Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:05:26
Deutsch
29.Mar 2024

Polynom = x^2+1423

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) = 1423 = 1423
f(1) = 89 = 89
f(2) = 1427 = 1427
f(3) = 179 = 179
f(4) = 1439 = 1439
f(5) = 181 = 181
f(6) = 1459 = 1459
f(7) = 23 = 23
f(8) = 1487 = 1487
f(9) = 47 = 47
f(10) = 1523 = 1523
f(11) = 193 = 193
f(12) = 1567 = 1567
f(13) = 199 = 199
f(14) = 1619 = 1619
f(15) = 103 = 103
f(16) = 1679 = 23*73
f(17) = 107 = 107
f(18) = 1747 = 1747
f(19) = 223 = 223
f(20) = 1823 = 1823
f(21) = 233 = 233
f(22) = 1907 = 1907
f(23) = 61 = 61
f(24) = 1999 = 1999
f(25) = 1 = 1
f(26) = 2099 = 2099
f(27) = 269 = 269
f(28) = 2207 = 2207
f(29) = 283 = 283
f(30) = 2323 = 23*101
f(31) = 149 = 149
f(32) = 2447 = 2447
f(33) = 157 = 157
f(34) = 2579 = 2579
f(35) = 331 = 331
f(36) = 2719 = 2719
f(37) = 349 = 349
f(38) = 2867 = 47*61
f(39) = 23 = 23
f(40) = 3023 = 3023
f(41) = 97 = 97
f(42) = 3187 = 3187
f(43) = 409 = 409
f(44) = 3359 = 3359
f(45) = 431 = 431
f(46) = 3539 = 3539
f(47) = 227 = 227
f(48) = 3727 = 3727
f(49) = 239 = 239
f(50) = 3923 = 3923
f(51) = 503 = 503
f(52) = 4127 = 4127
f(53) = 529 = 23*23
f(54) = 4339 = 4339
f(55) = 139 = 139
f(56) = 4559 = 47*97
f(57) = 73 = 73
f(58) = 4787 = 4787
f(59) = 613 = 613
f(60) = 5023 = 5023
f(61) = 643 = 643
f(62) = 5267 = 23*229
f(63) = 337 = 337
f(64) = 5519 = 5519
f(65) = 353 = 353
f(66) = 5779 = 5779
f(67) = 739 = 739
f(68) = 6047 = 6047
f(69) = 773 = 773
f(70) = 6323 = 6323
f(71) = 101 = 101
f(72) = 6607 = 6607
f(73) = 211 = 211
f(74) = 6899 = 6899
f(75) = 881 = 881
f(76) = 7199 = 23*313
f(77) = 919 = 919
f(78) = 7507 = 7507
f(79) = 479 = 479
f(80) = 7823 = 7823
f(81) = 499 = 499
f(82) = 8147 = 8147
f(83) = 1039 = 1039
f(84) = 8479 = 61*139
f(85) = 1081 = 23*47
f(86) = 8819 = 8819
f(87) = 281 = 281
f(88) = 9167 = 89*103
f(89) = 73 = 73
f(90) = 9523 = 89*107
f(91) = 1213 = 1213
f(92) = 9887 = 9887
f(93) = 1259 = 1259
f(94) = 10259 = 10259
f(95) = 653 = 653
f(96) = 10639 = 10639
f(97) = 677 = 677
f(98) = 11027 = 11027
f(99) = 1403 = 23*61
f(100) = 11423 = 11423

