2474:
88:
No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a
Hamming weight of one, and
319:
352:
192:
123:
149:
398:
372:
256:
232:
212:
576:
2498:
498:
430:
414:
77:
569:
1376:
562:
1371:
543:
468:
1386:
1366:
2079:
1659:
1381:
155:, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number.
2165:
51:
and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 101
1831:
1481:
1150:
943:
2007:
1866:
1697:
1511:
1501:
1155:
1135:
1836:
1956:
1579:
1421:
1336:
1145:
1127:
1021:
1011:
1001:
837:
1861:
269:
2084:
1629:
1250:
1036:
1031:
1026:
1016:
993:
1841:
1506:
1416:
1069:
259:
2195:
2160:
1946:
1856:
1730:
1705:
1614:
1604:
1326:
1216:
1198:
1118:
2455:
1725:
1599:
1230:
1006:
786:
713:
2419:
2059:
1710:
1564:
1491:
646:
514:
324:
164:
95:
32:
128:
89:
2352:
2246:
2210:
1951:
1674:
1654:
1471:
1140:
928:
1431:
900:
8:
2074:
1938:
1933:
1901:
1664:
1639:
1634:
1609:
1539:
1535:
1466:
1356:
1188:
984:
953:
377:
2477:
2231:
2226:
2140:
2114:
2012:
1991:
1763:
1644:
1594:
1516:
1486:
1426:
1193:
1173:
1104:
817:
357:
241:
217:
197:
1361:
2473:
2371:
2170:
2145:
2119:
1574:
1569:
1496:
1476:
1461:
1183:
1165:
1084:
1074:
1059:
822:
539:
464:
1896:
2407:
2200:
1786:
1758:
1748:
1740:
1624:
1589:
1584:
1551:
1245:
1208:
1099:
1094:
1089:
1079:
1051:
938:
885:
842:
781:
890:
2383:
2272:
2205:
2131:
2054:
2028:
1846:
1559:
1351:
1321:
1311:
1306:
972:
880:
827:
671:
611:
159:
39:, that is, there is a prime number of 1s when it is written as a binary number.
2388:
2256:
2241:
2105:
2069:
2044:
1920:
1891:
1876:
1753:
1649:
1619:
1346:
1301:
1178:
776:
771:
766:
738:
723:
636:
621:
599:
586:
263:
235:
28:
2492:
2311:
2295:
2236:
2190:
1886:
1871:
1781:
1064:
933:
895:
852:
733:
718:
708:
666:
656:
631:
554:
409:
152:
70:
3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, ... (sequence
20:
2347:
2336:
2251:
2089:
2064:
1981:
1881:
1851:
1826:
1810:
1715:
1682:
1405:
1316:
1255:
832:
728:
661:
641:
616:
36:
519:
Seventh
International Sumposium on Artificial Intelligence and Mathematics
2306:
2181:
1986:
1450:
1341:
1296:
1291:
1041:
948:
847:
676:
651:
626:
531:
488:
425:
354:; the binary representation of such a number consists of a prime number
2443:
2424:
1720:
1331:
456:
2049:
1976:
1968:
1773:
1687:
805:
2150:
2155:
1814:
1808:
428:
are numbers with an even number of 1s in their binary expansion (
412:
are numbers with an odd number of 1s in their binary expansion (
870:
492:
434:
418:
400:
zeros. Therefore, every even perfect number is pernicious.
