1989:
349:
99:
597:
1460:
conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers. The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot
380:
344:{\displaystyle {\begin{aligned}s_{0}&=k\\s_{n}&=s(s_{n-1})=\sigma _{1}(s_{n-1})-s_{n-1}\quad {\text{if}}\quad s_{n-1}>0\\s_{n}&=0\quad {\text{if}}\quad s_{n-1}=0\\s(0)&={\text{undefined}}\end{aligned}}}
385:
104:
1473:, 552, 564, 660, and 966. However, it is worth noting that 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1.
1274:
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence
1256:
1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (sequence
1332:
276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence
653:
Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is
1636:
1530:
1234:
1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (sequence
1579:
592:{\displaystyle {\begin{aligned}\sigma _{1}(10)-10&=5+2+1=8,\\\sigma _{1}(8)-8&=4+2+1=7,\\\sigma _{1}(7)-7&=1,\\\sigma _{1}(1)-1&=0.\end{aligned}}}
1599:
1550:
39:
Do all aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture)
1668:
364:
condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are
1314:
220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... (sequence
1843:
1932:
1742:
1436:
1339:
1321:
1303:
1281:
1263:
1241:
684:
612:
42:
605:
followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See (sequence
1461:
sequences have not been fully determined might be such a number. The first five candidate numbers are often called the
1978:
615:) for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:
60:
of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
1871:
2175:
1988:
1737:"Sequence A063769 (Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.)"
657:
Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called
2340:
1925:
2350:
1853:
Primzahlfamilien - Das
Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail
17:
2129:
2165:
1761:
2345:
2150:
1429:, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence
1896:
Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges)
1453:
1918:
2294:
2304:
2170:
2094:
647:
is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is
1804:
2155:
2114:
1483:
believe the
CatalanâDickson conjecture is false (so they conjecture some aliquot sequences are
1457:
1883:
2084:
1953:
1608:
1502:
1296:
25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... (sequence
2258:
2160:
1602:
1466:
633:
has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is
1555:
8:
2319:
2314:
2109:
2104:
2089:
2028:
1687:
623:
has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is
365:
2243:
2238:
2199:
2119:
2099:
1584:
1535:
1347:
1910:
1641:
2279:
2219:
1889:
1758:
1711:
1708:
1328:
Numbers whose aliquot sequence is not known to be finite or eventually periodic are
601:
Many aliquot sequences terminate at zero; all such sequences necessarily end with a
2309:
2284:
2204:
2124:
2008:
1968:
1821:
1484:
77:
1895:
1756:
2289:
2214:
2208:
2145:
2043:
2033:
1963:
1866:
1671:
640:
630:
1310:
Numbers whose aliquot sequence terminates in a cycle with length at least 2 are
69:
2299:
2253:
2079:
2063:
2053:
2023:
1848:. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201â206.
1675:
1496:
1480:
1476:
1289:
620:
57:
1825:
2334:
2248:
2048:
2038:
2018:
643:
has a repeating aliquot sequence of period 3 or greater. (Sometimes the term
2263:
2180:
2058:
2003:
1973:
1470:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
602:
1842:
Manuel Benito; Wolfgang
CreyaufmĂźller; Juan Luis Varona; Paul Zimmermann.
1732:
1370:
1366:
1362:
1346:
A number that is never the successor in an aliquot sequence is called an
89:
49:
56:
is a sequence of positive integers in which each term is the sum of the
1449:
1358:
1354:
31:
2013:
1877:
1766:
1716:
1248:
The final terms (excluding 1) of the aliquot sequences that start at
1867:
Current status of aliquot sequences with start term below 2 million
1783:
1958:
1490:
1292:, other than perfect numbers themselves (6, 28, 496, ...), are
2229:
1901:
1736:
1431:
1334:
1316:
1298:
1276:
1258:
1236:
1030:
30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0
688:
607:
1706:
1818:
Distributed cycle detection in large-scale sparse graphs
1940:
1288:
Numbers whose aliquot sequence known to terminate in a
1162:
42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0
1820:, SimpĂłsio Brasileiro de Pesquisa Operacional (SBPO),
1731:
1644:
1611:
1587:
1558:
1538:
1505:
383:
102:
1845:
Aliquot
Sequence 3630 Ends After Reaching 100 Digits
1816:
Rocha, Rodrigo
Caetano; Thatte, Bhalchandra (2015),
1270:Numbers whose aliquot sequence terminates in 1 are
1226:The lengths of the aliquot sequences that start at
368:, the limit of these sequences are usually 0 or 6.
