Inhaltsverzeichnis

Development of
Algorithmic Constructions

15:04:46
Deutsch
29.Mar 2024

Polynom = x^2+48x-311

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) = 311 = 311
f(1) = 131 = 131
f(2) = 211 = 211
f(3) = 79 = 79
f(4) = 103 = 103
f(5) = 23 = 23
f(6) = 13 = 13
f(7) = 37 = 37
f(8) = 137 = 137
f(9) = 101 = 101
f(10) = 269 = 269
f(11) = 169 = 13*13
f(12) = 409 = 409
f(13) = 241 = 241
f(14) = 557 = 557
f(15) = 317 = 317
f(16) = 713 = 23*31
f(17) = 397 = 397
f(18) = 877 = 877
f(19) = 481 = 13*37
f(20) = 1049 = 1049
f(21) = 569 = 569
f(22) = 1229 = 1229
f(23) = 661 = 661
f(24) = 1417 = 13*109
f(25) = 757 = 757
f(26) = 1613 = 1613
f(27) = 857 = 857
f(28) = 1817 = 23*79
f(29) = 961 = 31*31
f(30) = 2029 = 2029
f(31) = 1069 = 1069
f(32) = 2249 = 13*173
f(33) = 1181 = 1181
f(34) = 2477 = 2477
f(35) = 1297 = 1297
f(36) = 2713 = 2713
f(37) = 1417 = 13*109
f(38) = 2957 = 2957
f(39) = 1541 = 23*67
f(40) = 3209 = 3209
f(41) = 1669 = 1669
f(42) = 3469 = 3469
f(43) = 1801 = 1801
f(44) = 3737 = 37*101
f(45) = 1937 = 13*149
f(46) = 4013 = 4013
f(47) = 2077 = 31*67
f(48) = 4297 = 4297
f(49) = 2221 = 2221
f(50) = 4589 = 13*353
f(51) = 2369 = 23*103
f(52) = 4889 = 4889
f(53) = 2521 = 2521
f(54) = 5197 = 5197
f(55) = 2677 = 2677
f(56) = 5513 = 37*149
f(57) = 2837 = 2837
f(58) = 5837 = 13*449
f(59) = 3001 = 3001
f(60) = 6169 = 31*199
f(61) = 3169 = 3169
f(62) = 6509 = 23*283
f(63) = 3341 = 13*257
f(64) = 6857 = 6857
f(65) = 3517 = 3517
f(66) = 7213 = 7213
f(67) = 3697 = 3697
f(68) = 7577 = 7577
f(69) = 3881 = 3881
f(70) = 7949 = 7949
f(71) = 4069 = 13*313
f(72) = 8329 = 8329
f(73) = 4261 = 4261
f(74) = 8717 = 23*379
f(75) = 4457 = 4457
f(76) = 9113 = 13*701
f(77) = 4657 = 4657
f(78) = 9517 = 31*307
f(79) = 4861 = 4861
f(80) = 9929 = 9929
f(81) = 5069 = 37*137
f(82) = 10349 = 79*131
f(83) = 5281 = 5281
f(84) = 10777 = 13*829
f(85) = 5497 = 23*239
f(86) = 11213 = 11213
f(87) = 5717 = 5717
f(88) = 11657 = 11657
f(89) = 5941 = 13*457
f(90) = 12109 = 12109
f(91) = 6169 = 31*199
f(92) = 12569 = 12569
f(93) = 6401 = 37*173
f(94) = 13037 = 13037
f(95) = 6637 = 6637
f(96) = 13513 = 13513
f(97) = 6877 = 13*23*23
f(98) = 13997 = 13997
f(99) = 7121 = 7121
f(100) = 14489 = 14489

