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odious n.  and  palindromic n.

First numbers that belong to both sets:

1, 2, 4, 7, 8, 11, 22, 44, 55, 88, 121, 131, 151, 161, 171, 181, 191, 242, 262, 292, 313, 333, 434, 454, 484, 494, 505, 515, 535, 545, 555, 565, 575, 595, 626, 656, 666, 696, 707, 727, 737, 747, 757, 767, 787, 818, 838, 868, 878, 919, 929, 939, 949, 959, 979, 1001, 1221, 1441, 1881, 2002, 2332, 2442, 2552, 2882, 3003, 3113, 3223, 3333, 3553, 3773, 4004, 4114, 4334, 4664, 4884, 4994, 5005, 5665, 5995, 6006, 6226, 6446, 6556, 6666, 6776, 6996, 7117, 7337, 7557, 7777, 8008, 8228, 8558, 8668, 8778, 9119, 9229, 9449, 9669, 9779, 10101, 10201, 10301, 10401, 10501, 10601, 10901, 11111, 11211, 11411, 11611, 11711, 11811, 12121, 12221, 12421, 12521, 12621, 12721, 13031, 13431, 13531, 13831, 13931, 14041, 14341, 14441, 14641, 14941, 15051, 15251, 15451, 15551, 15751, 15851, 16061, 16261, 16361, 16461, 16561, 16661, 16761, 17271, 17371, 17871, 17971, 18381, 18481, 19091, 19291, 19391, 19591, 19691, 19791, 19891, 20102, 20202, 20302, 20402, 20502, 20602, 20802, 20902, 21012, 21112, 21312, 21412, 21612, 21812, 21912, 22222, 22422, 22522, 22822, 22922, 23032, 23332, 23432, 23832, 24142, 24242, 24342, 24442, 24842, 25352, 25552, 25852, 26062, 26162, 26862, 26962, 27372, 27472, 27572, 27772, 27872, 27972, 28082, 28382, 28682, 28882, 29092, 29192, 29392, 29692, 29892, 29992, 30003, 30303, 30403, 30703, 31213, 31513, 31913, 32223, 32323, 32423, 32623, 33133, 33533, 33733, 33833, 34043, 34143, 34243, 34343, 34443, 34943, 35453, 35553, 35653, 35953, 36163, 36263, 36363, 36563, 36863, 37173, 37273, 37573, 37673, 37873, 37973, 38083, 38383, 38683, 38983, 39193, 39493, 39793, 40104, 40204, 40404, 40604, 40804, 40904, 41214, 41414, 41614, 41914, 42024, 42224, 42324, 42524, 42624, 42824, 42924, 43134, 43234, 43334, 43534, 43734, 43834, 44144, 44444, 44544, 44844, 45054, 45254, 45354.

Some larger examples :