72:
47:
The first pernicious number is 3, since 3 = 11
487:
1534:
380:
360:
327:
272:
244:
220:
200:
167:
131:
98:
1919:
392:
366:
346:
313:
250:
226:
206:
186:
143:
117:
918:
2490:
804:
598:
584:
92:. On the other hand, every number of the form
570:
67:). The sequence of pernicious numbers begins
2406:
756:
512:
871:Possessing a specific set of other numbers
694:
577:
563:
194:has a binary representation consisting of
2334:
1281:
499:On-Line Encyclopedia of Integer Sequences
536:Perfect Numbers And Fibonacci Sequences
2491:
2442:
513:Colton, Simon; Dennis, Louise (2002),
483:
481:
479:
258:, and is therefore pernicious. By the
2441:
2405:
2369:
2333:
2293:
1918:
1807:
1533:
1448:
1403:
1280:
970:
917:
869:
803:
755:
693:
597:
558:
971:
455:
27:is a positive integer such that the
2370:
530:
476:
461:Mersenne Numbers And Fermat Numbers
90:one is not considered to be a prime
13:
2294:
403:
14:
2510:
463:, World Scientific, p. 263,
2499:Base-dependent integer sequences
2472:
2080:Perfect digit-to-digit invariant
1449:
538:, World Scientific, p. 50,
314:{\displaystyle 2^{n-1}(2^{n}-1)}
16:Number with prime Hamming weight
238:is a Mersenne number for prime
524:
515:"The NumbersWithNames Program"
506:
449:
308:
289:
1:
919:Expressible via specific sums
442:
214:ones, and is pernicious when
83:
7:
2008:Multiplicative digital root
42:
10:
2515:
1404:
489:Sloane, N. J. A.
2468:
2451:
2437:
2415:
2401:
2379:
2365:
2343:
2329:
2302:
2289:
2265:
2219:
2179:
2130:
2104:
2085:Perfect digital invariant
2037:
2021:
2000:
1967:
1932:
1928:
1914:
1822:
1803:
1772:
1739:
1696:
1673:
1660:Superior highly composite
1550:
1546:
1529:
1457:
1444:
1412:
1399:
1287:
1276:
1238:
1229:
1207:
1164:
1126:
1117:
1050:
992:
983:
979:
966:
924:
913:
876:
865:
813:
799:
762:
751:
704:
689:
607:
593:
1698:Euler's totient function
1482:Euler–Jacobi pseudoprime
757:Other polynomial numbers
1512:Somer–Lucas pseudoprime
1502:Lucas–Carmichael number
1337:Lazy caterer's sequence
493:"Sequence A052294"
347:{\displaystyle 2^{n}-1}
187:{\displaystyle 2^{n}-1}
118:{\displaystyle 2^{n}+1}
1387:Wedderburn–Etherington
787:Lucky numbers of Euler
394:
368:
348:
315:
252:
228:
208:
188:
145:
144:{\displaystyle n>1}
119:
1675:Prime omega functions
1492:Frobenius pseudoprime
1282:Combinatorial numbers
1151:Centered dodecahedral
944:Primary pseudoperfect
395:
374:of ones, followed by
369:
349:
321:for a Mersenne prime
316:
253:
229:
209:
189:
146:
120:
33:binary representation
2134:-composition related
1934:Arithmetic functions
1536:Arithmetic functions
1472:Elliptic pseudoprime
1156:Centered icosahedral
1136:Centered tetrahedral
378:
358:
325:
270:
260:Euclid–Euler theorem
242:
218:
198:
165:
129:
96:
55:, followed by 6 (110
2060:Kaprekar's constant
1580:Colossally abundant
1467:Catalan pseudoprime
1367:Schröder–Hipparchus
1146:Centered octahedral
1022:Centered heptagonal
1012:Centered pentagonal
1002:Centered triangular
602:and related numbers
393:{\displaystyle n-1}
2478:Mathematics portal
2420:Aronson's sequence
2166:Smarandache–Wellin
1923:-dependent numbers
1630:Primitive abundant
1517:Strong pseudoprime
1507:Perrin pseudoprime
1487:Fermat pseudoprime
1427:Wolstenholme prime