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+48x-311

f(0)=311
f(1)=131
f(2)=211
f(3)=79
f(4)=103
f(5)=23
f(6)=13
f(7)=37
f(8)=137
f(9)=101
f(10)=269
f(11)=1
f(12)=409
f(13)=241
f(14)=557
f(15)=317
f(16)=31
f(17)=397
f(18)=877
f(19)=1
f(20)=1049
f(21)=569
f(22)=1229
f(23)=661
f(24)=109
f(25)=757
f(26)=1613
f(27)=857
f(28)=1
f(29)=1
f(30)=2029
f(31)=1069
f(32)=173
f(33)=1181
f(34)=2477
f(35)=1297
f(36)=2713
f(37)=1
f(38)=2957
f(39)=67
f(40)=3209
f(41)=1669
f(42)=3469
f(43)=1801
f(44)=1
f(45)=149
f(46)=4013
f(47)=1
f(48)=4297
f(49)=2221
f(50)=353
f(51)=1
f(52)=4889
f(53)=2521
f(54)=5197
f(55)=2677
f(56)=1
f(57)=2837
f(58)=449
f(59)=3001
f(60)=199
f(61)=3169
f(62)=283
f(63)=257
f(64)=6857
f(65)=3517
f(66)=7213
f(67)=3697
f(68)=7577
f(69)=3881
f(70)=7949
f(71)=313
f(72)=8329
f(73)=4261
f(74)=379
f(75)=4457
f(76)=701
f(77)=4657
f(78)=307
f(79)=4861
f(80)=9929
f(81)=1
f(82)=1
f(83)=5281
f(84)=829
f(85)=239
f(86)=11213
f(87)=5717
f(88)=11657
f(89)=457
f(90)=12109
f(91)=1
f(92)=12569
f(93)=1
f(94)=13037
f(95)=6637
f(96)=13513
f(97)=1
f(98)=13997
f(99)=7121

b) Substitution of the polynom
The polynom f(x)=x^2+48x-311 could be written as f(y)= y^2-887 with x=y-24

