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apocalyptic numbers
A number n such that the pattern 666 is contained among the digits of 2n. more

The first 600 apocalyptic numbers :
157, 192, 218, 220, 222, 224, 226, 243, 245, 247, 251, 278, 285, 286, 287, 312, 355, 361, 366, 382, 384, 390, 394, 411, 434, 443, 478, 497, 499, 506, 508, 528, 529, 539, 540, 541, 564, 578, 580, 582, 583, 610, 612, 614, 620, 624, 635, 646, 647, 648, 649, 650, 660, 662, 664, 666, 667, 669, 671, 684, 686, 693, 700, 702, 704, 714, 718, 720, 723, 747, 748, 749, 787, 800, 807, 819, 820, 822, 823, 824, 826, 828, 836, 838, 840, 841, 842, 844, 846, 848, 850, 857, 859, 861, 864, 865, 866, 867, 868, 869, 871, 873, 875, 882, 884, 894, 898, 920, 922, 924, 925, 927, 928, 929, 931, 937, 970, 972, 975, 977, 979, 981, 983, 985, 994, 1002, 1004, 1018, 1019, 1026, 1031, 1040, 1056, 1058, 1060, 1062, 1065, 1075, 1080, 1081, 1086, 1087, 1088, 1090, 1092, 1102, 1103, 1105, 1128, 1133, 1135, 1136, 1138, 1139, 1140, 1141, 1155, 1157, 1158, 1160, 1162, 1164, 1166, 1170, 1191, 1192, 1194, 1199, 1200, 1202, 1212, 1217, 1219, 1224, 1230, 1232, 1240, 1241, 1243, 1245, 1246, 1247, 1257, 1259, 1261, 1265, 1267, 1270, 1271, 1272, 1277, 1279, 1280, 1281, 1285, 1286, 1289, 1290, 1298, 1301, 1303, 1304, 1305, 1310, 1312, 1316, 1318, 1321, 1333, 1338, 1339, 1340, 1341, 1343, 1345, 1347, 1349, 1351, 1359, 1372, 1374, 1382, 1384, 1394, 1399, 1403, 1411, 1413, 1415, 1416, 1418, 1420, 1421, 1422, 1424, 1429, 1432, 1434, 1435, 1438, 1449, 1452, 1464, 1465, 1466, 1467, 1468, 1470, 1472, 1478, 1480, 1482, 1483, 1485, 1487, 1488, 1490, 1492, 1493, 1494, 1495, 1497, 1499, 1501, 1503, 1510, 1515, 1522, 1524, 1525, 1527, 1535, 1539, 1541, 1546, 1550, 1552, 1556, 1560, 1561, 1562, 1565, 1568, 1572, 1574, 1578, 1584, 1586, 1587, 1588, 1589, 1590, 1591, 1596, 1611, 1623, 1625, 1626, 1627, 1628, 1629, 1630, 1631, 1633, 1634, 1636, 1638, 1639, 1640, 1642, 1644, 1651, 1652, 1653, 1656, 1658, 1661, 1663, 1665, 1666, 1667, 1669, 1670, 1674, 1679, 1682, 1684, 1687, 1688, 1690, 1692, 1695, 1701, 1706, 1708, 1713, 1715, 1722, 1723, 1739, 1743, 1744, 1750, 1751, 1753, 1755, 1758, 1760, 1762, 1766, 1769, 1772, 1774, 1775, 1776, 1777, 1783, 1784, 1785, 1792, 1797, 1798, 1800, 1802, 1803, 1804, 1805, 1806, 1807, 1809, 1811, 1812, 1813, 1817, 1819, 1820, 1821, 1823, 1825, 1827, 1829, 1831, 1833, 1837, 1839, 1843, 1849, 1851, 1853, 1855, 1857, 1860, 1862, 1864, 1866, 1868, 1870, 1871, 1880, 1881, 1882, 1883, 1884, 1885, 1887, 1889, 1891, 1893, 1895, 1896, 1899, 1903, 1910, 1912, 1914, 1916, 1917, 1919, 1925, 1929, 1933, 1940, 1942, 1944, 1946, 1947, 1953, 1954, 1957, 1963, 1968, 1969, 1972, 1977, 1979, 1980, 1987, 1990, 1992, 2000, 2001, 2002, 2004, 2005, 2006, 2007, 2008, 2009, 2012, 2015, 2019, 2021, 2024, 2025, 2032, 2033, 2034, 2039, 2043, 2045, 2047, 2049, 2052, 2053, 2054, 2055, 2057, 2059, 2064, 2066, 2067, 2068, 2069, 2070, 2071, 2072, 2073, 2074, 2076, 2077, 2078, 2079, 2081, 2083, 2092, 2095, 2097, 2099, 2101, 2103, 2105, 2113, 2114, 2115, 2117, 2118, 2120, 2123, 2125, 2130, 2131, 2132, 2133, 2134, 2138, 2143, 2148, 2150, 2155, 2157, 2158, 2160, 2161, 2163, 2168, 2170, 2180, 2185, 2186, 2187, 2189, 2195, 2200, 2202, 2204, 2206, 2210, 2212, 2214, 2216, 2222, 2223, 2224, 2225, 2226, 2227, 2229, 2231, 2234, 2238, 2239, 2240, 2242, 2243, 2245, 2246, 2247, 2250, 2251, 2252, 2254, 2259, 2260, 2262, 2265, 2267, 2269, 2270, 2271, 2272, 2273, 2274, 2275, 2276, 2277, 2278, 2279, 2281, 2283, 2284, 2285, 2286, 2287, 2290, 2291, 2292, 2293, 2294, 2296, 2298, 2300, 2301, 2302, 2303, 2304, 2306, 2308, 2310, 2311, 2315, 2316, 2317, 2318, 2319, 2320, 2321, 2322, 2323, 2324, 2325.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 26285 values, from 157 to 30000).

n\r 0  1 
21316213123 2 
3873087668789 3 
46562655266006571 4 
552615234526452635263 5 
6437743884407435343784382 6 
73748376237633756374537723739 7 
832823278330532953280327432953276 8 
9291329322922292329152920289429192947 9 
102639262126362617262826222613262826462635 10 
1123722401238224052381241423652398237924032385

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.