E126. Les progressions interdites *** Q

E126. Les progressions interdites ***

Q

Sait-on construire une suite de 2016 entiers distincts compris entre 1 et 100 000 tels que trois quelconques d’entre eux ne forment jamais une progression arithmétique?

Q

Même question que Q

avec 2050 entiers naturels distincts.

Q1 : Oui,

On commence par prendre les entiers 1 et 2

l'entier suivant ne peut être 3, par contre l'entier 4 fait l'affaire.

L'entier 5 fait aussi l'affaire.

Les entiers 6,7,8,9 sont à éliminer.

L'entier 10 fait l'affaire.

Nous avons alors une suite de 5 entiers tels que trois quelconques d’entre eux ne forment jamais une progression arithmétique

On poursuit de cette manière jusqu'à obtenir la suite de 2016 entiers ci-dessous :

1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 82, 83, 85, 86, 91, 92, 94, 95, 109, 110, 112, 113, 118, 119, 121, 122, 244, 245, 247, 248, 253, 254, 256, 257, 271, 272, 274, 275, 280, 281, 283, 284, 325, 326, 328, 329, 334, 335, 337, 338, 352, 353, 355, 356, 361, 362, 364, 365, 730, 731, 733, 734, 739, 740, 742, 743, 757, 758, 760, 761, 766, 767, 769, 770, 811, 812, 814, 815, 820, 821, 823, 824, 838, 839, 841, 842, 847, 848, 850, 851, 973, 974, 976, 977, 982, 983, 985, 986, 1000, 1001, 1003, 1004, 1009, 1010, 1012, 1013, 1054, 1055, 1057, 1058, 1063, 1064, 1066, 1067, 1081, 1082, 1084, 1085, 1090, 1091, 1093, 1094, 2188, 2189, 2191, 2192, 2197, 2198, 2200, 2201, 2215, 2216, 2218, 2219, 2224, 2225, 2227, 2228, 2269, 2270, 2272, 2273, 2278, 2279, 2281, 2282, 2296, 2297, 2299, 2300, 2305, 2306, 2308, 2309, 2431, 2432, 2434, 2435, 2440, 2441, 2443, 2444, 2458, 2459, 2461, 2462, 2467, 2468, 2470, 2471, 2512, 2513, 2515, 2516, 2521, 2522, 2524, 2525, 2539, 2540, 2542, 2543, 2548, 2549, 2551, 2552, 2917, 2918, 2920, 2921, 2926, 2927, 2929, 2930, 2944, 2945, 2947, 2948, 2953, 2954, 2956, 2957, 2998, 2999, 3001, 3002, 3007, 3008, 3010, 3011, 3025, 3026, 3028, 3029, 3034, 3035, 3037, 3038, 3160, 3161, 3163, 3164, 3169, 3170, 3172, 3173, 3187, 3188, 3190, 3191, 3196, 3197, 3199, 3200, 3241, 3242, 3244, 3245, 3250, 3251, 3253, 3254, 3268, 3269, 3271, 3272, 3277, 3278, 3280, 3281, 6562, 6563, 6565, 6566, 6571, 6572, 6574, 6575, 6589, 6590, 6592, 6593, 6598, 6599, 6601, 6602, 6643, 6644, 6646, 6647, 6652, 6653, 6655, 6656, 6670, 6671, 6673, 6674, 6679, 6680, 6682, 6683, 6805, 6806, 6808, 6809, 6814, 6815, 6817, 6818, 6832, 6833, 6835, 6836, 6841, 6842, 6844, 6845, 6886, 6887, 6889, 6890, 6895, 6896, 6898, 6899, 6913, 6914, 6916, 6917, 6922, 6923, 6925, 6926, 7291, 7292, 7294, 7295, 7300, 7301, 7303, 7304, 7318, 7319, 7321, 7322, 7327, 7328, 7330, 7331, 7372, 7373, 7375, 7376, 7381, 7382, 7384, 7385, 7399, 7400, 7402, 7403, 7408, 7409, 7411, 7412, 7534, 7535, 7537, 7538, 7543, 7544, 7546, 7547, 7561, 7562, 7564, 7565, 7570, 7571, 7573, 7574, 7615, 7616, 7618, 7619, 