1662:
1630:
1593:
1573:
1544:
1524:
591:
343:
2332:
649:1264460, 1547860, 1727636, 1305184, 1264460, ...
1638:represent sociable numbers within the interval
33:
1491:Systematically searching for aliquot sequences
1926:
1781:
1670:. Two special cases are loops that represent
1495:The aliquot sequence can be represented as a
1443:
1815:
371:For example, the aliquot sequence of 10 is
1933:
1919:
1581:denotes the sum of the proper divisors of
72:sequence starting with a positive integer
63:
1902:Active research site on aliquot sequences
1743:On-Line Encyclopedia of Integer Sequences
1805:What do we know about aliquot sequences?
1782:CreyaufmĂźller, Wolfgang (May 24, 2014).
1674:and cycles of length two that represent
76:can be defined formally in terms of the
1762:"Catalan's Aliquot Sequence Conjecture"
43:(more unsolved problems in mathematics)
14:
2333:
1890:Forum on calculating aliquot sequences
1914:
1757:
1707:
1941:Divisibility-based sets of integers
24:
25:
2362:
1979:Fundamental theorem of arithmetic
1860:
1855:. Stuttgart 2000 (3rd ed.), 327p.
1700:
1987:
1456:, sometimes called the Catalanâ
666:Aliquot sequences from 0 to 47
286:
280:
233:
227:
34:Unsolved problem in mathematics
27:Mathematical recursive sequence
1809:
1797:
1775:
1750:
1725:
1657:
1645:
1568:
1562:
1074:34, 20, 22, 14, 10, 8, 7, 1, 0
566:
560:
524:
518:
464:
458:
404:
398:
322:
316:
205:
186:
170:
151:
13:
1:
1835:
1206:46, 26, 16, 15, 9, 4, 3, 1, 0
7:
1681:
10:
2367:
1733:Sloane, N. J. A.
1444:CatalanâDickson conjecture
1118:38, 22, 14, 10, 8, 7, 1, 0
920:20, 22, 14, 10, 8, 7, 1, 0
2272:
2228:
2189:
2176:Superior highly composite
2138:
2072:
1996:
1985:
1946:
1826:10.13140/RG.2.1.1233.8640
1195:45, 33, 15, 9, 4, 3, 1, 0
986:26, 16, 15, 9, 4, 3, 1, 0
832:12, 16, 15, 9, 4, 3, 1, 0
2073:Constrained divisor sums
1880:(Wolfgang CreyaufmĂźller)
1872:Tables of Aliquot Cycles
1693:
1487:above (i.e., diverge)).
78:sum-of-divisors function
1631:{\displaystyle G_{n,s}}
1525:{\displaystyle G_{n,s}}
655:95, 25, 6, 6, 6, 6, ...
635:220, 284, 220, 284, ...
64:Definition and overview
1664:
1632:
1595:
1575:
1546:
1532:, for a given integer
1526:
942:22, 14, 10, 8, 7, 1, 0
593:
345:
96:in the following way:
1954:Integer factorization
1904:(Jean-Luc Garambois)
1665:
1633:
1596:
1576:
1547:
1527:
1063:33, 15, 9, 4, 3, 1, 0
876:16, 15, 9, 4, 3, 1, 0
594:
346:
2341:Arithmetic functions
1886:(Christophe Clavier)
1642:
1609:
1585:
1574:{\displaystyle s(k)}
1556:
1536:
1503:
1184:44, 40, 50, 43, 1, 0
964:24, 36, 55, 17, 1, 0
676:Aliquot sequence of
381:
100:
2351:Arithmetic dynamics
2166:Colossally abundant
1997:Factorization forms
1688:Arithmetic dynamics
667:
2151:Primitive abundant
2139:With many divisors
1851:W. CreyaufmĂźller.
