290:, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has
20:
2504:
479:
118:
329:
Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and
Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positive
207:
321: + 1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.
606:
261:
599:
418:
23:
Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2Ă—3Ă—11Ă—23Ă—31). Therefore the product, 47058, is primary pseudoperfect.
1406:
592:
389:
53:
1401:
1416:
1396:
510:
508:
Sondow, Jonathan; MacMillan, Kieren (2017), "Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation",
2109:
1689:
380:
1411:
2195:
149:
1511:
1861:
1180:
1896:
1866:
1541:
1531:
2533:
2037:
1451:
1185:
1165:
1727:
1891:
2528:
1986:
1609:
1366:
1175:
1157:
1051:
1041:
1031:
483:
1871:
2114:
1659:
1280:
1066:
1061:
1056:
1046:
1023:
1099:
1356:
2225:
2190:
1976:
1886:
1760:
1735:
1644:
1634:
1246:
1228:
1148:
269:
2485:
1755:
1629:
1260:
1036:
816:
743:
353:
1740:
1594:
1521:
676:
279:
It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any
2449:
2089:
2382:
2276:
2240:
1981:
1704:
1684:
1501:
1170:
958:
930:
287:
280:
2104:
1968:
1963:
1931:
1694:
1669:
1664:
1639:
1569:
1565:
1496:
1386:
1218:
1014:
983:
2503:
2507:
2261:
2256:
2170:
2144:
2042:
2021:
1793:
1674:
1624:
1546:
1516:
1456:
1223:
1203:
1134:
847:
537:
519:
401:
233:
1391:
570:
2401:
2346:
2200:
2175:
2149:
1926:
1604:
1599:
1526:
1506:
1491:
1213:
1195:
1114:
1104:
1089:
867:
852:
567:
44:
2437:
2230:
1816:
1788:
1778:
1770:
1654:
1619:
1614:
1581:
1275:
1238:
1129:
1124:
1119:
1109:
1081:
968:
920:
915:
872:
811:
529:
492:
393:
349:
273:
2413:
2302:
2235:
2161:
2084:
2058:
1876:
1589:
1446:
1381:
1351:
1341:
1336:
1002:
910:
857:
701:
641:
533:
2418:
2286:
2271:
2135:
2099:
2074:
1950:
1921:
1906:
1783:
1679:
1649:
1376:
1331:
1208:
806:
801:
796:
768:
753:
666:
651:
629:
616:
286:
The prime factors of primary pseudoperfect numbers sometimes may provide solutions to
2522:
2341:
2325:
2266:
2220:
1916:
1901:
1811:
1536:
1094:
963:
925:
882:
763:
748:
738:
696:
686:
661:
541:
32:
555:
2377:
2366:
2281:
2119:
2094:
2011:
1911:
1881:
1856:
1840:
1745:
1712:
1461:
1435:
1346:
1285:
862:
758:
691:
671:
646:
365:
124:
497:
254:, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086 (sequence
2336:
2211:
2016:
1480:
1371:
1326:
1321:
1071:
978:
877:
706:
681:
656:
251:
28:
19:
356:
6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).
