19333, 1, 213589, 2333, 214517, 859927, 215447, 1, 216379, 78853, 217313, 79193, 218249, 971, 219187, 2087, 1, 882391, 221069, 886163, 20183, 1, 20269, 8677, 13171, 897527, 3167, 12347, 225809, 905143, 13339, 82633, 227719, 82981, 228677, 13681, 229637, 3323, 2591, 924323, 231563, 1, 21139, 932051, 21227, 17659, 234467, 1, 235439, 1, 4007, 947603, 1, 86501, 1, 86857, 239347, 959351, 1, 963283, 241313, 1, 2399, 971171, 1, 6211, 1, 1, 245269, 983063, 2767, 987043, 247259, 991031, 248257, 1, 249257, 90821, 250259, 1033, 251263, 59239, 1, 1011091, 2459, 1015127, 23117, 1, 23209, 2239, 256313, 9091, 15137, 9127, 3539, 1035427, 1, 1, 1, 94873, 3581, 1047703, 3917, 1051811, 1, 3331, 4483, 1060051, 1, 1061, 2203, 1068323, 3769, 1072471, 268637, 63331, 1, 1080791, 270719, 1, 1, 99013, 272809, 1093331, 273857, 16381, 1, 12379, 1, 1105943, 25183, 4127, 1, 6673, 279127, 2999, 280187, 1087, 281249, 1127123, 282313, 1, 1129,