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+1423

f(0)=1423
f(1)=89
f(2)=1427
f(3)=179
f(4)=1439
f(5)=181
f(6)=1459
f(7)=23
f(8)=1487
f(9)=47
f(10)=1523
f(11)=193
f(12)=1567
f(13)=199
f(14)=1619
f(15)=103
f(16)=73
f(17)=107
f(18)=1747
f(19)=223
f(20)=1823
f(21)=233
f(22)=1907
f(23)=61
f(24)=1999
f(25)=1
f(26)=2099
f(27)=269
f(28)=2207
f(29)=283
f(30)=101
f(31)=149
f(32)=2447
f(33)=157
f(34)=2579
f(35)=331
f(36)=2719
f(37)=349
f(38)=1
f(39)=1
f(40)=3023
f(41)=97
f(42)=3187
f(43)=409
f(44)=3359
f(45)=431
f(46)=3539
f(47)=227
f(48)=3727
f(49)=239
f(50)=3923
f(51)=503
f(52)=4127
f(53)=1
f(54)=4339
f(55)=139
f(56)=1
f(57)=1
f(58)=4787
f(59)=613
f(60)=5023
f(61)=643
f(62)=229
f(63)=337
f(64)=5519
f(65)=353
f(66)=5779
f(67)=739
f(68)=6047
f(69)=773
f(70)=6323
f(71)=1
f(72)=6607
f(73)=211
f(74)=6899
f(75)=881
f(76)=313
f(77)=919
f(78)=7507
f(79)=479
f(80)=7823
f(81)=499
f(82)=8147
f(83)=1039
f(84)=1
f(85)=1
f(86)=8819
f(87)=281
f(88)=1
f(89)=1
f(90)=1
f(91)=1213
f(92)=9887
f(93)=1259
f(94)=10259
f(95)=653
f(96)=10639
f(97)=677
f(98)=11027
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+1423 could be written as f(y)= y^2+1423 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 > 38