7624, 7625, 7627, 7628, 7642, 7643, 7645, 7646, 7651, 7652, 7654, 7655, 8749, 8750, 8752, 8753, 8758, 8759, 8761, 8762, 8776, 8777, 8779, 8780, 8785, 8786, 8788, 8789, 8830, 8831, 8833, 8834, 8839, 8840, 8842, 8843, 8857, 8858, 8860, 8861, 8866, 8867, 8869, 8870, 8992, 8993, 8995, 8996, 9001, 9002, 9004, 9005, 9019, 9020, 9022, 9023, 9028, 9029, 9031, 9032, 9073, 9074, 9076, 9077, 9082, 9083, 9085, 9086, 9100, 9101, 9103, 9104, 9109, 9110, 9112, 9113, 9478, 9479, 9481, 9482, 9487, 9488, 9490, 9491, 9505, 9506, 9508, 9509, 9514, 9515, 9517, 9518, 9559, 9560, 9562, 9563, 9568, 9569, 9571, 9572, 9586, 9587, 9589, 9590, 9595, 9596, 9598, 9599, 9721, 9722, 9724, 9725, 9730, 9731, 9733, 9734, 9748, 9749, 9751, 9752, 9757, 9758, 9760, 9761, 9802, 9803, 9805, 9806, 9811, 9812, 9814, 9815, 9829, 9830, 9832, 9833, 9838, 9839, 9841, 9842, 19684, 19685, 19687, 19688, 19693, 19694, 19696, 19697, 19711, 19712, 19714, 19715, 19720, 19721, 19723, 19724, 19765, 19766, 19768, 19769, 19774, 19775, 19777, 19778, 19792, 19793, 19795, 19796, 19801, 19802, 19804, 19805, 19927, 19928, 19930, 19931, 19936, 19937, 19939, 19940, 19954, 19955, 19957, 19958, 19963, 19964, 19966, 19967, 20008, 20009, 20011, 20012, 20017, 20018, 20020, 20021, 20035, 20036, 20038, 20039, 20044, 20045, 20047, 20048, 20413, 20414, 20416, 20417, 20422, 20423, 20425, 20426, 20440, 20441, 20443, 20444, 20449, 20450, 20452, 20453, 20494, 20495, 20497, 20498, 20503, 20504, 20506, 20507, 20521, 20522, 20524, 20525, 20530, 20531, 20533, 20534, 20656, 20657, 20659, 20660, 20665, 20666, 20668, 20669, 20683, 20684, 20686, 20687, 20692, 20693, 20695, 20696, 20737, 20738, 20740, 20741, 20746, 20747, 20749, 20750, 20764, 20765, 20767, 20768, 20773, 20774, 20776, 20777, 21871, 21872, 21874, 21875, 21880, 21881, 21883, 21884, 21898, 21899, 21901, 21902, 21907, 21908, 21910, 21911, 21952, 21953, 21955, 21956, 21961, 21962, 21964, 21965, 21979, 21980, 21982, 21983, 21988, 21989, 21991, 21992, 22114, 22115, 22117, 22118, 22123, 22124, 22126, 22127, 22141, 22142, 22144, 22145, 22150, 22151, 22153, 22154, 22195, 22196, 22198, 22199, 22204, 22205, 22207, 22208, 22222, 22223, 22225, 22226, 22231, 22232, 22234, 22235, 22600, 22601, 22603, 22604, 22609, 22610, 22612, 22613, 22627, 22628, 22630, 22631, 22636, 22637, 22639, 22640, 22681, 22682, 22684, 22685, 22690, 22691, 22693, 22694, 22708, 22709, 22711, 22712, 22717, 22718, 22720, 22721, 22843, 22844, 22846, 22847, 22852, 22853, 22855, 22856, 22870, 22871, 22873, 22874, 22879, 22880, 22882, 22883, 22924, 22925, 22927, 22928, 22933, 22934, 22936, 22937, 22951, 22952, 22954, 22955, 22960, 22961, 22963, 22964, 26245, 26246, 26248, 26249, 26254, 26255, 26257, 26258, 26272, 26273, 26275, 26276, 26281, 26282, 26284, 26285, 26326, 26327, 26329, 26330, 26335, 26336, 26338, 26339, 26353, 26354, 26356, 26357, 26362, 26363, 26365, 26366, 