6. Sequence of the polynom (only primes)

349, 53, 73, 17, 211, 89, 503, 163, 11, 239, 101, 317, 1427, 397, 103, 479, 2083, 563, 2423, 59, 67, 827, 3491, 919, 3863, 1013, 4243, 1109, 421, 71, 457, 1307, 5431, 1409, 5843, 6263, 1619, 6691, 157, 7127, 167, 113, 1949, 2063, 499, 2179, 8951, 2297, 857, 2417, 2539, 2663, 10903, 2789, 11411, 2917, 11927, 277, 12451, 12983, 3313, 13523, 3449, 14071, 14627, 3727, 1381, 1433, 4013, 4159, 16931, 1031, 4457, 18131, 419, 18743, 433, 4919, 19991, 5077, 20627, 5237, 5399, 1993, 5563, 2053, 337, 23251, 5897, 6067, 24611, 367, 25303, 26003, 599, 26711, 27427, 6947, 28151, 7129, 1699, 2693, 7499, 251, 7687, 1831, 7877, 31891, 8069, 8263, 631, 769, 34231, 787, 35027, 521, 35831, 9059, 36643, 37463, 557, 9677, 3557, 9887, 39971, 10099, 40823, 10313, 571, 10529, 2503, 977, 43427, 997, 607, 2659, 46103, 887, 11867, 4357, 12097, 4441, 12329, 49783, 739, 50723, 12799, 13037, 52627, 53591, 1229, 54563, 13763, 829, 14009, 56531, 269, 57527, 313, 14759, 5413, 15013, 853, 15269, 3623, 15527, 62627, 15787, 63671, 1459, 1097, 1483, 281, 66851, 991, 67927, 17117, 69011, 17389, 6373, 1039, 6473, 17939, 18217, 727, 74551, 75683, 1733, 4519, 1759, 757, 1181, 19927, 4723, 20219, 81463, 683, 20809, 7621, 21107, 85027, 21407, 1627, 1277, 87443, 22013, 88663, 2029, 89891, 91127, 22937, 1301, 347, 373, 23563, 23879, 8741, 24197, 24517, 98711, 100003, 25163, 359, 102611, 2347, 1009, 2377, 105251, 26479, 2011, 26813, 107923, 1597, 967, 27487, 27827, 10181, 1657, 113363, 28513, 114743, 28859, 116131, 29207, 2687, 118931, 2719, 7079, 121763, 123191, 30977, 7331, 31337, 31699, 11593, 32063, 128983, 32429, 1787, 32797, 131927, 1951, 1499, 3049, 431, 3083, 136403, 2017, 137911, 34667, 2081, 2389, 12953, 13093, 8563, 36587, 881, 148691, 37369, 8839, 3433, 3469, 2161, 38557, 155027, 156631, 39359, 158243, 2339, 14533, 40169, 40577, 163127, 2411, 164771, 41399, 1021, 41813, 168083, 169751, 3877, 171427, 43067, 2609, 43913, 176503, 439, 953, 1487, 45197, 487, 443, 183383, 185123, 46499, 186871, 3559, 190391, 47819, 3257, 193943, 195731, 49157, 17957, 18121, 2833, 50513, 11939, 50969, 204791, 51427, 206627, 12263, 4759, 210323, 52813, 212183, 53279, 214051, 215927, 54217, 19801, 3217, 19973, 55163, 221603, 55639, 3301, 225427, 56597, 2251, 5189, 229283, 5233, 231223, 58049, 541, 58537, 13831, 237091, 1123, 60013, 1289, 60509, 243031, 61007, 4153, 61507, 247031, 1051, 5683, 251063, 3467, 63527, 255127, 64037, 257171, 3797, 65063, 23753, 65579, 265427, 66617, 267511, 67139, 15859, 271703, 6199, 2711, 6247, 16231, 69247, 278051, 69779, 280183, 70313, 2741, 70849, 2351, 71387, 4231, 593, 72469, 290963, 73013, 5531, 4327, 6737, 3343, 617, 299731, 75209, 301943, 304163, 1303, 28057, 77999, 2791, 1493, 317651, 79697, 319927, 7297, 7349, 324503, 4789, 1163, 329111, 82567, 2111, 30341, 30553, 84313, 338423, 85487, 343127, 86077, 20323, 7879, 643, 7933, 4933, 20743, 1669, 355027, 89057, 357431, 89659, 32713, 90263, 32933, 364691, 5381, 367127, 1297, 2213, 92699, 372023, 374483, 8539, 6389, 94547, 7159, 1613, 381911, 95789, 384403, 1439, 2069, 97039, 35401, 98297, 23203, 98929, 396983, 99563, 399523, 9109, 402071, 1693, 101477, 407191, 6007, 691, 102763, 412343, 103409, 6121, 37957, 104707, 7927, 105359, 5791, 106013, 425363, 106669, 427991, 9817, 433271, 108649, 435923, 109313, 25799, 1549, 441251, 110647, 40357, 111317, 3691, 2113, 449303, 112663, 2879, 454711, 7753, 10427, 6481, 4583, 468371, 117437, 1877, 118127, 118819, 2549, 119513, 5387, 120209, 482231, 120907, 121607, 487831, 11119, 490643, 493463, 123719, 496291, 124427, 499127, 501971, 45893, 1889, 46153, 7487, 510551, 127997, 513427, 128717, 129439, 3109, 11833, 9851, 131617, 5227, 132347, 530851, 1823, 1847, 133813, 2293, 3329, 136027, 5297, 136769, 8089, 551543, 12569, 554531, 12637, 1553, 8221, 560531, 1979, 563543, 141263, 566563, 142019, 2131, 52057, 143537, 33863, 144299, 145063, 1667, 145829, 34403, 13327, 587927, 13397, 2801, 148139, 594103, 148913, 149689, 600311, 4987, 151247, 823, 152029, 609683, 612823, 615971, 154387, 5479, 14107, 5507, 2617, 2657, 36979, 157559, 10709, 158357, 8699, 159157, 3413, 58313, 8831, 161569, 647891, 2287, 651127, 3079, 654371, 877, 14983, 2633, 1861, 664151, 9791, 667427, 167267, 670711, 168089, 863, 168913, 680611, 40231, 1697, 12967, 690583, 15733, 40819, 15809, 697271, 174737, 700627, 703991, 176419, 2099, 1721, 64613, 10477, 64921, 2671, 717527, 179807, 720931, 10627, 12277, 181513, 727763, 731191, 16657, 734627, 184087, 738071, 184949, 185813, 186679, 68041, 187547, 4021, 188417, 2383, 2593, 8527, 191039, 2447, 10837, 193679, 776483, 3671, 7723, 11497, 783571, 196337, 71881, 2957, 5059, 2803, 1801, 199909, 47143, 200807, 805027, 937, 47779, 819491, 205327, 823127, 206237, 75161, 3511, 6863, 12239, 3943, 841427, 12401, 845111, 19249, 848803, 19333, 213589, 2333, 214517, 859927, 215447, 216379, 78853, 217313, 79193, 218249, 971, 219187, 2087, 882391, 221069, 886163, 20183, 20269, 8677, 13171, 897527, 3167, 12347, 225809, 905143, 13339, 82633, 227719, 82981, 228677, 13681, 229637, 3323, 2591, 924323, 231563, 21139, 932051, 21227, 17659, 234467, 235439, 4007, 947603, 86501, 86857, 239347, 959351, 963283, 241313, 2399, 971171, 6211, 245269, 983063, 2767, 987043, 247259, 991031, 248257, 249257, 90821, 250259, 1033, 251263, 59239, 1011091, 2459, 1015127, 23117, 23209, 2239, 256313, 9091, 15137, 9127, 3539, 1035427, 94873, 3581, 1047703, 3917, 1051811, 3331, 4483, 1060051, 1061, 2203, 1068323, 3769, 1072471, 268637, 63331, 1080791, 270719, 99013, 272809, 1093331, 273857, 16381, 12379, 1105943, 25183, 4127, 6673, 279127, 2999, 280187, 1087, 281249, 1127123, 282313, 1129,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+136x-349 and
the reducible primes which appear as divisor for the first time
p | x^2+136x-349 and p < x^2+136x-349