1759:Weisstein, Eric W.
1746:. OEIS Foundation.
1712:"Aliquot Sequence"
1709:Weisstein, Eric W.
1660:
1628:
1591:
1571:
1542:
1522:
1348:untouchable number
854:14, 10, 8, 7, 1, 0
665:
589:
587:
341:
339:
2328:
2327:
1884:Aliquot sequences
1874:(J.O.M. Pedersen)
1594:{\displaystyle k}
1545:{\displaystyle n}
1224:
1223:
865:15, 9, 4, 3, 1, 0
335:
284:
231:
16:(Redirected from
2358:
2346:Divisor function
2305:Harmonic divisor
2191:Aliquot sequence
2171:Highly composite
2095:Multiply perfect
1991:
1969:Divisor function
1935:
1928:
1921:
1912:
1911:
1907:
1898:(Karsten Bonath)
1829:
1828:
1813:
1807:
1801:
1795:
1794:
1792:
1790:
1779:
1773:
1772:
1771:
1754:
1748:
1747:
1729:
1723:
1722:
1721:
1704:
1669:
1667:
1666:
1663:{\displaystyle }
1661:
1637:
1635:
1634:
1629:
1627:
1626:
1600:
1598:
1597:
1592:
1580:
1578:
1577:
1572:
1551:
1549:
1548:
1543:
1531:
1529:
1528:
1523:
1521:
1520:
1434:
1337:
1319:
1301:
1279:
1261:
1251:
1239:
1229:
1140:40, 50, 43, 1, 0
1096:36, 55, 17, 1, 0
898:18, 21, 11, 1, 0
691:
679:
673:
668:
664:
659:aspiring numbers
656:
650:
636:
626:
610:
598:
596:
595:
590:
588:
559:
558:
517:
516:
457:
456:
397:
396:
374:
363:
350:
348:
347:
342:
340:
336:
333:
302:
301:
285:
282:
269:
268:
249:
248:
232:
229:
226:
225:
204:
203:
185:
184:
169:
168:
140:
139:
116:
115:
95:
87:
75:
54:aliquot sequence
35:
21:
2366:
2365:
2361:
2360:
2359:
2357:
2356:
2355:
2331:
2330:
2329:
2324:
2268:
2224:
2185:
2156:Highly abundant
2134:
2115:Unitary perfect
2068:
1992:
1983:
1964:Unitary divisor
1942:
1939:
1905:
1892:(MersenneForum)
1863:
1858:
1838:
1833:
1832:
1814:
1810:
1803:A. S. Mosunov,
1802:
1798:
1788:
1786:
1780:
1776:
1755:
1751:
1730:
1726:
1705:
1701:
1696:
1684:
1672:perfect numbers
1643:
1640:
1639:
1616:
1612:
1610:
1607:
1606:
1586:
1583:
1582:
1557:
1554:
1553:
1537:
1534:
1533:
1510:
1506:
1504:
1501:
1500:
1493:
1446:
1430:
1333:
1315:
1297:
1275:
1257:
1249:
1235:
1227:
683:
677:
671:
654:
648:
645:sociable number
641:sociable number
634:
631:amicable number
625:6, 6, 6, 6, ...