26488, 26489, 26491, 26492, 26497, 26498, 26500, 26501, 26515, 26516, 26518, 26519, 26524, 26525, 26527, 26528, 26569, 26570, 26572, 26573, 26578, 26579, 26581, 26582, 26596, 26597, 26599, 26600, 26605, 26606, 26608, 26609, 26974, 26975, 26977, 26978, 26983, 26984, 26986, 26987, 27001, 27002, 27004, 27005, 27010, 27011, 27013, 27014, 27055, 27056, 27058, 27059, 27064, 27065, 27067, 27068, 27082, 27083, 27085, 27086, 27091, 27092, 27094, 27095, 27217, 27218, 27220, 27221, 27226, 27227, 27229, 27230, 27244, 27245, 27247, 27248, 27253, 27254, 27256, 27257, 27298, 27299, 27301, 27302, 27307, 27308, 27310, 27311, 27325, 27326, 27328, 27329, 27334, 27335, 27337, 27338, 28432, 28433, 28435, 28436, 28441, 28442, 28444, 28445, 28459, 28460, 28462, 28463, 28468, 28469, 28471, 28472, 28513, 28514, 28516, 28517, 28522, 28523, 28525, 28526, 28540, 28541, 28543, 28544, 28549, 28550, 28552, 28553, 28675, 28676, 28678, 28679, 28684, 28685, 28687, 28688, 28702, 28703, 28705, 28706, 28711, 28712, 28714, 28715, 28756, 28757, 28759, 28760, 28765, 28766, 28768, 28769, 28783, 28784, 28786, 28787, 28792, 28793, 28795, 28796, 29161, 29162, 29164, 29165, 29170, 29171, 29173, 29174, 29188, 29189, 29191, 29192, 29197, 29198, 29200, 29201, 29242, 29243, 29245, 29246, 29251, 29252, 29254, 29255, 29269, 29270, 29272, 29273, 29278, 29279, 29281, 29282, 29404, 29405, 29407, 29408, 29413, 29414, 29416, 29417, 29431, 29432, 29434, 29435, 29440, 29441, 29443, 29444, 29485, 29486, 29488, 29489, 29494, 29495, 29497, 29498, 29512, 29513, 29515, 29516, 29521, 29522, 29524, 29525, 59050, 59051, 59053, 59054, 59059, 59060, 59062, 59063, 59077, 59078, 59080, 59081, 59086, 59087, 59089, 59090, 59131, 59132, 59134, 59135, 59140, 59141, 59143, 59144, 59158, 59159, 59161, 59162, 59167, 59168, 59170, 59171, 59293, 59294, 59296, 59297, 59302, 59303, 59305, 59306, 59320, 59321, 59323, 59324, 59329, 59330, 59332, 59333, 59374, 59375, 59377, 59378, 59383, 59384, 59386, 59387, 59401, 59402, 59404, 59405, 59410, 59411, 59413, 59414, 59779, 59780, 59782, 59783, 59788, 59789, 59791, 59792, 59806, 59807, 59809, 59810, 59815, 59816, 59818, 59819, 59860, 59861, 59863, 59864, 59869, 59870, 59872, 59873, 59887, 59888, 59890, 59891, 59896, 59897, 59899, 59900, 60022, 60023, 60025, 60026, 60031, 60032, 60034, 60035, 60049, 60050, 60052, 60053, 60058, 60059, 60061, 60062, 60103, 60104, 60106, 60107, 60112, 60113, 60115, 60116, 60130, 60131, 60133, 60134, 60139, 60140, 60142, 60143, 61237, 61238, 61240, 61241, 61246, 61247, 61249, 61250, 61264, 61265, 61267, 61268, 61273, 61274, 61276, 61277, 61318, 61319, 61321, 61322, 61327, 61328, 61330, 61331, 61345, 61346, 61348, 61349, 61354, 61355, 61357, 61358, 61480, 61481, 61483, 61484, 61489, 61490, 61492, 61493, 61507, 61508, 61510, 61511, 61516, 61517, 61519, 61520, 61561, 61562, 61564, 61565, 61570, 61571, 61573,