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+24
f'(x)>2x+47

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

311, 131, 211, 79, 103, 23, 13, 37, 137, 101, 269, 1, 409, 241, 557, 317, 31, 397, 877, 1, 1049, 569, 1229, 661, 109, 757, 1613, 857, 1, 1, 2029, 1069, 173, 1181, 2477, 1297, 2713, 1, 2957, 67, 3209, 1669, 3469, 1801, 1, 149, 4013, 1, 4297, 2221, 353, 1, 4889, 2521, 5197, 2677, 1, 2837, 449, 3001, 199, 3169, 283, 257, 6857, 3517, 7213, 3697, 7577, 3881, 7949, 313, 8329, 4261, 379, 4457, 701, 4657, 307, 4861, 9929, 1, 1, 5281, 829, 239, 11213, 5717, 11657, 457, 12109, 1, 12569, 1, 13037, 6637, 13513, 1, 13997, 7121, 14489, 7369, 1153, 7621, 15497, 7877, 1, 1, 719, 271, 1, 8669, 17609, 8941, 1, 709, 18713, 9497, 521, 9781, 863, 10069, 659, 797, 21017, 10657, 21613, 10957, 1709, 11261, 617, 503, 179, 1, 24077, 12197, 1901, 12517, 25357, 12841, 839, 1013, 26669, 587, 27337, 1, 1, 14177, 28697, 1117, 29389, 14869, 30089, 491, 1, 421, 31513, 15937, 32237, 16301, 32969, 1, 2593, 17041, 34457, 17417, 1531, 1, 35977, 18181, 36749, 599, 37529, 1, 38317, 1489, 39113, 859, 223, 20161, 1, 1, 41549, 20981, 1367, 21397, 547, 21817, 3389, 967, 44909, 22669, 1237, 1777, 1, 23537, 47513, 23977, 48397, 24421, 2143, 1913, 1619, 25321, 1381, 1, 4001, 26237, 52937, 26701, 523, 1, 2383, 1, 4289, 907, 56713, 28597, 1, 2237, 263, 29569, 59629, 1307, 60617, 30557, 61613, 2389, 62617, 853, 63629, 32069, 4973, 1051, 65677, 1439, 66713, 33617, 67757, 34141, 1, 937, 641, 35201, 70937, 2749, 1, 36277, 1091, 36821, 74189, 37369, 75289, 2917, 1, 1, 77513, 1, 1, 1, 79769, 40169, 80909, 1, 2647, 1, 1, 41897, 84377, 1847, 85549, 3313, 86729, 43661, 87917, 44257, 89113, 1447, 2441, 1, 91529, 2003, 677, 46681, 7229, 47297, 95213, 47917, 96457, 48541, 1, 49169, 331, 49801, 100237, 1627, 101513, 3929, 102797, 51721, 104089, 52369, 105389, 1433, 4639, 4129, 108013, 811, 3527, 55001, 8513, 1, 1109, 1, 113357, 1, 114713, 57697, 8929, 739, 1, 59069, 359, 4597, 1, 60457, 3923, 2659, 123017, 61861, 124429, 4813, 1, 63281, 127277, 63997, 9901, 64717, 5659, 2111, 3557, 66169, 133069, 1, 1, 1, 136013, 1, 383, 1, 6043, 1, 3797, 70621, 953, 1, 143513, 1, 145037, 72901, 1423, 3203, 11393, 74441, 1373, 75217, 151213, 75997, 152777, 76781, 1, 77569, 1, 3407, 2351, 6089, 159113, 2161, 929, 80761, 1, 81569, 2447, 6337, 1, 1, 167213, 84017, 419, 2293, 170509, 85669, 172169, 86501, 1327, 87337, 1, 88177, 2243, 89021, 178889, 1, 431, 1, 182297, 91577, 184013, 4019, 185737, 7177, 187469, 94169, 439, 941, 1, 95917, 1871, 96797, 443, 1, 1, 1, 15233, 1259, 947, 100357, 5449, 7789, 203417, 102161, 8923, 103069, 6679, 103981, 208877, 8069, 210713, 105817, 212557, 1, 16493, 1607, 9403, 487, 218137, 109537, 220013, 110477, 1, 1663, 7219, 3037, 225689, 1, 227597, 114277, 1, 115237, 1163, 116201, 2141, 9013, 235309, 1, 237257, 5179, 18401, 120097, 241177, 121081, 243149, 122069, 245129, 1129, 19009, 1, 10831, 1583, 251117, 9697, 1, 4099, 255149, 1, 257177, 129097, 259213, 10009, 1, 1, 1471, 132169, 1, 133201, 8627, 1, 269513, 135277, 271597, 5927, 1, 137369, 3491, 1, 277897, 10729, 280013, 1, 4211, 141601, 284269, 