268:
The first four of these numbers are one less than the corresponding numbers in
2473:
2454:
1750:
1361:
559:
415:
Butske, William; Jaje, Lynda M.; Mayernik, Daniel R. (2000), "On the equation
336:
up to 8, there exists exactly one primary pseudoperfect number with precisely
299:
247:
243:
584:
2079:
2006:
1998:
1803:
1717:
835:
575:
378:
Anne, Premchand (1998), "Egyptian fractions and the inheritance problem",
2180:
2185:
1844:
474:{\displaystyle \scriptstyle \sum _{p|N}{\frac {1}{p}}+{\frac {1}{N}}=1}
405:
330:
397:
524:
113:{\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,}
2471:
2435:
2399:
2363:
2323:
1948:
1837:
1563:
1478:
1433:
1310:
1000:
947:
899:
833:
785:
723:
627:
588:
256:
216: = 2, this expression gives a representation for
481:, pseudoperfect numbers, and perfectly weighted graphs",
422:
344:
th known primary pseudoperfect number. Those with 2 ≤
421:
152:
56:
202:{\displaystyle 1+\sum _{p\,|\;\!N}{\frac {N}{p}}=N.}
2295:
2249:
2209:
2160:
2134:
2067:
2051:
2030:
1997:
1962:
1802:
1769:
1726:
1703:
1580:
1268:
1259:
1237:
1194:
1156:
1147:
1080:
1022:
1013:
473:
239:The eight known primary pseudoperfect numbers are
201:
143:is a primary pseudoperfect number if it satisfies
112:
174:
85:
224:. Therefore, each primary pseudoperfect number
600:
8:
212:Except for the primary pseudoperfect number
2468:
2432:
2396:
2360:
2320:
1994:
1959:
1945:
1834:
1577:
1560:
1475:
1430:
1307:
1265:
1153:
1019:
1010:
997:
944:
901:Possessing a specific set of other numbers
896:
830:
782:
720:
624:
607:
593:
585:
173:
84:
523:
496:
454:
441:
431:
427:
420:
180:
168:
167:
163:
151:
91:
79:
78:
74:
57:
55:
18:
340:(distinct) prime factors, namely, the
313:is one less than a prime number, then
7:
534:10.4169/amer.math.monthly.124.3.232
390:Mathematical Association of America
220:as the sum of distinct divisors of
309:If a primary pseudoperfect number
14:
511:The American Mathematical Monthly
302:that the same is true for larger
2502:
2110:Perfect digit-to-digit invariant
381:The College Mathematics Journal
283:primary pseudoperfect numbers.
123:where the sum is over only the
571:"Primary Pseudoperfect Number"
432:
169:
80:
1:
949:Expressible via specific sums
498:10.1090/S0025-5718-99-01088-1
556:Primary Pseudoperfect Number
41:primary pseudoperfect number
2038:Multiplicative digital root
2550:
484:Mathematics of Computation
2498:
2481:
2467:
2445:
2431:
2409:
2395:
2373:
2359:
2332:
2319:
2115:Perfect digital invariant
1958:
1944:
1852:
1833:
1690:Superior highly composite
1576:
1559:
1487:
1474:
1442:
1429:
1317:
1306:
1009:
996:
954:
943:
906:
895:
843:
829:
792:
781:
734:
719:
637:
623:
1728:Euler's totient function
1512:Euler–Jacobi pseudoprime
787:Other polynomial numbers
232: = 2) is also
1542:Somer–Lucas pseudoprime
1532:Lucas–Carmichael number
1367:Lazy caterer's sequence
294:primes in it, for each
1417:Wedderburn–Etherington
817:Lucky numbers of Euler
475:
354:arithmetic progression
203:
114:
31:, and particularly in
24:
1705:Prime omega functions
1522:Frobenius pseudoprime
1312:Combinatorial numbers
1181:Centered dodecahedral
974:Primary pseudoperfect
476:
204:
115:
22:
2164:-composition related
1964:Arithmetic functions
1566:Arithmetic functions
1502:Elliptic pseudoprime
1186:Centered icosahedral
1166:Centered tetrahedral
419:
270:Sylvester's sequence
150:
54:
43:if it satisfies the
2090:Kaprekar's constant
1610:Colossally abundant
1497:Catalan pseudoprime
1397:Schröder–Hipparchus
1176:Centered octahedral
1052:Centered heptagonal
1042:Centered pentagonal
1032:Centered triangular
632:and related numbers
272:, but then the two
2534:Egyptian fractions
2508:Mathematics portal
2450:Aronson's sequence
2196:Smarandache–Wellin
1953:-dependent numbers
1660:Primitive abundant
1547:Strong pseudoprime
1537:Perrin pseudoprime
1517:Fermat pseudoprime
1457:Wolstenholme prime
1281:Squared triangular
1067:Centered decagonal
1062:Centered nonagonal
1057:Centered octagonal
1047:Centered hexagonal
568:Weisstein, Eric W.