624:
606:
586:
585:
575:
554:
550:
547:
546:
533:
512:
508:
505:
504:
473:
452:
448:
445:
444:
413:
392:
388:
384:
382:
379:
378:
372:
361:
352:
338:
337:
332:
325:
310:
309:
291:
287:
281:
270:
264:
260:
257:
256:
238:
234:
228:
215:
211:
193:
189:
180:
176:
158:
154:
141:
135:
131:
128:
127:
117:
111:
107:
103:
101:
98:
97:
93:
86:
80:
73:
66:
58:proper divisors
46:
45:
40:
37:
28:
23:
22:
15:
12:
11:
5:
2364:
2354:
2353:
2348:
2343:
2326:
2325:
2323:
2322:
2317:
2312:
2307:
2302:
2297:
2292:
2287:
2282:
2276:
2274:
2270:
2269:
2267:
2266:
2261:
2256:
2251:
2246:
2241:
2235:
2233:
2226:
2225:
2223:
2222:
2217:
2212:
2202:
2196:
2194:
2187:
2186:
2184:
2183:
2178:
2173:
2168:
2163:
2158:
2153:
2148:
2142:
2140:
2136:
2135:
2133:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2085:Almost perfect
2082:
2076:
2074:
2070:
2069:
2067:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
2000:
1998:
1994:
1993:
1986:
1984:
1982:
1981:
1976:
1971:
1966:
1961:
1956:
1950:
1948:
1944:
1943:
1938:
1937:
1930:
1923:
1915:
1909:
1908:
1899:
1893:
1887:
1881:
1875:
1869:
1862:
1861:External links
1859:
1857:
1856:
1849:
1839:
1837:
1834:
1831:
1830:
1808:
1796:
1774:
1749:
1724:
1698:
1697:
1695:
1692:
1691:
1690:
1683:
1680:
1676:amicable pairs
1659:
1656:
1653:
1650:
1647:
1625:
1622:
1619:
1615:
1590:
1570:
1567:
1564:
1561:
1541:
1519:
1516:
1513:
1509:
1497:directed graph
1492:
1489:
1445:
1442:
1441:
1440:
1344:
1343:
1326:
1325:
1308:
1307:
1290:perfect number
1286:
1285:
1268:
1267:
1246:
1245:
1222:
1221:
1218:
1215:
1211:
1210:
1207:
1204:
1200:
1199:
1196:
1193:
1189:
1188:
1185:
1182:
1178:
1177:
1174:
1171:
1167:
1166:
1163:
1160:
1156:
1155:
1152:
1149:
1145:
1144:
1141:
1138:
1134:
1133:
1130:
1127:
1123:
1122:
1119:
1116:
1112:
1111:
1108:
1105:
1101:
1100:
1097:
1094:
1090:
1089:
1086:
1083:
1079:
1078:
1075:
1072:
1068:
1067:
1064:
1061:
1057:
1056:
1053:
1050:
1046:
1045:
1042:
1039:
1035:
1034:
1031:
1028:
1024:
1023:
1020:
1017:
1013:
1012:
1009:
1006:
1002:
1001:
998:
995:
991:
990:
987:
984:
980:
979:
976:
973:
969:
968:
965:
962:
958:
957:
954:
951:
947:
946:
943:
940:
936:
935:
932:
929:
925:
924:
921:
918:
914:
913:
910:
907:
903:
902:
899:
896:
892:
891:
888:
885:
881:
880:
877:
874:
870:
869:
866:
863:
859:
858:
855:
852:
848:
847:
844:
841:
837:
836:
833:
830:
826:
825:
822:
819:
815:
814:
811:
810:10, 8, 7, 1, 0
808:
804:
803:
800:
797:
793:
792:
789:
786:
782:
781:
778:
775:
771:
770:
767:
764:
760:
759:
756:
753:
749:
748:
745:
742:
738:
737:
734:
731:
727:
726:
723:
720:
716:
715:
712:
709:
705:
704:
701:
698:
694:
693:
680:
674:
663:
662:
651:
637:
627:
621:perfect number
584:
581:
578:
576:
574:
571:
568:
565:
562:
557:
553:
549:
548:
545:
542:
539:
536:
534:
532:
529:
526:
523:
520:
515:
511:
507:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
474:
472:
469:
466:
463:
460:
455:
451:
447:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
414:
412:
409:
406:
403:
400:
395:
391:
387:
386:
373:10, 8, 7, 1, 0
356:
331:
328:
326:
324:
321:
318:
315:
312:
311:
308:
305:
300:
297:
294:
290:
279:
276:
273:
271:
267:
263:
259:
258:
255:
252:
247:
244:
241:
237:
224:
221:
218:
214:
210:
207:
202:
199:
196:
192:
188:
183:
179:
175:
172:
167:
164:
161:
157:
153:
150:
147:
144:
142:
138:
134:
130:
129:
126:
123:
120:
118:
114:
110:
106:
105:
84:
65:
62:
41:
38:
32:
26:
9:
6:
4:
3:
2:
2363:
2352:
2349:
2347:
2344:
2342:
2339:
2338:
2336:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2277:
2275:
2271:
2265:
2262:
2260:
2259:Polydivisible
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2236:
2234:
2231:
2227:
2221:
2218:
2216:
2213:
2210:
2206:
2203:
2201:
2198:
2197:
2195:
2192:
2188:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2161:Superabundant
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2143:
2141:
2137:
2131:
2130:ErdĹsâNicolas
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2077:
2075:
2071:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2039:Perfect power
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2001:
1999:
1995:
1990:
1980:
1977:
1975:
1972:
1970:
1967:
1965:
1962:
1960:
1957:
1955:
1952:
1951:
1949:
1945:
1936:
1931:
1929:
1924:
1922:
1917:
1916:
1913:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1868:
1865:
1864:
1854:
1850:
1847:
1846:
1841:
1840:
1827:
1823:
1819:
1812:
1806:
1800:
1785:
1784:"Lehmer Five"
1778:
1769:
1768:
1763:
1760:
1753:
1745:
1744:
1738:
1734:
1728:
1719:
1718:
1713:
1710:
1703:
1699:
1689:
1686:
1685:
1679:
1677:
1673:
1654:
1651:
1648:
1623:
1620:
1617:
1613:
1604:
1588:
1565:
1559:
1539:
1517:
1514:
1511:
1507:
1498:
1488:
1486:
1482:
1478:
1474:
1472:
1468:
1465:(named after
1464:
1459:
1455:
1451:
1448:An important
1438:
1433:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1353:
1352:
1351:
1349:
1341:
1336:
1331:
1330:
1329:
1323:
1318:
1313:
1312:
1311:
1305:
1300:
1295:
1294:
1293:
1291:
1283:
1278:
1273:
1272:
1271:
1265:
1260:
1255:
1254:
1253:
1243:
1238:
1233:
1232:
1231:
1219:
1216:
1213:
1212:
1208:
1205:
1202:
1201:
1197:
1194:
1191:
1190:
1186:
1183:
1180:
1179:
1175:
1172:
1169:
1168:
1164:
1161:
1158:
1157:
1153:
1150:
1147:
1146:
1142:
1139:
1136:
1135:
1131:
1128:
1125:
1124:
1120:
1117:
1114:
1113:
1109:
1106:
1103:
1102:
1098:
1095:
1092:
1091:
1087:
1084:
1081:
1080:
1076:
1073:
1070:
1069:
1065:
1062:
1059:
1058:
1054:
1051:
1048:
1047:
1043:
1040:
1037:
1036:
1032:
1029:
1026:
1025:
1021:
1018:
1015:
1014:
1010:
1007:
1004:
1003:
999:
996:
993:
992:
988:
985:
982:
981:
977:
974:
971:
970:
966:
963:
960:
959:
955:
952:
949:
948:
944:
941:
938:
937:
933:
930:
927:
926:
922:
919:
916:
915:
911:
908:
905:
904:
900:
897:
894:
893:
889:
886:
883:
882:
878:
875:
872:
871:
867:
864:
861:
860:
856:
853:
850:
849:
845:
842:
839:
838:
834:
831:
828:
827:
823:
820:
817:
816:
812:
809:
806:
805:
801:
799:9, 4, 3, 1, 0
798:
795:
794:
790:
787:
784:
783:
779:
776:
773:
772:
768:
765:
762:
761:
757:
754:
751:
750:
746:
743:
740:
739:
735:
732:
729:
728:
724:
721:
718:
717:
713:
710:
707:
706:
702:
699:
696:
695:
690:
686:
681:
675:
670:
669:
660:
652:
646:
642:
638:
632:
628:
622:
618:
617:
616:
614:
609:
604:
599:
582:
579:
577:
572:
569:
563:
555:
551:
543:
540:
537:
535:
530:
527:
521:
513:
509:
501:
498:
495:
492:
489:
486:
483:
480:
477:
475:
470:
467:
461:
453:
449:
441:
438:
435:
432:
429:
426:
423:
420:
417:
415:
410:
407:
401:
393:
389:
376:
369:
367:
359:
355:
329:
327:
319:
313:
306:
303:
298:
295:
292:
288:
277:
274:
272:
265:
261:
253:
250:
245:
242:
239:
235:
222:
219:
216:
212:
208:
200:
197:
194:
190:
181:
177:
173:
165:
162:
159:
155:
148:
145:
143:
136:
132:
124:
121:
119:
112:
108:
91:
83:
79:
71:
61:
59:
55:
51:
44:
30:
19:
2320:Superperfect
2315:Refactorable
2190:
2110:Superperfect
2105:Hyperperfect
2090:Quasiperfect
1974:Prime factor
1878:Aliquot Page
1852:
1844:
1817:
1811:
1799:
1787:. Retrieved
1777:
1765:
1752:
1740:
1727:
1715:
1702:
1494:
1475:
1462:
1447:
1417:, 262, 268,
1345:
1327:
1309:
1287:
1269:
1247:
1225:
1129:39, 17, 1, 0
1085:35, 13, 1, 0
1052:32, 31, 1, 0
997:27, 13, 1, 0
931:21, 11, 1, 0
658:
644:
603:prime number
600:
377:
370:
357:
353:
81:
67:
53:
47:
29:
2244:Extravagant
2239:Equidigital
2200:Untouchable
2120:Semiperfect
2100:Hemiperfect
2029:Square-free
1906:(in French)
1467:D.H. Lehmer
1463:Lehmer five
90:aliquot sum
50:mathematics
18:Lehmer five
2335:Categories
2280:Arithmetic
2273:Other sets
2232:-dependent
1836:References
1450:conjecture
788:8, 7, 1, 0
744:4, 3, 1, 0
366:convergent
2310:Descartes
2285:Deficient
2220:Betrothed
2125:Practical
2014:Semiprime
2009:Composite
1767:MathWorld
1717:MathWorld
1485:unbounded
1481:Selfridge
570:−
552:σ
528:−
510:σ
468:−
450:σ
408:−
390:σ
375:because:
334:undefined
296:−
243:−
220:−
209:−
198:−
178:σ
163:−
92:function
2295:Solitary
2290:Friendly
2215:Sociable
2205:Amicable
2193:-related
2146:Abundant
2044:Achilles
2034:Powerful
1947:Overview
1789:June 14,
1682:See also
1552:, where
1217:47, 1, 0
1173:43, 1, 0
1151:41, 1, 0
1107:37, 1, 0
1041:31, 1, 0
1019:29, 1, 0
953:23, 1, 0
909:19, 1, 0
887:17, 1, 0
843:13, 1, 0
821:11, 1, 0
682:Length (
2300:Sublime
2254:Harshad
2080:Perfect
2064:Unusual
2054:Regular
2024:Sphenic
1959:Divisor
1735:(ed.).