6203, 9239, 11057, 2857, 144817, 4339, 145897, 1733, 1427, 295049, 563, 12923, 149161, 299417, 1, 23201, 151357, 303817, 152461, 306029, 11813, 308249, 154681, 13499, 155797, 1747, 4241, 314957, 12157, 317209, 159169, 319469, 5171, 24749, 7019, 1, 162577, 1213, 1621, 328589, 1, 25453, 166021, 333197, 167177, 1, 1, 337837, 169501, 340169, 170669, 9257, 1, 344857, 13309, 3371, 174197, 15199, 1609, 27073, 1, 354329, 177761, 1, 2671, 1, 1, 27809, 181361, 15823, 2311, 3361, 1, 1543, 184997, 371213, 6007, 373657, 187441, 1, 631, 378569, 189901, 381037, 191137, 29501, 192377, 385997, 5233, 388489, 194869, 390989, 8527, 30269, 6367, 3023, 198637, 1, 15377, 401069, 5437, 2333, 202441, 17659, 2017, 408713, 1, 13267, 206281, 413849, 207569, 1, 1, 1879, 210157, 1, 211457, 1499, 1553, 32833, 214069, 429449, 215381, 1, 1, 1, 9479, 437357, 1, 440009, 1481, 6607, 17077, 1, 223337, 448013, 224677, 1, 1, 1, 1, 3061, 691, 458797, 230077, 1, 1, 464237, 232801, 1, 18013, 1949, 235541, 472457, 236917, 2131, 7687, 477977, 1, 480749, 241069, 21023, 6553, 37409, 243857, 489113, 245257, 4513, 246661, 15959, 1, 38273, 10847, 1, 6781, 503213, 1493, 2543, 253741, 508909, 2341, 1667, 1873, 514637, 1, 517513, 1, 16787, 260921, 40253, 262369, 526189, 263821, 529097, 265277, 23131, 743, 41149, 4003, 14537, 8699, 540809, 20857, 6883, 1523, 5413, 4091, 23899, 275581, 552649, 21313, 15017, 278561, 3229, 2719, 43201, 281557, 564617, 1, 1, 1, 570649, 1, 44129, 287597, 2393, 2207, 579757, 1, 582809, 12703, 18899, 293701, 588937, 295237, 2251, 1, 595097, 1, 598189, 2969, 2011, 787, 604397, 302977, 1721, 304537, 1, 8273, 1, 307669, 616909, 309241, 26959, 23909, 4549, 312397, 626377, 313981, 1, 315569, 6143, 1, 635917, 13859, 9539, 1, 49409, 321961, 645529, 1, 1, 325181, 1, 1889, 3877, 1, 1, 330041, 4441, 823, 1, 4219, 3167, 1, 18149, 336577, 29339, 26017, 6221, 2281, 2851, 341521, 1699, 343177, 688013, 1, 691337, 346501, 30203, 1, 53693, 349841, 701357, 351517, 704713, 1, 8963, 354881, 2273, 1, 23059, 1, 1, 27689, 721613, 1, 7039, 363361, 1, 365069, 1931, 1, 2861, 11887, 738713, 370217, 57089, 371941, 745609, 373669, 749069, 1, 32719, 377137, 3583, 1, 4243, 380621, 762989, 1, 7589, 12391, 769997, 385877, 1, 883, 21001, 389401, 1, 1, 784109, 887, 60589, 394717, 25523, 17239, 6067, 30637, 21577, 400069, 10151, 401861, 805517, 3919, 1, 31189, 812717, 17707, 816329, 409069, 63073, 1, 1, 1, 827213, 2083, 2621, 1867, 2791, 418169, 838169, 420001, 4703, 1, 7757, 1, 849197, 425521, 852889, 427369, 37243, 1, 860297, 431077, 864013, 11701, 66749, 3989, 13007, 1, 875209, 1, 878957, 14207, 67901, 1, 1, 444181, 1, 34313, 3299, 447961, 897817, 19559, 901613, 1, 29207, 34897, 1, 4423, 24677, 1, 70529, 459397, 40031, 461317, 924557, 3109, 928409, 465169, 71713, 467101, 25301, 469037, 30323, 36229, 41039, 472921, 6361, 6011, 951689, 4721, 2887, 2833, 2137, 480737, 5569, 1, 74413, 1, 971309, 486641, 975257, 488617, 3089, 3581, 75629, 13313, 2027, 21503, 991129, 38197, 995117, 498557, 12647, 1, 9739, 1, 1007129, 1, 43963, 1, 3067, 7591, 78401, 510617, 1023257, 512641, 1, 514669, 1, 516701, 3463, 518737, 1039513, 1, 1043597, 1,

