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cyclic n.  and  pancake n.

First numbers that belong to both sets:

1, 2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, 407, 631, 667, 781, 821, 947, 991, 1129, 1177, 1327, 1541, 1597, 1831, 2017, 2081, 2279, 2347, 2557, 2629, 2851, 2927, 3161, 3487, 3571, 3917, 4187, 4279, 4561, 4657, 4951, 5051, 5357, 5461, 5779, 6217, 6329, 7141, 7261, 7627, 7751, 8129, 8257, 8647, 8779, 9181, 9731, 9871, 10441, 11027, 11477, 11629, 12091, 12247, 12721, 12881, 13367, 14029, 14197, 14707, 14879, 15401, 15577, 16111, 16291, 16837, 17021, 17579, 17767, 18337, 19111, 19307, 20101, 20707, 20911, 21529, 21737, 22367, 22579, 23221, 23437, 24091, 24311, 24977, 25201, 26107, 26797, 27029, 27731, 27967, 28681, 28921, 29891, 30629, 30877, 31627, 31879, 32897, 33671, 33931, 34717, 34981, 35779, 36047, 36857, 37129, 37951, 39341, 40471, 41329, 41617, 42487, 42779, 43661, 43957, 44851, 45151, 46057, 46361, 47279, 47587, 49771, 50087, 51041, 51361, 52651, 53629, 53957, 54947, 55279, 56281, 56617, 57971, 58997, 59341, 60727, 61777, 62129, 63191, 63547, 64621, 66067, 66431, 67529, 67897, 69007, 69379, 70501, 70877, 72011, 72391, 73537, 73921, 75079, 76637, 77029, 78607, 79801, 81407, 81811, 83029, 83437, 84667, 85079, 86321, 87991, 88411, 89677, 90101, 91379, 91807, 93097, 93529, 94831, 95267, 96581, 97021, 98347, 98791, 100129, 100577, 102379, 103741, 104197, 105571, 106031, 107417, 107881, 109279, 111157, 111629, 113051, 113527, 114961, 115441, 116887, 117371, 118829, 121279, 122761, 126757, 127261, 129287, 130817, 131329, 132871, 133387, 134941, 135461, 137027, 137551, 141247, 141779, 143917, 145531, 148241, 149879, 150427, 152077, 152629, 154291, 154847, 156521, 157081, 158767, 159331, 161029, 161597, 163307, 163879, 165601, 166177, 167911, 168491, 170237, 170821, 172579, 173167, 174937, 175529, 177311, 177907, 179701, 180301, 182107, 182711, 184529, 185137, 189421, 190037, 191891, 194377, 195001, 196879, 197507, 199397, 200029, 201931, 202567, 204481, 207047, 207691, 210277, 212227, 212879, 214841, 215497, 217471.

Some larger examples :