61574, 61588, 61589, 61591, 61592, 61597, 61598, 61600, 61601, 61966, 61967, 61969, 61970, 61975, 61976, 61978, 61979, 61993, 61994, 61996, 61997, 62002, 62003, 62005, 62006, 62047, 62048, 62050, 62051, 62056, 62057, 62059, 62060, 62074, 62075, 62077, 62078, 62083, 62084, 62086, 62087, 62209, 62210, 62212, 62213, 62218, 62219, 62221, 62222, 62236, 62237, 62239, 62240, 62245, 62246, 62248, 62249, 62290, 62291, 62293, 62294, 62299, 62300, 62302, 62303, 62317, 62318, 62320, 62321, 62326, 62327, 62329, 62330, 65611, 65612, 65614, 65615, 65620, 65621, 65623, 65624, 65638, 65639, 65641, 65642, 65647, 65648, 65650, 65651, 65692, 65693, 65695, 65696, 65701, 65702, 65704, 65705, 65719, 65720, 65722, 65723, 65728, 65729, 65731, 65732, 65854, 65855, 65857, 65858, 65863, 65864, 65866, 65867, 65881, 65882, 65884, 65885, 65890, 65891, 65893, 65894, 65935, 65936, 65938, 65939, 65944, 65945, 65947, 65948, 65962, 65963, 65965, 65966, 65971, 65972, 65974, 65975, 66340, 66341, 66343, 66344, 66349, 66350, 66352, 66353, 66367, 66368, 66370, 66371, 66376, 66377, 66379, 66380, 66421, 66422, 66424, 66425, 66430, 66431, 66433, 66434, 66448, 66449, 66451, 66452, 66457, 66458, 66460, 66461, 66583, 66584, 66586, 66587, 66592, 66593, 66595, 66596, 66610, 66611, 66613, 66614, 66619, 66620, 66622, 66623, 66664, 66665, 66667, 66668, 66673, 66674, 66676, 66677, 66691, 66692, 66694, 66695, 66700, 66701, 66703, 66704, 67798, 67799, 67801, 67802, 67807, 67808, 67810, 67811, 67825, 67826, 67828, 67829, 67834, 67835, 67837, 67838, 67879, 67880, 67882, 67883, 67888, 67889, 67891, 67892, 67906, 67907, 67909, 67910, 67915, 67916, 67918, 67919, 68041, 68042, 68044, 68045, 68050, 68051, 68053, 68054, 68068, 68069, 68071, 68072, 68077, 68078, 68080, 68081, 68122, 68123, 68125, 68126, 68131, 68132, 68134, 68135, 68149, 68150, 68152, 68153, 68158, 68159, 68161, 68162, 68527, 68528, 68530, 68531, 68536, 68537, 68539, 68540, 68554, 68555, 68557, 68558, 68563, 68564, 68566, 68567, 68608, 68609, 68611, 68612, 68617, 68618, 68620, 68621, 68635, 68636, 68638, 68639, 68644, 68645, 68647, 68648, 68770, 68771, 68773, 68774, 68779, 68780, 68782, 68783, 68797, 68798, 68800, 68801, 68806, 68807, 68809, 68810, 68851, 68852, 68854, 68855, 68860, 68861, 68863, 68864, 68878, 68879, 68881, 68882, 68887, 68888, 68890, 68891, 78733, 78734, 78736, 78737, 78742, 78743, 78745, 78746, 78760, 78761, 78763, 78764, 78769, 78770, 78772, 78773, 78814, 78815, 78817, 78818, 78823, 78824, 78826, 78827, 78841, 78842, 78844, 78845, 78850, 78851, 78853, 78854, 78976, 78977, 78979, 78980, 78985, 78986, 78988, 78989, 79003, 79004, 79006, 79007, 79012, 79013, 79015, 79016, 79057, 79058, 79060, 79061, 79066, 79067, 79069, 79070, 79084, 79085, 79087, 79088, 79093, 79094, 79096, 79097, 79462, 79463, 79465, 79466, 79471, 79472, 79474, 79475, 79489, 79490, 79492, 79493, 79498, 79499, 79501, 79502, 79543, 79544, 79546, 79547, 79552, 79553, 79555, 79556, 79570, 79571, 79573, 79574, 79579, 79580, 79582, 79583, 79705, 79706, 79708, 79709, 79714, 79715, 79717, 