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apocalyptic n.  and  Duffinian n.

First numbers that belong to both sets:

243, 245, 247, 355, 361, 497, 529, 539, 578, 583, 635, 649, 667, 669, 671, 723, 749, 800, 841, 865, 867, 869, 871, 873, 875, 925, 927, 975, 979, 981, 985, 1058, 1075, 1081, 1105, 1133, 1135, 1141, 1157, 1191, 1219, 1241, 1243, 1247, 1261, 1265, 1271, 1281, 1285, 1333, 1343, 1349, 1351, 1359, 1403, 1411, 1413, 1415, 1421, 1465, 1467, 1495, 1497, 1501, 1525, 1535, 1539, 1541, 1565, 1587, 1589, 1591, 1623, 1629, 1631, 1633, 1639, 1651, 1661, 1679, 1682, 1687, 1713, 1739, 1751, 1769, 1775, 1803, 1805, 1807, 1809, 1813, 1817, 1819, 1821, 1825, 1829, 1837, 1839, 1843, 1849, 1853, 1855, 1857, 1883, 1893, 1899, 1903, 1919, 1925, 1929, 1953, 1957, 1963, 1969, 2005, 2007, 2015, 2019, 2021, 2025, 2033, 2047, 2057, 2059, 2071, 2073, 2077, 2101, 2105, 2117, 2123, 2125, 2133, 2155, 2163, 2187, 2189, 2227, 2231, 2271, 2279, 2285, 2291, 2303, 2304, 2315, 2317, 2321, 2323, 2327, 2329, 2331, 2335, 2345, 2359, 2361, 2363, 2387, 2401, 2407, 2425, 2431, 2449, 2469, 2471, 2479, 2487, 2489, 2491, 2493, 2500, 2501, 2507, 2511, 2527, 2533, 2537, 2561, 2563, 2569, 2575, 2581, 2603, 2605, 2627, 2635, 2641, 2649, 2653, 2665, 2669, 2681, 2704, 2705, 2709, 2721, 2723, 2735, 2738, 2743, 2757, 2775, 2795, 2809, 2817, 2839, 2847, 2849, 2863, 2881, 2899, 2901, 2911, 2921, 2935, 2943, 2947, 2959, 2973, 2975, 2981, 2987, 3007, 3013, 3033, 3035, 3047, 3053, 3055, 3065, 3071, 3073, 3081, 3101, 3113, 3117, 3131, 3139, 3141, 3143, 3149, 3155, 3161, 3171, 3173, 3175, 3179, 3189, 3193, 3197, 3207, 3211, 3215, 3223, 3225, 3233, 3239, 3261, 3263, 3265, 3267, 3269, 3275, 3279, 3281, 3287, 3297, 3303, 3305, 3355, 3365, 3369, 3379, 3383, 3385, 3387, 3403, 3409, 3421.

Some larger examples :