1251:Squared triangular
1037:Centered decagonal
1032:Centered nonagonal
1027:Centered octagonal
1017:Centered hexagonal
390:
364:
344:
311:
248:
224:
204:
184:
151:, including every
141:
115:
2486:
2485:
2464:
2463:
2433:
2432:
2397:
2396:
2361:
2360:
2325:
2324:
2285:
2284:
2281:
2280:
2100:
2099:
1910:
1909:
1799:
1798:
1795:
1794:
1741:Aliquot sequences
1552:Divisor functions
1525:
1524:
1497:Lucas pseudoprime
1477:Euler pseudoprime
1462:Carmichael number
1440:
1439:
1395:
1394:
1272:
1271:
1268:
1267:
1264:
1263:
1225:
1224:
1113:
1112:
1070:Square triangular
962:
961:
909:
908:
861:
860:
795:
794:
747:
746:
685:
684:
502:, OEIS Foundation
367:{\displaystyle n}
251:{\displaystyle n}
227:{\displaystyle n}
207:{\displaystyle n}
25:pernicious number
2506:
2476:
2439:
2438:
2408:Natural language
2403:
2402:
2367:
2366:
2335:Generated via a
2331:
2330:
2291:
2290:
2196:Digit-reassembly
2161:Self-descriptive
1965:
1964:
1930:
1929:
1916:
1915:
1867:Lucas–Carmichael
1857:Harmonic divisor
1805:
1804:
1731:Sparsely totient
1706:Highly cototient
1615:Multiply perfect
1605:Highly composite
1548:
1547:
1531:
1530:
1446:
1445:
1401:
1400:
1382:Telephone number
1278:
1277:
1236:
1235:
1217:Square pyramidal
1199:Stella octangula
1124:
1123:
990:
989:
981:
980:
973:Figurate numbers
968:
967:
915:
914:
867:
866:
801:
800:
753:
752:
691:
690:
595:
594:
579:
572:
565:
556:
555:
549:
548:
528:
522:
521:
510:
504:
503:
485:
474:
473:
453:
437:
421:
399:
397:
396:
391:
373:
371:
370:
365:
353:
351:
350:
345:
337:
336:
320:
318:
317:
312:
301:
300:
288:
287:
257:
255:
254:
249:
234:is prime. Every
233:
231:
230:
225:
213:
211:
210:
205:
193:
191:
190:
185:
177:
176:
150:
148:
147:
142:
124:
122:
121:
116:
108:
107:
75:
2514:
2513:
2509:
2508:
2507:
2505:
2504:
2503:
2489:
2488:
2487:
2482:
2460:
2456:Strobogrammatic
2447:
2429:
2411:
2393:
2375:
2357:
2339:
2321:
2298:
2277:
2261:
2220:Divisor-related
2215:
2175:
2126:
2096:
2033:
2017:
1996:
1963:
1936:
1924:
1906:
1818:
1817:related numbers
1791:
1768:
1735:
1726:Perfect totient
1692:
1669:
1600:Highly abundant
1542:
1521:
1453:
1436:
1408:
1391:
1377:Stirling second
1283:
1260:
1221:
1203:
1160:
1109:
1046:
1007:Centered square
975:
958:
920:
905:
872:
857:
809:
808:defined numbers
791:
758:
743:
714:Double Mersenne
700:
681:
603:
589:
587:natural numbers
583:
553:
552:
546:
529:
525:
511:
507:
486:
477:
471:
454:
450:
445:
429:
413:
406:
404:Related numbers
379:
376:
375:
359:
356:
355:
332:
328:
326:
323:
322:
296:
292:
277:
273:
271:
268:
267:
264:perfect numbers
243:
240:
239:
219:
216:
215:
199:
196:
195:
172:
168:
166:
163:
162:
160:Mersenne number
130:
127:
126:
103:
99:
97:
94:
93:
86:
81:
71:
66:
62:
58:
54:
50:
45:
17:
12:
11:
5:
2512:
2502:
2501:
2484:
2483:
2481:
2480:
2469:
2466:
2465:
2462:
2461:
2459:
2458:
2452:
2449:
2448:
2435:
2434:
2431:
2430:
2428:
2427:
2422:
2416:
2413:
2412:
2399:
2398:
2395:
2394:
2392:
2391:
2389:Sorting number
2386:
2384:Pancake number
2380:
2377:
2376:
2363:
2362:
2359:
2358:
2356:
2355:
2350:
2344:
2341:
2340:
2327:
2326:
2323:
2322:
2320:
2319:
2314:
2309:
2303:
2300:
2299:
2296:Binary numbers
2287:
2286:
2283:
2282:
2279:
2278:
2276:
2275:
2269:
2267:
2263:
2262:
2260:
2259:
2254:
2249:
2244:
2239:
2234:
2229:
2223:
2221:
2217:
2216:
2214:
2213:
2208:
2203:
2198:
2193:
2187:
2185:
2177:
2176:
2174:
2173:
2168:
2163:
2158:
2153:
2148:
2143:
2137:
2135:
2128:
2127:
2125:
2124:
2123:
2122:
2111:
2109:
2106:P-adic numbers
2102:
2101:
2098:
2097:
2095:
2094:
2093:
2092:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2041:
2039:
2035:
2034:
2032:
2031:
2025:
2023:
2022:Coding-related
2019:
2018:
2016:
2015:
2010:
2004:
2002:
1998:
1997:
1995:
1994:
1989:
1984:
1979:
1973:
1971:
1962:
1961:
1960:
1959:
1957:Multiplicative
1954:
1943:
1941:
1926:
1925:
1921:Numeral system
1912:
1911:
1908:
1907:
1905:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1823:
1820:
1819:
1801:
1800:
1797:
1796:
1793:
1792:
1790:
1789:
1784:
1778:
1776:
1770:
1769:
1767:
1766:
1761:
1756:
1751:
1745:
1743:
1737:
1736:
1734:
1733:
1728:
1723:
1718:
1713:
1711:Highly totient
1708:
1702:
1700:
1694:
1693:
1691:
1690:
1685:
1679:
1677:
1671:
1670:
1668:
1667:
1662:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1565:Almost perfect
1562:
1556:
1554:
1544:
1543:
1527:
1526:
1523:
1522:
1520:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1458:
1455:
1454:
1442:
1441:
1438:
1437:
1435:
1434:
1429:
1424:
1419:
1413:
1410:
1409:
1397:
1396:
1393:
1392:
1390:
1389:
1384:
1379:
1374:
1372:Stirling first
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1288:
1285:
1284:
1274:
1273:
1270:
1269:
1266:
1265:
1262:
1261:
1259:
1258:
1253:
1248:
1242:
1240:
1233:
1227:
1226:
1223:
1222:
1220:
1219:
1213:
1211:
1205:
1204:
1202:
1201:
1196:
1191:
1186:
1181:
1176:
1170:
1168:
1162:
1161:
1159:
1158:
1153:
1148:
1143:
1138:
1132:
1130:
1121:
1115:
1114:
1111:
1110:
1108:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1056:
1054:
1048:
1047:
1045:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
998:
996:
987:
977:
976:
964:
963:
960:
959:
957:
956:
951:
946:
941:
936:
931:
925:
922:
921:
911:
910:
907:
906:
904:
903:
898:
893:
888:
883:
877:
874:
873:
863:
862:
859:
858:
856:
855:
850:
845:
840:
835:
830:
825:
820:
814:
811:
810:
797:
796:
793:
792:
790:
789:
784:
779:
774:
769:
763:
760:
759:
749:
748:
745:
744:
742:
741:
736:
731:
726:
721:
716:
711:
705:
702:
701:
687:
686:
683:
682:
680:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
605:
604:
591:
590:
582:
581:
574:
567:
559:
551:
550:
545:978-9811244094
544:
523:
505:
475:
470:978-9811230332
469:
447:
446:
444:
441:
440:
439:
423:
410:Odious numbers
405:
402:
389:
386:
383:
363:
343:
340:
335:
331:
310:
307:
304:
299:
295:
291:
286:
283:
280:
276:
266:take the form
247:
236:Mersenne prime
223:
203:
183:
180:
175:
171:
140:
137:
134:
114:
111:
106:
102:
85:
82:
69:
64:
60:
56:
52:
48:
44:
41:
29:Hamming weight
15:
9:
6:
4:
3:
2:
2511:
2500:
2497:
2496:
2494:
2479:
2475:
2471:
2470:
2467:
2457:
2454:
2453:
2450:
2445:
2440:
2436:
2426:
2423:
2421:
2418:
2417:
2414:
2409:
2404:
2400:
2390:
2387:
2385:
2382:
2381:
2378:
2373:
2368:
2364:
2354:
2351:
2349:
2346:
2345:
2342:
2338:
2332:
2328:
2318:
2315:
2313:
2310:
2308:
2305:
2304:
2301:
2297:
2292:
2288:
2274:
2271:
2270:
2268:
2264:
2258:
2255:
2253:
2250:
2248:
2247:Polydivisible
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2224:
2222:
2218:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2188:
2186:
2183:
2178:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2138:
2136:
2133:
2129:
2121:
2118:
2117:
2116:
2113:
2112:
2110:
2107:
2103:
2091:
2088:
2087:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2042:
2040:
2036:
2030:
2027:
2026:
2024:
2020:
2014:
2011:
2009:
2006:
2005:
2003:
2001:Digit product
1999:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1974:
1972:
1970:
1966:
1958:
1955:
1953:
1950:
1949:
1948:
1945:
1944:
1942:
1940:
1935:
1931:
1927:
1922:
1917:
1913:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1837:Erdős–Nicolas
1835:
1833:
1830:
1828:
1825:
1824:
1821:
1816:
1812:
1806:
1802:
1788:
1785:
1783:
1780:
1779:
1777:
1775:
1771:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1746:
1744:
1742:
1738:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1703:
1701:
1699:
1695:
1689:
1686:
1684:
1681:
1680:
1678:
1676:
1672:
1666:
1663:
1661:
1658:
1656:
1655:Superabundant
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1557:
1555:
1553:
1549:
1545:
1541:
1537:
1532:
1528:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1459:
1456:
1452:
1447:
1443:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1414:
1411:
1407:
1402:
1398:
1388:
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1289:
1286:
1279:
1275:
1257:
1254:
1252:
1249:
1247:
1244:
1243:
1241:
1237:
1234:
1232:
1231:4-dimensional
1228:
1218:
1215:
1214:
1212:
1210:
1206:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1171:
1169:
1167:
1163:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1141:Centered cube
1139:
1137:
1134:
1133:
1131:
1129:
1125:
1122:
1120:
1119:3-dimensional
1116:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1057:
1055:
1053:
1049:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
999:
997:
995:
991:
988:
986:
985:2-dimensional
982:
978:
974:
969:
965:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
930:
929:Nonhypotenuse
927:
926:
923:
916:
912:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
878:
875:
868:
864:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
815:
812:
807:
802:
798:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
764:
761:
754:
750:
740:
737:
735:
732:
730:
727:
725:
722:
720:
717:
715:
712:
710:
707:
706:
703:
698:
692:
688:
678:
675:
673:
670:
668:
667:Perfect power
665:
663:
660:
658:
657:Seventh power
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
606:
601:
596:
592:
588:
580:
575:
573:
568:
566:
561:
560:
557:
547:
541:
537:
533:
527:
520:
516:
509:
501:
500:
494:
490:
484:
482:
480:
472:
466:
462:
458:
452:
448:
436:
432:
427:
424:
420:
416:
411:
408:
407:
401:
387:
384:
381:
361:
341:
338:
333:
329:
305:
302:
297:
293:
284:
281:
278:
274:
265:
261:
245:
237:
221:
201:
181:
178:
173:
169:
161:
156:
154:
153:Fermat number
138:
135:
132:
112:
109:
104:
100:
91:
79:
74:
68:
63:) and 9 (1001
40:
38:
34:
30:
26:
22:
21:number theory
2316:
2211:Transposable
2075:Narcissistic
1982:Digital root
1902:Super-Poulet
1862:Jordan–Pólya
1811:prime factor
1716:Noncototient
1683:Almost prime
