Sum of Digits Cubed to the Rescue

Just as I struggled with finding something of interest about 27164, as reported in my post titled Circulant Matrix to the Rescue, I similarly struggled in finding something of interest about 27168. This really bugged me but try as I may I could not find anything really interesting after several days of trying.

Finally however, after playing around with the individual digits of the numbers on either side of 27168, I noticed that the sum of digits cubed ( ${\text{SOD}}^3$) were both prime. Specifically I discovered that:$$\begin{array}{rl}27167\to 2^3+7^3+1^3+6^3+7^3& =911\\ 27169\to 2^3+7^3+1^3+6^3+9^3& =1297\end{array}$$Both 911 and 1297 are prime but the ${\text{SOD}}^3$ of 27168 is not because we have:$$27168\to 2^3+7^3+1^3+6^3+8^3=1080$$It is very rare to have three numbers in a row whose ${\text{SOD}}^3$ are prime. In the range up to one million, there are only four such groups of three numbers with the middle numbers being 1100, 10100, 100100 and 110000 and all having 2 as their ${\text{SOD}}^3$. For example, take the group 1099, 1100 and 1101 where we have:$$\begin{array}{rl}1099\to 1^3+0^3+9^3+9^3& =1459\\ 1100\to 1^3+1^3+0^3+0^3& =2\\ 1101\to 1^3+1^3+0^3+1^3& =3\end{array}$$So having three numbers in a row whose ${\text{SOD}}^3$s are prime is hardly an interesting sequence but what about numbers like 27168 with adjacent numbers that have this property.  Well, up to one million, there are 22045 such numbers representing 2.2045% of the range (permalink). I won't list them all here but up to 40,000, the 1137 numbers are (permalink):

112, 114, 123, 147, 165, 183, 213, 222, 237, 255, 264, 282, 288, 297, 327, 336, 354, 363, 417, 462, 486, 495, 525, 534, 552, 567, 576, 585, 615, 624, 633, 642, 657, 693, 756, 783, 813, 822, 828, 846, 855, 873, 888, 927, 945, 963, 996, 1012, 1014, 1023, 1047, 1065, 1083, 1102, 1104, 1111, 1113, 1122, 1128, 1131, 1146, 1173, 1203, 1212, 1218, 1272, 1311, 1377, 1407, 1416, 1443, 1494, 1500, 1593, 1605, 1713, 1722, 1737, 1782, 1803, 1872, 1944, 1953, 2013, 2022, 2037, 2055, 2064, 2082, 2088, 2097, 2103, 2112, 2118, 2172, 2202, 2226, 2271, 2307, 2343, 2400, 2433, 2442, 2475, 2505, 2556, 2574, 2583, 2604, 2712, 2721, 2745, 2754, 2802, 2808, 2853, 2907, 3027, 3036, 3054, 3063, 3111, 3177, 3207, 3243, 3306, 3375, 3384, 3423, 3498, 3504, 3603, 3663, 3672, 3717, 3735, 3762, 3834, 3900, 3948, 3993, 4017, 4062, 4086, 4095, 4107, 4116, 4143, 4194, 4200, 4233, 4242, 4275, 4323, 4398, 4413, 4422, 4581, 4602, 4725, 4806, 4851, 4905, 4914, 4938, 4992, 5025, 5034, 5052, 5067, 5076, 5085, 5100, 5193, 5205, 5256, 5274, 5283, 5304, 5481, 5502, 5526, 5553, 5571, 5607, 5667, 5706, 5724, 5751, 5805, 5823, 5841, 5913, 6015, 6024, 6033, 6042, 6057, 6093, 6105, 6204, 6303, 6363, 6372, 6402, 6507, 6567, 6633, 6657, 6666, 6675, 6684, 6732, 6765, 6864, 6903, 7056, 7083, 7113, 7122, 7137, 7182, 7212, 7221, 7245, 7254, 7317, 7335, 7362, 7425, 7506, 7524, 7551, 7632, 7665, 7803, 7812, 7887, 8013, 8022, 8028, 8046, 8055, 8073, 8088, 8103, 8172, 8202, 8208, 8253, 8334, 8400, 8406, 8451, 8505, 8523, 8541, 8664, 8703, 8712, 8787, 8808, 8877, 9027, 