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cyclic n.  and  evil n.

First numbers that belong to both sets:

3, 5, 15, 17, 23, 29, 33, 43, 51, 53, 65, 71, 77, 83, 85, 89, 95, 101, 113, 119, 123, 139, 141, 149, 159, 163, 177, 187, 197, 209, 215, 221, 235, 249, 255, 257, 263, 267, 269, 277, 281, 287, 293, 303, 311, 317, 323, 329, 335, 337, 347, 349, 353, 359, 365, 371, 373, 377, 383, 389, 393, 401, 407, 411, 413, 427, 435, 437, 447, 449, 455, 461, 467, 469, 473, 479, 485, 503, 509, 519, 523, 533, 537, 547, 553, 559, 561, 571, 573, 581, 591, 593, 599, 619, 629, 643, 649, 667, 673, 679, 683, 685, 691, 697, 703, 705, 717, 739, 745, 751, 753, 763, 771, 773, 785, 795, 797, 807, 811, 821, 839, 843, 851, 853, 857, 863, 869, 879, 881, 887, 893, 907, 917, 933, 937, 943, 951, 957, 965, 977, 983, 989, 1003, 1013, 1031, 1037, 1043, 1049, 1059, 1061, 1073, 1079, 1091, 1093, 1097, 1103, 1105, 1111, 1115, 1117, 1121, 1133, 1139, 1141, 1145, 1151, 1157, 1167, 1169, 1181, 1195, 1203, 1217, 1223, 1229, 1235, 1237, 1241, 1247, 1253, 1257, 1271, 1277, 1283, 1285, 1289, 1295, 1297, 1303, 1307, 1309, 1313, 1319, 1333, 1337, 1343, 1345, 1351, 1357, 1363, 1381, 1385, 1391, 1393, 1399, 1403, 1409, 1415, 1427, 1429, 1433, 1439, 1457, 1463, 1469, 1481, 1487, 1489, 1495, 1499, 1501, 1511, 1517, 1523, 1529, 1535, 1541, 1551, 1553, 1559, 1563, 1565, 1579, 1589, 1601, 1607, 1613, 1619, 1621, 1631, 1637, 1649, 1661, 1671, 1685, 1689, 1695, 1699, 1723, 1733, 1745, 1757, 1761, 1779, 1781, 1793, 1799, 1811, 1817, 1823, 1829, 1841, 1847, 1851, 1853, 1861, 1865, 1871, 1873, 1879, 1883, 1885, 1889, 1895, 1901, 1907, 1909, 1913, 1919, 1923.

Some larger examples :