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

1423, 89, 1427, 179, 1439, 181, 1459, 23, 1487, 47, 1523, 193, 1567, 199, 1619, 103, 73, 107, 1747, 223, 1823, 233, 1907, 61, 1999, 1, 2099, 269, 2207, 283, 101, 149, 2447, 157, 2579, 331, 2719, 349, 1, 1, 3023, 97, 3187, 409, 3359, 431, 3539, 227, 3727, 239, 3923, 503, 4127, 1, 4339, 139, 1, 1, 4787, 613, 5023, 643, 229, 337, 5519, 353, 5779, 739, 6047, 773, 6323, 1, 6607, 211, 6899, 881, 313, 919, 7507, 479, 7823, 499, 8147, 1039, 1, 1, 8819, 281, 1, 1, 1, 1213, 9887, 1259, 10259, 653, 10639, 677, 11027, 1, 11423, 1453, 11827, 1, 12239, 389, 12659, 1609, 569, 1663, 13523, 859, 13967, 887, 14419, 1831, 14879, 1889, 1, 487, 15823, 251, 709, 2069, 1, 2131, 17299, 1097, 17807, 1129, 1, 1, 401, 2389, 19379, 307, 19919, 631, 1, 2593, 21023, 2663, 21587, 1367, 22159, 1, 22739, 2879, 23327, 2953, 509, 757, 24527, 1, 1093, 3181, 25759, 3259, 26387, 1669, 443, 1709, 379, 3499, 28319, 3581, 28979, 1, 1289, 937, 30323, 3833, 1, 3919, 31699, 2003, 1, 1, 33107, 1, 1, 4273, 1, 1091, 35279, 557, 1, 4549, 36767, 4643, 1, 1, 38287, 2417, 1, 4931, 39839, 1, 40627, 641, 1801, 1307, 42227, 1, 1, 5431, 719, 2767, 44687, 2819, 45523, 5743, 1, 5849, 2053, 1489, 48079, 1, 48947, 6173, 49823, 1, 50707, 1, 51599, 3253, 1117, 6619, 53407, 6733, 54323, 1, 547, 1741, 56179, 1, 57119, 1, 58067, 3659, 59023, 3719, 1, 7559, 1297, 7681, 2693, 1951, 62927, 991, 659, 8053, 64927, 8179, 1, 4153, 66959, 4217, 67987, 8563, 3001, 8693, 70067, 1103, 71119, 2239, 811, 1, 823, 1, 74323, 4679, 75407, 1, 1, 9631, 1063, 9769, 78707, 2477, 79823, 1, 1327, 1, 1, 10331, 83219, 5237, 1, 5309, 85523, 1, 3769, 10909, 853, 691, 1, 2801, 90227, 11353, 91423, 11503, 92627, 5827, 877, 5903, 4133, 11959, 1319, 12113, 97523, 3067, 1, 1553, 100019, 1, 101279, 12739, 102547, 6449, 1, 6529, 105107, 13219, 1033, 13381, 107699, 1693, 1787, 1, 110323, 13873, 1151, 1, 112979, 7103, 114319, 7187, 1, 14543, 117023, 14713, 118387, 1, 119759, 941, 121139, 1, 122527, 1, 123923, 7789, 5449, 7877, 126739, 1, 128159, 1, 129587, 1, 131023, 1, 953, 16649, 133919, 16831, 1, 1, 1, 8599, 138323, 17383, 1, 17569, 1399, 1, 142799, 2243, 144307, 18133, 145823, 1, 147347, 9257, 6473, 1, 1, 18899, 151967, 1, 153523, 2411, 155087, 4871, 156659, 19681, 1, 1, 6949, 10039, 1, 10139, 163027, 20479, 2699, 20681, 166259, 1, 167887, 1, 169523, 1, 171167, 21499, 3677, 10853, 1171, 10957, 1, 22123, 177823, 971, 2459, 1409, 181199, 5689, 182899, 1, 184607, 1, 8101, 11699, 4001, 11807, 1879, 23831, 191519, 24049, 1231, 6067, 195023, 3061, 1, 24709, 1, 1, 2251, 12577, 202127, 12689, 3343, 25603, 1, 1123, 2843, 3257, 209359, 6571, 211187, 26513, 213023, 1, 214867, 13487, 216719, 1, 218579, 1193, 220447, 27673, 222323, 6977, 1613, 1759, 226099, 1, 1, 28619, 1543, 1, 231823, 14549, 233747, 29339, 235679, 29581, 237619, 1, 239567, 7517, 10501, 30313, 243487, 30559, 1, 1, 247439, 15527, 249427, 1361, 2441, 1, 253427, 7951, 1627, 4007, 257459, 32309, 5521, 32563, 261523, 1, 263567, 1, 265619, 