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 4 5 1.125 0.5 0.625
4 16 17 6 11 1.0625 0.375 0.6875
5 32 31 10 21 0.96875 0.3125 0.65625
6 64 59 20 39 0.921875 0.3125 0.609375
7 128 111 37 74 0.8671875 0.2890625 0.578125
8 256 221 68 153 0.86328125 0.265625 0.59765625
9 512 430 126 304 0.83984375 0.24609375 0.59375
10 1024 845 225 620 0.82519531 0.21972656 0.60546875
11 2048 1654 402 1252 0.80761719 0.19628906 0.61132813
12 4096 3246 738 2508 0.79248047 0.18017578 0.61230469
13 8192 6361 1366 4995 0.77648926 0.16674805 0.60974121
14 16384 12608 2491 10117 0.76953125 0.15203857 0.61749268
15 32768 25053 4594 20459 0.76455688 0.14019775 0.62435913
16 65536 49660 8623 41037 0.75775146 0.13157654 0.62617493
17 131072 98765 16131 82634 0.75351715 0.12306976 0.63044739
18 262144 196464 30266 166198 0.74945068 0.11545563 0.63399506
19 524288 390893 57055 333838 0.74556923 0.10882378 0.63674545
20 1048576 778762 107908 670854 0.74268532 0.10290909 0.63977623
21 2097152 1552359 204516 1347843 0.74022245 0.09752083 0.64270163
22 4194304 3094860 388889 2705971 0.73787212 0.09271836 0.64515376
23 8388608 6171691 741240 5430451 0.7357229 0.08836269 0.64736021
24 16777216 12310607 1415438 10895169 0.73376936 0.08436668 0.64940268


8. Check for existing Integer Sequences by OEIS

Found in Database : 349, 53, 73, 17, 211, 89, 503, 163, 11, 239, 101, 317, 1427, 397, 103, 479, 2083, 563, 2423, 59,
Found in Database : 349, 53, 73, 17, 211, 89, 503, 163, 11, 239, 101, 317, 1427, 397, 103, 479, 2083, 563, 2423, 59, 67, 827, 3491, 919, 3863, 1013, 4243, 1109, 421, 71, 457, 1307, 5431, 1409, 5843, 6263, 1619,
Found in Database : 11, 17, 53, 59, 67, 71, 73, 89, 101, 103, 113,