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cyclic n.  and  Duffinian n.

First numbers that belong to both sets:

35, 65, 77, 85, 115, 119, 133, 143, 161, 185, 187, 209, 215, 217, 221, 235, 247, 259, 265, 299, 319, 323, 329, 335, 341, 365, 371, 377, 391, 403, 407, 413, 415, 427, 437, 451, 469, 481, 485, 493, 511, 515, 517, 527, 533, 535, 551, 553, 559, 565, 583, 589, 595, 611, 623, 629, 635, 649, 667, 671, 685, 697, 707, 713, 721, 731, 749, 763, 767, 779, 781, 785, 793, 799, 803, 805, 815, 817, 835, 851, 865, 869, 871, 893, 899, 901, 913, 917, 923, 943, 949, 959, 965, 985, 989, 1003, 1007, 1037, 1043, 1057, 1067, 1073, 1079, 1099, 1105, 1111, 1115, 1121, 1133, 1135, 1141, 1147, 1157, 1159, 1165, 1177, 1189, 1207, 1211, 1219, 1241, 1243, 1247, 1253, 1261, 1271, 1273, 1285, 1295, 1309, 1313, 1315, 1333, 1337, 1343, 1349, 1351, 1357, 1363, 1385, 1387, 1391, 1393, 1397, 1403, 1411, 1415, 1417, 1457, 1463, 1465, 1469, 1495, 1501, 1507, 1513, 1517, 1529, 1535, 1537, 1541, 1565, 1577, 1585, 1589, 1591, 1603, 1631, 1633, 1639, 1643, 1645, 1649, 1651, 1661, 1679, 1685, 1687, 1691, 1727, 1735, 1739, 1763, 1765, 1769, 1781, 1793, 1799, 1807, 1817, 1819, 1829, 1835, 1837, 1841, 1843, 1853, 1855, 1865, 1883, 1897, 1903, 1909, 1915, 1919, 1921, 1927, 1937, 1939, 1943, 1955, 1957, 1961, 1963, 1969, 1981, 1985, 1991, 2021, 2033, 2047, 2059, 2071, 2077, 2101, 2117, 2119, 2123, 2159, 2165, 2171, 2173, 2177, 2183, 2191, 2201, 2215, 2219, 2227, 2231, 2249, 2257, 2261, 2263, 2279, 2285, 2291, 2315, 2317, 2321, 2323, 2327, 2335, 2345, 2363, 2369, 2387, 2405, 2407, 2413, 2419, 2429, 2431, 2435, 2449, 2453, 2461, 2471, 2479, 2483, 2489, 2491, 2497, 2501, 2507, 2509, 2513, 2515, 2519, 2533.

Some larger examples :