6. Sequence of the polynom (only primes)

311, 131, 211, 79, 103, 23, 13, 37, 137, 101, 269, 409, 241, 557, 317, 31, 397, 877, 1049, 569, 1229, 661, 109, 757, 1613, 857, 2029, 1069, 173, 1181, 2477, 1297, 2713, 2957, 67, 3209, 1669, 3469, 1801, 149, 4013, 4297, 2221, 353, 4889, 2521, 5197, 2677, 2837, 449, 3001, 199, 3169, 283, 257, 6857, 3517, 7213, 3697, 7577, 3881, 7949, 313, 8329, 4261, 379, 4457, 701, 4657, 307, 4861, 9929, 5281, 829, 239, 11213, 5717, 11657, 457, 12109, 12569, 13037, 6637, 13513, 13997, 7121, 14489, 7369, 1153, 7621, 15497, 7877, 719, 271, 8669, 17609, 8941, 709, 18713, 9497, 521, 9781, 863, 10069, 659, 797, 21017, 10657, 21613, 10957, 1709, 11261, 617, 503, 179, 24077, 12197, 1901, 12517, 25357, 12841, 839, 1013, 26669, 587, 27337, 14177, 28697, 1117, 29389, 14869, 30089, 491, 421, 31513, 15937, 32237, 16301, 32969, 2593, 17041, 34457, 17417, 1531, 35977, 18181, 36749, 599, 37529, 38317, 1489, 39113, 859, 223, 20161, 41549, 20981, 1367, 21397, 547, 21817, 3389, 967, 44909, 22669, 1237, 1777, 23537, 47513, 23977, 48397, 24421, 2143, 1913, 1619, 25321, 1381, 4001, 26237, 52937, 26701, 523, 2383, 4289, 907, 56713, 28597, 2237, 263, 29569, 59629, 1307, 60617, 30557, 61613, 2389, 62617, 853, 63629, 32069, 4973, 1051, 65677, 1439, 66713, 33617, 67757, 34141, 937, 641, 35201, 70937, 2749, 36277, 1091, 36821, 74189, 37369, 75289, 2917, 77513, 79769, 40169, 80909, 2647, 41897, 84377, 1847, 85549, 3313, 86729, 43661, 87917, 44257, 89113, 1447, 2441, 91529, 2003, 677, 46681, 7229, 47297, 95213, 47917, 96457, 48541, 49169, 331, 49801, 100237, 1627, 101513, 3929, 102797, 51721, 104089, 52369, 105389, 1433, 4639, 4129, 108013, 811, 3527, 55001, 8513, 1109, 113357, 114713, 57697, 8929, 739, 59069, 359, 4597, 60457, 3923, 2659, 123017, 61861, 124429, 4813, 63281, 127277, 63997, 9901, 64717, 5659, 2111, 3557, 66169, 133069, 136013, 383, 6043, 3797, 70621, 953, 143513, 145037, 72901, 1423, 3203, 11393, 74441, 1373, 75217, 151213, 75997, 152777, 76781, 77569, 3407, 2351, 6089, 159113, 2161, 929, 80761, 81569, 2447, 6337, 167213, 84017, 419, 2293, 170509, 85669, 172169, 86501, 1327, 87337, 88177, 2243, 89021, 178889, 431, 182297, 91577, 184013, 4019, 185737, 7177, 187469, 94169, 439, 941, 95917, 1871, 96797, 443, 15233, 1259, 947, 100357, 5449, 7789, 203417, 102161, 8923, 103069, 6679, 103981, 208877, 8069, 210713, 105817, 212557, 16493, 1607, 9403, 487, 218137, 109537, 220013, 110477, 1663, 7219, 3037, 225689, 227597, 114277, 115237, 1163, 116201, 2141, 9013, 235309, 237257, 5179, 18401, 120097, 241177, 121081, 243149, 122069, 245129, 1129, 19009, 10831, 1583, 251117, 9697, 4099, 255149, 257177, 129097, 259213, 10009, 1471, 132169, 133201, 8627, 269513, 135277, 271597, 5927, 137369, 3491, 277897, 10729, 280013, 4211, 141601, 284269, 6203, 9239, 11057, 2857, 144817, 4339, 145897, 1733, 1427, 295049, 563, 12923, 149161, 299417, 23201, 151357, 303817, 152461, 306029, 11813, 308249, 154681, 13499, 155797, 1747, 4241, 314957, 12157, 317209, 159169, 319469, 5171, 24749, 7019, 162577, 1213, 1621, 328589, 25453, 166021, 333197, 167177, 337837, 169501, 340169, 170669, 9257, 344857, 13309, 3371, 174197, 15199, 1609, 27073, 354329, 177761, 2671, 27809, 181361, 15823, 2311, 3361, 1543, 184997, 371213, 6007, 373657, 187441, 631, 378569, 189901, 381037, 191137, 29501, 192377, 385997, 5233, 388489, 194869, 390989, 8527, 30269, 6367, 3023, 198637, 15377, 401069, 5437, 2333, 202441, 17659, 2017, 408713, 13267, 206281, 413849, 207569, 1879, 210157, 211457, 1499, 1553, 32833, 214069, 429449, 215381, 9479, 437357, 440009, 1481, 6607, 17077, 223337, 448013, 224677, 3061, 691, 458797, 230077, 464237, 232801, 18013, 1949, 235541, 472457, 236917, 2131, 7687, 477977, 480749, 241069, 21023, 6553, 37409, 243857, 489113, 245257, 4513, 246661, 15959, 38273, 10847, 6781, 503213, 1493, 2543, 253741, 508909, 2341, 1667, 1873, 514637, 517513, 16787, 260921, 40253, 262369, 526189, 263821, 529097, 265277, 23131, 743, 41149, 4003, 14537, 8699, 540809, 20857, 6883, 1523, 5413, 4091, 23899, 275581, 552649, 21313, 15017, 278561, 3229, 2719, 43201, 281557, 564617, 570649, 44129, 287597, 2393, 2207, 579757, 582809, 12703, 18899, 293701, 588937, 295237, 2251, 595097, 598189, 2969, 2011, 787, 604397, 302977, 1721, 304537, 8273, 307669, 616909, 309241, 26959, 23909, 4549, 312397, 626377, 313981, 315569, 6143, 635917, 13859, 9539, 49409, 321961, 645529, 325181, 1889, 3877, 330041, 4441, 823, 4219, 3167, 18149, 336577, 29339, 26017, 6221, 2281, 2851, 341521, 1699, 343177, 688013, 691337, 346501, 30203, 53693, 349841, 701357, 351517, 704713, 8963, 354881, 2273, 23059, 27689, 721613, 7039, 363361, 365069, 1931, 2861, 11887, 738713, 370217, 57089, 371941, 745609, 373669, 749069, 32719, 377137, 3583, 4243, 380621, 762989, 7589, 12391, 769997, 385877, 883, 21001, 389401, 784109, 887, 60589, 394717, 25523, 17239, 6067, 30637, 21577, 400069, 10151, 401861, 805517, 3919, 31189, 812717, 17707, 816329, 409069, 63073, 827213, 2083, 2621, 1867, 2791, 418169, 838169, 420001, 4703, 7757, 849197, 425521, 852889, 427369, 37243, 860297, 431077, 864013, 11701, 66749, 3989, 13007, 875209, 878957, 14207, 67901, 444181, 34313, 3299, 447961, 897817, 19559, 901613, 29207, 34897, 4423, 24677, 70529, 459397, 40031, 461317, 924557, 3109, 928409, 465169, 71713, 467101, 25301, 469037, 30323, 36229, 41039, 472921, 6361, 6011, 951689, 4721, 2887, 2833, 2137, 480737, 5569, 74413, 971309, 486641, 975257, 488617, 3089, 3581, 75629, 13313, 2027, 21503, 991129, 38197, 995117, 498557, 12647, 9739, 1007129, 43963, 3067, 7591, 78401, 510617, 1023257, 512641, 514669, 516701, 3463, 518737, 1039513, 1043597,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+48x-311 and
the reducible primes which appear as divisor for the first time
p | x^2+48x-311 and p < x^2+48x-311