1458:Dickson
1454:Catalan
1452:due to
1435:in the
1432:A005114
1338:in the
1335:A131884
1320:in the
1317:A121507
1302:in the
1299:A063769
1280:in the
1277:A080907
1262:in the
1259:A115350
1240:in the
1237:A044050
777:7, 1, 0
755:5, 1, 0
733:3, 1, 0
722:2, 1, 0
689:A098007
687::
611:in the
608:A080907
351:If the
88:or the
70:aliquot
2249:Frugal
2209:Triple
2049:Smooth
2019:Pronic
1603:Cycles
2264:Smith
2181:Weird
2059:Rough
2004:Prime
1694:Notes
975:25, 6
52:, an
2230:Base
1791:2015
1741:The
1479:and
1437:OEIS
1340:OEIS
1322:OEIS
1304:OEIS
1282:OEIS
1264:OEIS
1252:are
1242:OEIS
1230:are
711:1, 0
685:OEIS
613:OEIS
251:>
68:The
1822:doi
1605:in
1477:Guy
1471:276
1469:):
1427:290
1423:288
1419:276
1415:248
1411:246
1407:238
1403:216
1399:210
1395:206
1391:188
1387:162
1383:146
1379:124
1375:120
1214:47
1203:46
1192:45
1181:44
1170:43
1165:15
1159:42
1148:41
1137:40
1126:39
1115:38
1104:37
1093:36
1082:35
1071:34
1060:33
1049:32
1038:31
1033:16
1027:30
1016:29
1005:28
994:27
983:26
972:25
961:24
950:23
939:22
928:21
917:20
906:19
895:18
884:17
873:16
862:15
851:14
840:13
829:12
818:11
807:10
629:An
362:= 0
48:In
2337::
1764:.
1739:.
1714:.
1678:.
1601:.
1499:,
1425:,
1421:,
1413:,
1409:,
1405:,
1401:,
1397:,
1393:,
1389:,
1385:,
1381:,
1377:,
1373:,
1371:96
1369:,
1367:88
1365:,
1363:52
1361:,
1357:,
1350:.
1220:3
1209:9
1198:8
1187:6
1176:3
1154:3
1143:5
1132:4
1121:8
1110:3
1099:5
1088:4
1077:9
1066:7
1055:4
1044:3
1022:3
1011:1
1008:28
1000:4
989:8
978:2
967:6
956:3
945:7
934:4
923:8
912:3
901:5
890:3
879:7
868:6
857:6
846:3
835:8
824:3
813:5
802:5
796:9
791:4
785:8
780:3
774:7
769:1
763:6
758:3
752:5
747:4
741:4
736:3
730:3
725:3
719:2
714:2
708:1
703:1
697:0
692:)
639:A
619:A
583:0.
411:10
402:10
360:-1
283:if
230:if
2211:)
2207:(
1934:e
1927:t
1920:v
1824::
1793:.
1770:.
1720:.
1658:]
1655:n
1652:,
1649:1
1646:[
1624:s
1621:,
1618:n
1614:G
1589:k
1569:)
1566:k
1563:(
1560:s
1540:n
1518:s
1515:,
1512:n
1508:G
1439:)
1359:5
1355:2
1342:)
1324:)
1306:)
1284:)
1266:)
1250:n
1244:)
1228:n
766:6
700:0
678:n
672:n
661:.
580:=
573:1
567:)
564:1
561:(
556:1
544:,
541:1
538:=
531:7
525:)
522:7
519:(
514:1
502:,
499:7
496:=
493:1
490:+
487:2
484:+
481:4
478:=
471:8
465:)
462:8
459:(
454:1
442:,
439:8
436:=
433:1
430:+
427:2
424:+
421:5
418:=
405:)
399:(
394:1
358:n
354:s
330:=
323:)
320:0
317:(
314:s
307:0
304:=
299:1
293:n
289:s
278:0
275:=
266:n
262:s
254:0
246:1
240:n
236:s
223:1
217:n
213:s
206:)
201:1
195:n
191:s
187:(
182:1
174:=
171:)
166:1
160:n
156:s
152:(
149:s
146:=
137:n
133:s
125:k
122:=
113:0
109:s
94:s
85:1
82:Ď
74:k
36::
20:)
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