9045, 9063, 9096, 9144, 9153, 9207, 9348, 9393, 9405, 9414, 9438, 9492, 9513, 9603, 9900, 9906, 9933, 9942, 10012, 10014, 10023, 10047, 10065, 10083, 10102, 10104, 10111, 10113, 10122, 10128, 10131, 10146, 10173, 10203, 10212, 10218, 10272, 10311, 10377, 10407, 10416, 10443, 10494, 10500, 10593, 10605, 10713, 10722, 10737, 10782, 10803, 10872, 10944, 10953, 11002, 11004, 11011, 11013, 11022, 11028, 11031, 11046, 11073, 11101, 11103, 11112, 11118, 11121, 11127, 11145, 11163, 11172, 11202, 11208, 11211, 11217, 11244, 11301, 11334, 11343, 11367, 11400, 11406, 11415, 11424, 11433, 11442, 11451, 11499, 11541, 11592, 11613, 11637, 11697, 11703, 11712, 11895, 11952, 11967, 11985, 12003, 12012, 12018, 12072, 12102, 12108, 12111, 12117, 12144, 12225, 12261, 12333, 12342, 12348, 12393, 12399, 12414, 12432, 12438, 12474, 12582, 12621, 12678, 12687, 12702, 12744, 12768, 12852, 12867, 12885, 12900, 12933, 13011, 13077, 13101, 13134, 13143, 13167, 13200, 13233, 13242, 13248, 13293, 13314, 13323, 13332, 13413, 13422, 13428, 13446, 13455, 13473, 13491, 13545, 13617, 13662, 13707, 13743, 13923, 13941, 14007, 14016, 14043, 14094, 14100, 14106, 14115, 14124, 14133, 14142, 14151, 14199, 14214, 14232, 14238, 14274, 14313, 14322, 14328, 14346, 14355, 14373, 14391, 14403, 14412, 14436, 14445, 14454, 14463, 14487, 14496, 14511, 14535, 14544, 14643, 14685, 14724, 14733, 14847, 14865, 14883, 14904, 14931, 14946, 14991, 15093, 15099, 15141, 15192, 15282, 15345, 15411, 15435, 15444, 15576, 15756, 15774, 15822, 15888, 15897, 15903, 15912, 15987, 16005, 16113, 16137, 16197, 16221, 16278, 16287, 16317, 16362, 16443, 16485, 16599, 16632, 16683, 16728, 16773, 16791, 16827, 16845, 16863, 16896, 16917, 16971, 16986, 17013, 17022, 17037, 17082, 17103, 17112, 17202, 17244, 17268, 17307, 17343, 17424, 17433, 17556, 17574, 17628, 17673, 17691, 17754, 17763, 17778, 17802, 17961, 18003, 18072, 18195, 18252, 18267, 18285, 18300, 18447, 18465, 18483, 18522, 18588, 18597, 18627, 18645, 18663, 18696, 18702, 18825, 18843, 18858, 18885, 18915, 18957, 18966, 19044, 19053, 19152, 19167, 19185, 19233, 19299, 19323, 19341, 19404, 19431, 19446, 19491, 19503, 19512, 19587, 19617, 19671, 19686, 19761, 19815, 19857, 19866, 19941, 19998, 20013, 20022, 20037, 20055, 20064, 20082, 20088, 20097, 20103, 20112, 20118, 20172, 20202, 20226, 20271, 20307, 20343, 20400, 20433, 20442, 20475, 20505, 20556, 20574, 20583, 20604, 20712, 20721, 20745, 20754, 20802, 20808, 20853, 20907, 21003, 21012, 21018, 21072, 21102, 21108, 21111, 21117, 21144, 21225, 21261, 21333, 21342, 21348, 21393, 21399, 21414, 21432, 21438, 21474, 21582, 21621, 21678, 21687, 21702, 21744, 21768, 21852, 21867, 21885, 21900, 21933, 22002, 22026, 22071, 22125, 22161, 22206, 22215, 22224, 22251, 22284, 22332, 22383, 22446, 22455, 22473, 22482, 22521, 22545, 22578, 22581, 22611, 22701, 22743, 22758, 22824, 22833, 22842, 22851, 23007, 23043, 23100, 23133, 23142, 23148, 23193, 23199, 23232, 23283, 23313, 23322, 23331, 23403, 23412, 23418, 23445, 23454, 23487, 23496, 23544, 23553, 23676, 23700, 23766, 23799, 23823, 23847, 23892, 23913, 23946, 23982, 24033, 24042, 24075, 24114, 24132, 