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 16 8 8 1 0.5 0.5
5 32 29 13 16 0.90625 0.40625 0.5
6 64 56 23 33 0.875 0.359375 0.515625
7 128 111 41 70 0.8671875 0.3203125 0.546875
8 256 221 75 146 0.86328125 0.29296875 0.5703125
9 512 428 133 295 0.8359375 0.25976563 0.57617188
10 1024 841 227 614 0.82128906 0.22167969 0.59960938
11 2048 1650 407 1243 0.80566406 0.19873047 0.60693359
12 4096 3239 757 2482 0.79077148 0.18481445 0.60595703
13 8192 6425 1377 5048 0.78430176 0.16809082 0.61621094
14 16384 12695 2541 10154 0.77484131 0.15509033 0.61975098
15 32768 25150 4704 20446 0.76751709 0.14355469 0.6239624
16 65536 49922 8715 41207 0.76174927 0.13298035 0.62876892
17 131072 99333 16097 83236 0.75785065 0.12281036 0.63504028
18 262144 197523 30224 167299 0.75349045 0.11529541 0.63819504
19 524288 393490 56726 336764 0.75052261 0.10819626 0.64232635
20 1048576 783568 106900 676668 0.74726868 0.10194778 0.64532089
21 2097152 1561326 202936 1358390 0.74449825 0.09676743 0.64773083
22 4194304 3111389 386432 2724957 0.74181294 0.09213257 0.64968038
23 8388608 6204663 736337 5468326 0.73965347 0.08777821 0.65187526
24 16777216 12375139 1405572 10969567 0.73761576 0.08377862 0.65383714


8. Check for existing Integer Sequences by OEIS

Found in Database : 311, 131, 211, 79, 103, 23, 13, 37, 137, 101, 269, 1, 409, 241, 557, 317, 31, 397, 877, 1,
Found in Database : 311, 131, 211, 79, 103, 23, 13, 37, 137, 101, 269, 409, 241, 557, 317, 31, 397, 877, 1049, 569, 1229, 661, 109, 757, 1613, 857, 2029, 1069, 173, 1181, 2477, 1297, 2713, 2957, 67,
Found in Database : 13, 23, 31, 37, 67, 79, 101, 103, 109, 131, 137, 149,