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cyclic n.  and  palindromic n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 11, 33, 77, 101, 131, 141, 151, 161, 181, 191, 303, 313, 323, 353, 373, 383, 393, 515, 535, 545, 565, 595, 707, 717, 727, 757, 767, 787, 797, 919, 929, 949, 959, 989, 1001, 1111, 1441, 1551, 1661, 1991, 3113, 3223, 3333, 3443, 3553, 7117, 7447, 7667, 7777, 7887, 9119, 9229, 9669, 9889, 10001, 10301, 10401, 10501, 10601, 10801, 11111, 11311, 11411, 11611, 11911, 12021, 12121, 12421, 12521, 12721, 12821, 13031, 13231, 13331, 13631, 13831, 13931, 14041, 14141, 14241, 14341, 14441, 14741, 14941, 15051, 15151, 15251, 15451, 15551, 15751, 16061, 16161, 16261, 16361, 16561, 16661, 16861, 17071, 17171, 17371, 17471, 17671, 17771, 17971, 18181, 18281, 18381, 18481, 18581, 18781, 18881, 19291, 19391, 19591, 19891, 19991, 30103, 30203, 30403, 30503, 30703, 30803, 30903, 31013, 31313, 31513, 31613, 31813, 31913, 32023, 32123, 32223, 32323, 32423, 32623, 32723, 32923, 33133, 33233, 33433, 33533, 33733, 33833, 34043, 34343, 34543, 34843, 34943, 35053, 35153, 35353, 35653, 35753, 35953, 36263, 36563, 36763, 36863, 37073, 37273, 37473, 37573, 37673, 37973, 38083, 38183, 38383, 38483, 38683, 38783, 38983, 39193, 39293, 39493, 39693, 39793, 39893, 39993, 50005, 50205, 50405, 51115, 51215, 51515, 51715, 53035, 53135, 53335, 53435, 53935, 54345, 54445, 54545, 54745, 54845, 55155, 55255, 56065, 56165, 56365, 56665, 56765, 56965, 58085, 58285, 58385, 58585, 58885, 59095, 59195, 59395, 59495, 59595, 59695, 59795, 70007, 70107, 70207, 70307, 70507, 70607, 70807, 70907, 71017, 71317, 71417, 71617, 71717, 71917, 72127, 72227, 72527, 72727, 72827, 73037, 73237, 73337, 73537, 73637, 73837, 73937, 74047, 74147, 74247, 74447, 74647, 74747, 74947, 75457, 75557, 75757, 75857, 76067, 76267, 76367, 76467, 76567, 76667, 76767, 76867, 77177, 77377, 77477, 77677, 77777, 77977, 78187, 78387, 78487, 78587, 78787, 78887, 78987, 79097, 79197.

Some larger examples :