On the 6th of August 2023, I made a post titled Hidden Beast Numbers in which I considered numbers whose sum of prime factors contained the digit sequence "666", the so-called "number of the beast". The prime factors could be counted with or without multiplicity. An example of the former would be:$$998515=5 \times 7 \times 47 \times 607\\ \text{where } 5 + 7 + 47 +607 = 666$$An example of the latter would be:$$11898=2 \times 3 \times 3 \times 661 \\ \text{where } 2 + 3 + 661 =666$$Today when looking for properties for the number 27534, I noticed that when expressed to base 8 it contains the digit 6 three times but the digits are not contiguous. We have:$$27534_{10}=65616_8$$This led me to investigate decimal numbers that, when expressed in base 8, contain the digit sequence "666".
It turns out that there are 197 numbers with this property in the range up to 40000. These numbers are (permalink):
438, 950, 1462, 1974, 2486, 2998, 3504, 3505, 3506, 3507, 3508, 3509, 3510, 3511, 4022, 4534, 5046, 5558, 6070, 6582, 7094, 7600, 7601, 7602, 7603, 7604, 7605, 7606, 7607, 8118, 8630, 9142, 9654, 10166, 10678, 11190, 11696, 11697, 11698, 11699, 11700, 11701, 11702, 11703, 12214, 12726, 13238, 13750, 14262, 14774, 15286, 15792, 15793, 15794, 15795, 15796, 15797, 15798, 15799, 16310, 16822, 17334, 17846, 18358, 18870, 19382, 19888, 19889, 19890, 19891, 19892, 19893, 19894, 19895, 20406, 20918, 21430, 21942, 22454, 22966, 23478, 23984, 23985, 23986, 23987, 23988, 23989, 23990, 23991, 24502, 25014, 25526, 26038, 26550, 27062, 27574, 28032, 28033, 28034, 28035, 28036, 28037, 28038, 28039, 28040, 28041, 28042, 28043, 28044, 28045, 28046, 28047, 28048, 28049, 28050, 28051, 28052, 28053, 28054, 28055, 28056, 28057, 28058, 28059, 28060, 28061, 28062, 28063, 28064, 28065, 28066, 28067, 28068, 28069, 28070, 28071, 28072, 28073, 28074, 28075, 28076, 28077, 28078, 28079, 28080, 28081, 28082, 28083, 28084, 28085, 28086, 28087, 28088, 28089, 28090, 28091, 28092, 28093, 28094, 28095, 28598, 29110, 29622, 30134, 30646, 31158, 31670, 32176, 32177, 32178, 32179, 32180, 32181, 32182, 32183, 32694, 33206, 33718, 34230, 34742, 35254, 35766, 36272, 36273, 36274, 36275, 36276, 36277, 36278, 36279, 36790, 37302, 37814, 38326, 38838, 39350, 39862
An example would be:$$39862_{10} = 115666_{\,8} $$In base 7, there are 296 such numbers and they are (permalink):
342, 685, 1028, 1371, 1714, 2057, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2743, 3086, 3429, 3772, 4115, 4458, 4795, 4796, 4797, 4798, 4799, 4800, 4801, 5144, 5487, 5830, 6173, 6516, 6859, 7196, 7197, 7198, 7199, 7200, 7201, 7202, 7545, 7888, 8231, 8574, 8917, 9260, 9597, 9598, 9599, 9600, 9601, 9602, 9603, 9946, 10289, 10632, 10975, 11318, 11661, 11998, 11999, 12000, 12001, 12002, 12003, 12004, 12347, 12690, 13033, 13376, 13719, 14062, 14399, 14400, 14401, 14402, 14403, 14404, 14405, 14748, 15091, 15434, 15777, 16120, 16463, 16758, 16759, 16760, 16761, 16762, 16763, 16764, 16765, 16766, 16767, 16768, 16769, 16770, 16771, 16772, 16773, 16774, 16775, 16776, 16777, 16778, 16779, 16780, 16781, 16782, 16783, 16784, 16785, 16786, 16787, 16788, 16789, 16790, 16791, 16792, 16793, 16794, 16795, 16796, 16797, 16798, 16799, 16800, 16801, 16802, 16803, 16804, 16805, 16806, 17149, 17492, 17835, 18178, 18521, 18864, 19201, 19202, 19203, 19204, 19205, 19206, 19207, 19550, 19893, 20236, 20579, 20922, 21265, 21602, 21603, 21604, 21605, 21606, 21607, 21608, 21951, 22294, 22637, 22980, 23323, 23666, 24003, 24004, 24005, 24006, 24007, 24008, 24009, 24352, 24695, 25038, 25381, 25724, 26067, 26404, 26405, 26406, 26407, 26408, 26409, 26410, 26753, 27096, 27439, 27782, 28125, 28468, 28805, 28806, 28807, 28808, 28809, 28810, 28811, 29154, 29497, 29840, 30183, 30526, 30869, 31206, 31207, 31208, 31209, 31210, 31211, 31212, 31555, 31898, 32241, 32584, 32927, 33270, 33565, 33566, 33567, 33568, 33569, 33570, 33571, 33572, 33573, 33574, 33575, 33576, 33577, 33578, 33579, 33580, 33581, 33582, 33583, 33584, 33585, 33586, 33587, 33588, 33589, 33590, 33591, 33592, 33593, 33594, 33595, 33596, 33597, 33598, 33599, 33600, 33601, 33602, 33603, 33604, 33605, 33606, 33607, 33608, 33609, 33610, 33611, 33612, 33613, 33956, 34299, 34642, 34985, 35328, 35671, 36008, 36009, 36010, 36011, 36012, 36013, 36014, 36357, 36700, 37043, 37386, 37729, 38072, 38409, 38410, 38411, 38412, 38413, 38414, 38415, 38758, 39101, 39444, 39787
An example would be:$$39787_{10} = 223666_{ \,7} $$In base 9, there are 103 such numbers and they are (permalink):
546, 1275, 2004, 2733, 3462, 4191, 4914, 4915, 4916, 4917, 4918, 4919, 4920, 4921, 4922, 5649, 6378, 7107, 7836, 8565, 9294, 10023, 10752, 11475, 11476, 11477, 11478, 11479, 11480, 11481, 11482, 11483, 12210, 12939, 13668, 14397, 15126, 15855, 16584, 17313, 18036, 18037, 18038, 18039, 18040, 18041, 18042, 18043, 18044, 18771, 19500, 20229, 20958, 21687, 22416, 23145, 23874, 24597, 24598, 24599, 24600, 24601, 24602, 24603, 24604, 24605, 25332, 26061, 26790, 27519, 28248, 28977, 29706, 30435, 31158, 31159, 31160, 31161, 31162, 31163, 31164, 31165, 31166, 31893, 32622, 33351, 34080, 34809, 35538, 36267, 36996, 37719, 37720, 37721, 37722, 37723, 37724, 37725, 37726, 37727, 38454, 39183, 39912
An example would be:$$39912_{10} = 60666_{\,9}$$
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