1665:Superperfect
1640:Refactorable
1635:Quasiperfect
1610:Hyperperfect
1451:Pseudoprimes
1422:Wall–Sun–Sun
1357:Ordered Bell
1327:Fuss–Catalan
1239:non-centered
1189:Dodecahedral
1166:non-centered
1052:non-centered
954:Wolstenholme
699:× 2 ± 1
696:
695:Of the form
662:Eighth power
642:Fourth power
535:
532:Cai, Tianxin
526:
518:
508:
496:
460:
451:
426:Evil numbers
157:
87:
46:
24:
18:
2232:Extravagant
2227:Equidigital
2182:permutation
2141:Palindromic
2115:Automorphic
2013:Sum-product
1992:Sum-product
1947:Persistence
1842:Erdős–Woods
1764:Untouchable
1645:Semiperfect
1595:Hemiperfect
1256:Tesseractic
1194:Icosahedral
1174:Tetrahedral
1105:Dodecagonal
806:Recursively
677:Prime power
652:Sixth power
647:Fifth power
627:Power of 10
585:Classes of
457:Deza, Elena
262:, the even
2444:Graphemics
2317:Pernicious
2171:Undulating
2146:Pandigital
2120:Trimorphic
1721:Nontotient
1570:Arithmetic
1184:Octahedral
1085:Heptagonal
1075:Pentagonal
1060:Triangular
901:Sierpiński
823:Jacobsthal
622:Power of 3
617:Power of 2
443:References
84:Properties
2201:Parasitic
2050:Factorion
1977:Digit sum
1969:Digit sum
1787:Fortunate
1774:Primorial
1688:Semiprime
1625:Practical
1590:Descartes
1585:Deficient
1575:Betrothed
1417:Wieferich
1246:Pentatope
1209:pyramidal
1100:Decagonal
1095:Nonagonal
1090:Octagonal
1080:Hexagonal
939:Practical
886:Congruent
818:Fibonacci
782:Loeschian
385:−
339:−
303:−
282:−
179:−
59:), 7 (111
2493:Category
2273:Friedman
2206:Primeval
2151:Repdigit
2108:-related
2055:Kaprekar
2029:Meertens
1952:Additive
1939:dynamics
1847:Friendly
1759:Sociable
1749:Amicable
1560:Abundant
1540:dynamics
1362:Schröder
1352:Narayana
1322:Eulerian
1312:Delannoy
1307:Dedekind
1128:centered
994:centered
881:Amenable
838:Narayana
828:Leonardo
724:Mersenne
672:Powerful
612:Achilles
534:(2022),
459:(2021),
43:Examples
2446:related
2410:related
2374:related
2372:Sorting
2257:Vampire
2242:Harshad
2184:related
2156:Repunit
2070:Lychrel
2045:Dudeney
1897:Størmer
1892:Sphenic
1877:Regular
1815:divisor
1754:Perfect
1650:Sublime
1620:Perfect
1347:Motzkin
1302:Catalan
843:Padovan
777:Leyland
772:Idoneal
767:Hilbert
739:Woodall
491:(ed.),
435:A001969
433::
419:A000069
417::
76:in the
73:A052294
31:of its
2312:Odious
2237:Frugal
2191:Cyclic
2180:Digit-
1887:Smooth
1872:Pronic
1832:Cyclic
1809:Other
1782:Euclid
1432:Wilson
1406:Primes
1065:Square
934:Polite
896:Riesel
891:Knödel
853:Perrin
734:Thabit
719:Fermat
709:Cullen
632:Square
600:Powers
542:
467:
2353:Prime
2348:Lucky
2337:sieve
2266:Other
2252:Smith
2132:Digit
2090:Happy
2065:Keith
2038:Other
1882:Rough
1852:Giuga
1317:Euler
1179:Cubic
833:Lucas
729:Proth
125:with
37:prime
2307:Evil
1987:Self
1937:and
1827:Blum
1538:and
1342:Lobb
1297:Cake
1292:Bell
1042:Star
949:Ulam
848:Pell
637:Cube
540:ISBN
497:The
465:ISBN
431:OEIS
415:OEIS
136:>
78:OEIS
23:, a
2425:Ban
1813:or
1332:Lah
35:is
19:In
2495::
517:,
495:,
478:^
438:).
422:).
158:A
80:).
697:a
578:e
571:t
564:v
388:1
382:n
362:n
342:1
334:n
330:2
309:)
306:1
298:n
294:2
290:(
285:1
279:n
275:2
246:n
222:n
202:n
182:1
174:n
170:2
139:1
133:n
113:1
110:+
105:n
101:2
65:2
61:2
57:2
53:2
49:2
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