33331, 267679, 33589, 2521, 4231, 1187, 8527, 11909, 34369, 275999, 34631, 1, 1, 280207, 17579, 2741, 35423, 284447, 1, 1601, 1, 12553, 1, 1607, 36493, 1637, 1, 295187, 18517, 297359, 1, 299539, 37579, 1667, 37853, 1, 2383, 2861, 9601, 1, 1, 310559, 38959, 1, 1, 1493, 19759, 317267, 39799, 319519, 1, 321779, 10091, 1, 5081, 326323, 40933, 5387, 1, 330899, 20753, 3299, 20897, 335507, 42083, 1, 42373, 1, 5333, 1721, 10739, 344819, 1, 347167, 43543, 1811, 1, 351887, 22067, 354259, 1, 1, 44729, 359027, 11257, 4951, 2833, 7741, 1, 366239, 1997, 1, 23117, 371087, 23269, 1877, 1, 375967, 47149, 16453, 1483, 380879, 11941, 383347, 48073, 8209, 48383, 4363, 1, 4391, 1, 393299, 1, 17209, 49633, 398323, 12487, 1597, 1, 1777, 50581, 4019, 2213, 4211, 25609, 2957, 1, 413587, 51859, 416159, 52181, 418739, 6563, 6907, 1, 1901, 2311, 426527, 53479, 429139, 26903, 431759, 27067, 434387, 1, 19001, 1, 439667, 13781, 442319, 1733, 444979, 1, 1873, 56123, 1, 28229, 1, 1, 19813, 57131, 458399, 1, 1979, 3613, 463823, 14537, 466547, 2543, 469279, 58831, 472019, 29587, 2129, 29759, 477523, 59863, 480287, 60209, 7919, 15139, 10337, 1, 488627, 61253, 491423, 61603, 1579, 30977, 1, 31153, 1, 62659, 502687, 1, 505523, 1, 508367, 1, 1, 64081, 514079, 64439, 516947, 1, 1, 32579, 522707, 65519, 525599, 65881, 1, 16561, 2341, 1, 534323, 66973, 5021, 67339, 540179, 1, 5273, 1, 546067, 68443, 549023, 68813, 1, 1, 554959, 17389, 7643, 69929, 1, 1, 4057, 35339, 1, 35527, 569939, 1, 2039, 71809, 575987, 18047, 579023, 1, 582067, 72949, 585119, 73331, 1, 36857, 1, 37049, 9743, 1, 597407, 74869, 1, 1, 4051, 18911, 6007, 1, 2267, 76423, 612947, 1, 616079, 38603, 2017, 1, 622367, 3391, 13309, 19597, 628687, 1, 631859, 79181, 635039, 79579, 27749, 39989, 7207, 40189, 7243, 80779, 647839, 81181, 1, 5099, 13921, 1, 657523, 82393, 28729, 82799, 664019, 41603, 10939, 1, 4271, 1, 2381, 3671, 677107, 21211, 680399, 10657, 683699, 85669, 687007, 86083, 690323, 1, 7151, 43457, 2591, 3797, 700319, 1867, 6967, 1, 1, 22147, 1, 89009, 31033, 89431, 717139, 44927, 720527, 45139, 723923, 90703, 727327, 91129, 3191, 1, 734159, 5749, 32069, 92413, 10151, 1, 1, 1, 747919, 46853, 751379, 4093, 1, 94573, 758323, 2969, 761807, 1, 3049, 95881, 768799, 1, 2729, 1, 775823, 2113, 779347, 1, 16657, 98081, 786419, 24631, 789967, 1, 34501, 1, 1, 99859, 800659, 50153, 804239, 50377, 4513, 101203, 4483, 101653, 17341, 12763, 35593, 25639, 822259, 1, 1, 1, 4583, 1, 8089, 2269, 8627, 104831, 840479, 1, 6073, 26437, 2237, 3319, 851507, 106669, 855199, 1, 858899, 2339, 862607, 1, 3083, 2309, 870047, 1, 2179, 6841, 38153, 27481, 14447, 1, 885023, 110863, 8629, 55667, 892559, 55903, 8377, 112279, 2671, 2399, 39301, 28307, 907727, 1, 4723, 1, 1, 114659, 2777, 2503, 923023, 57809, 926867, 116099, 930719, 1, 934579, 14633, 938447, 29387, 1, 118033, 946207, 5153, 950099, 1, 4943, 59747, 1, 119983, 1, 120473, 1, 30241, 969679, 7591, 1, 121949, 977567, 122443, 981523, 61469, 985487, 61717, 2803, 123931, 1, 124429, 997427, 1,

