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 |
2 | 13162 | 13123 | 2 |
3 | 8730 | 8766 | 8789 | 3 |
4 | 6562 | 6552 | 6600 | 6571 | 4 |
5 | 5261 | 5234 | 5264 | 5263 | 5263 | 5 |
6 | 4377 | 4388 | 4407 | 4353 | 4378 | 4382 | 6 |
7 | 3748 | 3762 | 3763 | 3756 | 3745 | 3772 | 3739 | 7 |
8 | 3282 | 3278 | 3305 | 3295 | 3280 | 3274 | 3295 | 3276 | 8 |
9 | 2913 | 2932 | 2922 | 2923 | 2915 | 2920 | 2894 | 2919 | 2947 | 9 |
10 | 2639 | 2621 | 2636 | 2617 | 2628 | 2622 | 2613 | 2628 | 2646 | 2635 | 10 |
11 | 2372 | 2401 | 2382 | 2405 | 2381 | 2414 | 2365 | 2398 | 2379 | 2403 | 2385 |
A pictorial representation of the table above
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.