24138, 24174, 24246, 24255, 24273, 24282, 24303, 24312, 24318, 24345, 24354, 24387, 24396, 24402, 24426, 24435, 24462, 24471, 24486, 24525, 24534, 24594, 24642, 24705, 24714, 24723, 24741, 24798, 24822, 24837, 24846, 24882, 24936, 24954, 24978, 25005, 25056, 25074, 25083, 25182, 25221, 25245, 25278, 25281, 25344, 25353, 25425, 25434, 25494, 25506, 25533, 25599, 25704, 25728, 25773, 25803, 25812, 25821, 25887, 25944, 26004, 26121, 26178, 26187, 26211, 26376, 26442, 26718, 26736, 26772, 26817, 27012, 27021, 27045, 27054, 27102, 27144, 27168, 27201, 27243, 27258, 27366, 27405, 27414, 27423, 27441, 27498, 27504, 27528, 27573, 27618, 27636, 27672, 27753, 27762, 27948, 28002, 28008, 28053, 28152, 28167, 28185, 28224, 28233, 28242, 28251, 28323, 28347, 28392, 28422, 28437, 28446, 28482, 28503, 28512, 28521, 28587, 28617, 28815, 28842, 28857, 28884, 28899, 28932, 29007, 29133, 29313, 29346, 29382, 29436, 29454, 29478, 29544, 29748, 29832, 29997, 30027, 30036, 30054, 30063, 30111, 30177, 30207, 30243, 30306, 30375, 30384, 30423, 30498, 30504, 30603, 30663, 30672, 30717, 30735, 30762, 30834, 30900, 30948, 30993, 31011, 31077, 31101, 31134, 31143, 31167, 31200, 31233, 31242, 31248, 31293, 31314, 31323, 31332, 31413, 31422, 31428, 31446, 31455, 31473, 31491, 31545, 31617, 31662, 31707, 31743, 31923, 31941, 32007, 32043, 32100, 32133, 32142, 32148, 32193, 32199, 32232, 32283, 32313, 32322, 32331, 32403, 32412, 32418, 32445, 32454, 32487, 32496, 32544, 32553, 32676, 32700, 32766, 32799, 32823, 32847, 32892, 32913, 32946, 32982, 33006, 33075, 33084, 33114, 33123, 33132, 33213, 33222, 33231, 33312, 33321, 33354, 33381, 33453, 33468, 33495, 33534, 33543, 33558, 33600, 33648, 33675, 33684, 33699, 33705, 33765, 33804, 33831, 33864, 33945, 34023, 34098, 34113, 34122, 34128, 34146, 34155, 34173, 34191, 34203, 34212, 34218, 34245, 34254, 34287, 34296, 34353, 34368, 34395, 34416, 34425, 34443, 34452, 34467, 34494, 34515, 34524, 34533, 34542, 34557, 34593, 34638, 34647, 34665, 34683, 34713, 34791, 34827, 34863, 34908, 34911, 34926, 34935, 34944, 34953, 34971, 35004, 35145, 35244, 35253, 35334, 35343, 35358, 35415, 35424, 35433, 35442, 35457, 35493, 35523, 35538, 35547, 35565, 35583, 35655, 35682, 35688, 35778, 35853, 35862, 35868, 35943, 36003, 36063, 36072, 36117, 36162, 36276, 36300, 36348, 36375, 36384, 36399, 36438, 36447, 36465, 36483, 36555, 36582, 36588, 36603, 36612, 36645, 36663, 36687, 36702, 36726, 36735, 36771, 36834, 36843, 36852, 36858, 36867, 36894, 36984, 37017, 37035, 37062, 37107, 37143, 37266, 37299, 37305, 37365, 37413, 37491, 37578, 37602, 37626, 37635, 37671, 37758, 37761, 37776, 37941, 38034, 38100, 38223, 38247, 38292, 38304, 38331, 38364, 38427, 38463, 38553, 38562, 38568, 38634, 38643, 38652, 38658, 38667, 38694, 38700, 38883, 38922, 38964, 39048, 39093, 39099, 39123, 39141, 39213, 39246, 39282, 39345, 39408, 39411, 39426, 39435, 39444, 39453, 39471, 39543, 39684, 39741, 39822, 39864, 39903, 39996

So next time I get stuck on a number with seemingly no interesting properties I have this ${\text{SOD}}^3$ property to call on as well as the determinant of the number's circulant matrix.