6. Sequence of the polynom (only primes)

1423, 89, 1427, 179, 1439, 181, 1459, 23, 1487, 47, 1523, 193, 1567, 199, 1619, 103, 73, 107, 1747, 223, 1823, 233, 1907, 61, 1999, 2099, 269, 2207, 283, 101, 149, 2447, 157, 2579, 331, 2719, 349, 3023, 97, 3187, 409, 3359, 431, 3539, 227, 3727, 239, 3923, 503, 4127, 4339, 139, 4787, 613, 5023, 643, 229, 337, 5519, 353, 5779, 739, 6047, 773, 6323, 6607, 211, 6899, 881, 313, 919, 7507, 479, 7823, 499, 8147, 1039, 8819, 281, 1213, 9887, 1259, 10259, 653, 10639, 677, 11027, 11423, 1453, 11827, 12239, 389, 12659, 1609, 569, 1663, 13523, 859, 13967, 887, 14419, 1831, 14879, 1889, 487, 15823, 251, 709, 2069, 2131, 17299, 1097, 17807, 1129, 401, 2389, 19379, 307, 19919, 631, 2593, 21023, 2663, 21587, 1367, 22159, 22739, 2879, 23327, 2953, 509, 757, 24527, 1093, 3181, 25759, 3259, 26387, 1669, 443, 1709, 379, 3499, 28319, 3581, 28979, 1289, 937, 30323, 3833, 3919, 31699, 2003, 33107, 4273, 1091, 35279, 557, 4549, 36767, 4643, 38287, 2417, 4931, 39839, 40627, 641, 1801, 1307, 42227, 5431, 719, 2767, 44687, 2819, 45523, 5743, 5849, 2053, 1489, 48079, 48947, 6173, 49823, 50707, 51599, 3253, 1117, 6619, 53407, 6733, 54323, 547, 1741, 56179, 57119, 58067, 3659, 59023, 3719, 7559, 1297, 7681, 2693, 1951, 62927, 991, 659, 8053, 64927, 8179, 4153, 66959, 4217, 67987, 8563, 3001, 8693, 70067, 1103, 71119, 2239, 811, 823, 74323, 4679, 75407, 9631, 1063, 9769, 78707, 2477, 79823, 1327, 10331, 83219, 5237, 5309, 85523, 3769, 10909, 853, 691, 2801, 90227, 11353, 91423, 11503, 92627, 5827, 877, 5903, 4133, 11959, 1319, 12113, 97523, 3067, 1553, 100019, 101279, 12739, 102547, 6449, 6529, 105107, 13219, 1033, 13381, 107699, 1693, 1787, 110323, 13873, 1151, 112979, 7103, 114319, 7187, 14543, 117023, 14713, 118387, 119759, 941, 121139, 122527, 123923, 7789, 5449, 7877, 126739, 128159, 129587, 131023, 953, 16649, 133919, 16831, 8599, 138323, 17383, 17569, 1399, 142799, 2243, 144307, 18133, 145823, 147347, 9257, 6473, 18899, 151967, 153523, 2411, 155087, 4871, 156659, 19681, 6949, 10039, 10139, 163027, 20479, 2699, 20681, 166259, 167887, 169523, 171167, 21499, 3677, 10853, 1171, 10957, 22123, 177823, 971, 2459, 1409, 181199, 5689, 182899, 184607, 8101, 11699, 4001, 11807, 1879, 23831, 191519, 24049, 1231, 6067, 195023, 3061, 24709, 2251, 12577, 202127, 12689, 3343, 25603, 1123, 2843, 3257, 209359, 6571, 211187, 26513, 213023, 214867, 13487, 216719, 218579, 1193, 220447, 27673, 222323, 6977, 1613, 1759, 226099, 28619, 1543, 231823, 14549, 233747, 29339, 235679, 29581, 237619, 239567, 7517, 10501, 30313, 243487, 30559, 247439, 15527, 249427, 1361, 2441, 253427, 7951, 1627, 4007, 257459, 32309, 5521, 32563, 261523, 263567, 265619, 33331, 267679, 33589, 2521, 4231, 1187, 8527, 11909, 34369, 275999, 34631, 280207, 17579, 2741, 35423, 284447, 1601, 12553, 1607, 36493, 1637, 295187, 18517, 297359, 299539, 37579, 1667, 37853, 2383, 2861, 9601, 310559, 38959, 1493, 19759, 317267, 39799, 319519, 321779, 10091, 5081, 326323, 40933, 5387, 330899, 20753, 3299, 20897, 335507, 42083, 42373, 5333, 1721, 10739, 344819, 347167, 43543, 1811, 351887, 22067, 354259, 44729, 359027, 11257, 4951, 2833, 7741, 366239, 1997, 23117, 371087, 23269, 1877, 375967, 47149, 16453, 1483, 380879, 11941, 383347, 48073, 8209, 48383, 4363, 4391, 393299, 17209, 49633, 398323, 12487, 1597, 1777, 50581, 4019, 2213, 4211, 25609, 2957, 413587, 51859, 416159, 52181, 418739, 6563, 6907, 1901, 2311, 426527, 53479, 429139, 26903, 431759, 27067, 434387, 19001, 439667, 13781, 442319, 1733, 444979, 1873, 56123, 28229, 19813, 57131, 458399, 1979, 3613, 463823, 14537, 466547, 2543, 469279, 58831, 472019, 29587, 2129, 29759, 477523, 59863, 480287, 60209, 7919, 15139, 10337, 488627, 61253, 491423, 61603, 1579, 30977, 31153, 62659, 502687, 505523, 508367, 64081, 514079, 64439, 516947, 32579, 522707, 65519, 525599, 65881, 16561, 2341, 534323, 66973, 5021, 67339, 540179, 5273, 546067, 68443, 549023, 68813, 554959, 17389, 7643, 69929, 4057, 35339, 35527, 569939, 2039, 71809, 575987, 18047, 579023, 582067, 72949, 585119, 73331, 36857, 37049, 9743, 597407, 74869, 4051, 18911, 6007, 2267, 76423, 612947, 616079, 38603, 2017, 622367, 3391, 13309, 19597, 628687, 631859, 79181, 635039, 79579, 27749, 39989, 7207, 40189, 7243, 80779, 647839, 81181, 5099, 13921, 657523, 82393, 28729, 82799, 664019, 41603, 10939, 4271, 2381, 3671, 677107, 21211, 680399, 10657, 683699, 85669, 687007, 86083, 690323, 7151, 43457, 2591, 3797, 700319, 1867, 6967, 22147, 89009, 31033, 89431, 717139, 44927, 720527, 45139, 723923, 90703, 727327, 91129, 3191, 734159, 5749, 32069, 92413, 10151, 747919, 46853, 751379, 4093, 94573, 758323, 2969, 761807, 3049, 95881, 768799, 2729, 775823, 2113, 779347, 16657, 98081, 786419, 24631, 789967, 34501, 99859, 800659, 50153, 804239, 50377, 4513, 101203, 4483, 101653, 17341, 12763, 35593, 25639, 822259, 4583, 8089, 2269, 8627, 104831, 840479, 6073, 26437, 2237, 3319, 851507, 106669, 855199, 858899, 2339, 862607, 3083, 2309, 870047, 2179, 6841, 38153, 27481, 14447, 885023, 110863, 8629, 55667, 892559, 55903, 8377, 112279, 2671, 2399, 39301, 28307, 907727, 4723, 114659, 2777, 2503, 923023, 57809, 926867, 116099, 930719, 934579, 14633, 938447, 29387, 118033, 946207, 5153, 950099, 4943, 59747, 119983, 120473, 30241, 969679, 7591, 121949, 977567, 122443, 981523, 61469, 985487, 61717, 2803, 123931, 124429, 997427,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+1423 and
the reducible primes which appear as divisor for the first time
p | x^2x+1423 and p < x^2x+1423