79718, 79732, 79733, 79735, 79736, 79741, 79742, 79744, 79745, 79786, 79787, 79789, 79790, 79795, 79796, 79798, 79799, 79813, 79814, 79816, 79817, 79822, 79823, 79825, 79826, 80920, 80921, 80923, 80924, 80929, 80930, 80932, 80933, 80947, 80948, 80950, 80951, 80956, 80957, 80959, 80960, 81001, 81002, 81004, 81005, 81010, 81011, 81013, 81014, 81028, 81029, 81031, 81032, 81037, 81038, 81040, 81041, 81163, 81164, 81166, 81167, 81172, 81173, 81175, 81176, 81190, 81191, 81193, 81194, 81199, 81200, 81202, 81203, 81244, 81245, 81247, 81248, 81253, 81254, 81256, 81257, 81271, 81272, 81274, 81275, 81280, 81281, 81283, 81284, 81649, 81650, 81652, 81653, 81658, 81659, 81661, 81662, 81676, 81677, 81679, 81680, 81685, 81686, 81688, 81689, 81730, 81731, 81733, 81734, 81739, 81740, 81742, 81743, 81757, 81758, 81760, 81761, 81766, 81767, 81769, 81770, 81892, 81893, 81895, 81896, 81901, 81902, 81904, 81905, 81919, 81920, 81922, 81923, 81928, 81929, 81931, 81932, 81973, 81974, 81976, 81977, 81982, 81983, 81985, 81986, 82000, 82001, 82003, 82004, 82009, 82010, 82012, 82013, 85294, 85295, 85297, 85298, 85303, 85304, 85306, 85307, 85321, 85322, 85324, 85325, 85330, 85331, 85333, 85334, 85375, 85376, 85378, 85379, 85384, 85385, 85387, 85388, 85402, 85403, 85405, 85406, 85411, 85412, 85414, 85415, 85537, 85538, 85540, 85541, 85546, 85547, 85549, 85550, 85564, 85565, 85567, 85568, 85573, 85574, 85576, 85577, 85618, 85619, 85621, 85622, 85627, 85628, 85630, 85631, 85645, 85646, 85648, 85649, 85654, 85655, 85657, 85658, 86023, 86024, 86026, 86027, 86032, 86033, 86035, 86036, 86050, 86051, 86053, 86054, 86059, 86060, 86062, 86063, 86104, 86105, 86107, 86108, 86113, 86114, 86116, 86117, 86131, 86132, 86134, 86135, 86140, 86141, 86143, 86144, 86266, 86267, 86269, 86270, 86275, 86276, 86278, 86279, 86293, 86294, 86296, 86297, 86302, 86303, 86305, 86306, 86347, 86348, 86350, 86351, 86356, 86357, 86359, 86360, 86374, 86375, 86377, 86378, 86383, 86384, 86386, 86387, 87481, 87482, 87484, 87485, 87490, 87491, 87493, 87494, 87508, 87509, 87511, 87512, 87517, 87518, 87520, 87521, 87562, 87563, 87565, 87566, 87571, 87572, 87574, 87575, 87589, 87590, 87592, 87593, 87598, 87599, 87601, 87602, 87724, 87725, 87727, 87728, 87733, 87734, 87736, 87737, 87751, 87752, 87754, 87755, 87760, 87761, 87763, 87764, 87805, 87806, 87808, 87809, 87814, 87815, 87817, 87818, 87832, 87833, 87835, 87836, 87841, 87842, 87844, 87845, 88210, 88211, 88213, 88214, 88219, 88220, 88222, 88223, 88237, 88238, 88240, 88241, 88246, 88247, 88249, 88250, 88291, 88292, 88294, 88295, 88300, 88301, 88303, 88304, 88318, 88319, 88321, 88322, 88327, 88328, 88330, 88331

Q2 : non

La suite ci-dessus ( A003278) à été étudiée par Ralf Stephan il propose la récurrence suivante :

a(n) = b(n+1) avec b(0)=1, b(2n)=3b(n)-2, b(2n+1)=3b(n)-1.

a(n) est le prochain entier à prendre sachant que a(1), a(2) , …….. a(n-1) sont déjà pris.

Le tableau ci-dessous montre que pour construire une suite 2050 nombres il faudrait deux entiers plus grands que

100000