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 17 8 9 1.0625 0.5 0.5625
5 32 32 15 17 1 0.46875 0.53125
6 64 59 28 31 0.921875 0.4375 0.484375
7 128 113 52 61 0.8828125 0.40625 0.4765625
8 256 213 89 124 0.83203125 0.34765625 0.484375
9 512 414 163 251 0.80859375 0.31835938 0.49023438
10 1024 814 287 527 0.79492188 0.28027344 0.51464844
11 2048 1585 528 1057 0.77392578 0.2578125 0.51611328
12 4096 3126 950 2176 0.76318359 0.23193359 0.53125
13 8192 6223 1698 4525 0.75964355 0.20727539 0.55236816
14 16384 12372 3125 9247 0.75512695 0.19073486 0.56439209
15 32768 24533 5804 18729 0.74868774 0.17712402 0.57156372
16 65536 48828 10751 38077 0.74505615 0.16404724 0.58100891
17 131072 97206 20128 77078 0.74162292 0.15356445 0.58805847
18 262144 193655 37564 156091 0.7387352 0.14329529 0.59543991
19 524288 385795 70907 314888 0.73584557 0.13524437 0.6006012
20 1048576 769134 134046 635088 0.73350334 0.12783623 0.60566711
21 2097152 1533551 254240 1279311 0.7312541 0.12123108 0.61002302
22 4194304 3058994 483491 2575503 0.729321 0.11527324 0.61404777
23 8388608 6103940 920612 5183328 0.72764635 0.1097455 0.61790085
24 16777216 12181387 1759207 10422180 0.72606725 0.10485691 0.62121034


8. Check for existing Integer Sequences by OEIS

Found in Database : 1423, 89, 1427, 179, 1439, 181, 1459, 23, 1487, 47, 1523, 193, 1567, 199, 1619, 103, 73, 107, 1747, 223,
Found in Database : 1423, 89, 1427, 179, 1439, 181, 1459, 23, 1487, 47, 1523, 193, 1567, 199, 1619, 103, 73, 107, 1747, 223, 1823, 233, 1907, 61, 1999, 2099, 269, 2207, 283, 101, 149, 2447, 157, 2579, 331, 2719, 349,
Found in Database : 23, 47, 61, 73, 89, 97, 101, 103, 107, 139, 149,