471:
470:
440:
348:≤ 8, when reduced
199:
179:
110:
90:
25:
2529:Integer sequences
2516:
2515:
2494:
2493:
2463:
2462:
2427:
2426:
2391:
2390:
2355:
2354:
2315:
2314:
2311:
2310:
2130:
2129:
1940:
1939:
1829:
1828:
1825:
1824:
1771:Aliquot sequences
1582:Divisor functions
1555:
1554:
1527:Lucas pseudoprime
1507:Euler pseudoprime
1492:Carmichael number
1470:
1469:
1425:
1424:
1302:
1301:
1298:
1297:
1294:
1293:
1255:
1254:
1143:
1142:
1100:Square triangular
992:
991:
939:
938:
891:
890:
825:
824:
777:
776:
715:
714:
462:
449:
423:
188:
159:
99:
70:
65:
45:Egyptian fraction
2541:
2506:
2469:
2438:Natural language
2433:
2397:
2365:Generated via a
2361:
2321:
2226:Digit-reassembly
2191:Self-descriptive
1995:
1960:
1946:
1897:Lucas–Carmichael
1887:Harmonic divisor
1835:
1761:Sparsely totient
1736:Highly cototient
1645:Multiply perfect
1635:Highly composite
1578:
1561:
1476:
1431:
1412:Telephone number
1308:
1266:
1247:Square pyramidal
1229:Stella octangula
1154:
1020:
1011:
1003:Figurate numbers
998:
945:
897:
831:
783:
721:
625:
609:
602:
595:
586:
581:
580:
544:
527:
501:
500:
480:
478:
477:
472:
463:
455:
450:
442:
439:
435:
408:
317: Ă— (
259:
208:
206:
205:
200:
189:
181:
178:
172:
119:
117:
116:
111:
100:
92:
89:
83:
66:
58:
2549:
2548:
2544:
2543:
2542:
2540:
2539:
2538:
2519:
2518:
2517:
2512:
2490:
2486:Strobogrammatic
2477:
2459:
2441:
2423:
2405:
2387:
2369:
2351:
2328:
2307:
2291:
2250:Divisor-related
2245:
2205:
2156:
2126:
2063:
2047:
2026:
1993:
1966:
1954:
1936:
1848:
1847:related numbers
1821:
1798:
1765:
1756:Perfect totient
1722:
1699:
1630:Highly abundant
1572:
1551:
1483:
1466:
1438:
1421:
1407:Stirling second
1313:
1290:
1251:
1233:
1190:
1139:
1076:
1037:Centered square
1005:
988:
950:
935:
902:
887:
839:
838:defined numbers
821:
788:
773:
744:Double Mersenne
730:
711:
633:
619:
617:natural numbers
613:
566:
565:
552:
507:
417:
416:
414:
398:10.2307/2687685
377:
374:
362:
327:
255:
148:
147:
137:
52:
51:
17:
12:
11:
5:
2547:
2545:
2537:
2536:
2531:
2521:
2520:
2514:
2513:
2511:
2510:
2499:
2496:
2495:
2492:
2491:
2489:
2488:
2482:
2479:
2478:
2472:
2465:
2464:
2461:
2460:
2458:
2457:
2452:
2446:
2443:
2442:
2436:
2429:
2428:
2425:
2424:
2422:
2421:
2419:Sorting number
2416:
2414:Pancake number
2410:
2407:
2406:
2400:
2393:
2392:
2389:
2388:
2386:
2385:
2380:
2374:
2371:
2370:
2364:
2357:
2356:
2353:
2352:
2350:
2349:
2344:
2339:
2333:
2330:
2329:
2326:Binary numbers
2324:
2317:
2316:
2313:
2312:
2309:
2308:
2306:
2305:
2299:
2297:
2293:
2292:
2290:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2253:
2251:
2247:
2246:
2244:
2243:
2238:
2233:
2228:
2223:
2217:
2215:
2207:
2206:
2204:
2203:
2198:
2193:
2188:
2183:
2178:
2173:
2167:
2165:
2158:
2157:
2155:
2154:
2153:
2152:
2141:
2139:
2136:P-adic numbers
2132:
2131:
2128:
2127:
2125:
2124:
2123:
2122:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2071:
2069:
2065:
2064:
2062:
2061:
2055:
2053:
2052:Coding-related
2049:
2048:
2046:
2045:
2040:
2034:
2032:
2028:
2027:
2025:
2024:
2019:
2014:
2009:
2003:
2001:
1992:
1991:
1990:
1989:
1987:Multiplicative
1984:
1973:
1971:
1956:
1955:
1951:Numeral system
1949:
1942:
1941:
1938:
1937:
1935:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1853:
1850:
1849:
1838:
1831:
1830:
1827:
1826:
1823:
1822:
1820:
1819:
1814:
1808:
1806:
1800:
1799:
1797:
1796:
1791:
1786:
1781:
1775:
1773:
1767:
1766:
1764:
1763:
1758:
1753:
1748:
1743:
1741:Highly totient
1738:
1732:
1730:
1724:
1723:
1721:
1720:
1715:
1709:
1707:
1701:
1700:
1698:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1595:Almost perfect
1592:
1586:
1584:
1574:
1573:
1564:
1557:
1556:
1553:
1552:
1550:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1488:
1485:
1484:
1479:
1472:
1471:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1443:
1440:
1439:
1434:
1427:
1426:
1423:
1422:
1420:
1419:
1414:
1409:
1404:
1402:Stirling first
1399:
1394:
1389:
1384:
1379:
1374:
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1324:
1318:
1315:
1314:
1311:
1304:
1303:
1300:
1299:
1296:
1295:
1292:
1291:
1289:
1288:
1283:
1278:
1272:
1270:
1263:
1257:
1256:
1253:
1252:
1250:
1249:
1243:
1241:
1235:
1234:
1232:
1231:
1226:
1221:
1216:
1211:
1206:
1200:
1198:
1192:
1191:
1189:
1188:
1183:
1178:
1173:
1168:
1162:
1160:
1151:
1145:
1144:
1141:
1140:
1138:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1086:
1084:
1078:
1077:
1075:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1028:
1026:
1017:
1007:
1006:
1001:
994:
993:
990:
989:
987:
986:
981:
976:
971:
966:
961:
955:
952:
951:
948:
941:
940:
937:
936:
934:
933:
928:
923:
918:
913:
907:
904:
903:
900:
893:
892:
889:
888:
886:
885:
880:
875:
870:
865:
860:
855:
850:
844:
841:
840:
834:
827:
826:
823:
822:
820:
819:
814:
809:
804:
799:
793:
790:
789:
786:
779:
778:
775:
774:
772:
771:
766:
761:
756:
751:
746:
741:
735:
732:
731:
724:
717:
716:
713:
712:
710:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
638:
635:
634:
628:
621:
620:
614:
612:
611:
604:
597:
589:
583:
582:
563:
551:
550:External links
548:
547:
546:
518:(3): 232–240,
504:
503:
469:
466:
461:
458:
453:
448:
445:
438:
434:
430:
426:
411:
410:
373:
370:
369:
368:
361:
358:
352:288, form the
326:
323:
288:Znám's problem
266:
265:
210:
209:
198:
195:
192:
187:
184:
177:
171:
166:
162:
158:
155:
139:Equivalently,
136:
133:
125:prime divisors
121:
120:
109:
106:
103:
98:
95:
88:
82:
77:
73:
69:
64:
61:
16:Type of number
15:
13:
10:
9:
6:
4:
3:
2:
2546:
2535:
2532:
2530:
2527:
2526:
2524:
2509:
2505:
2501:
2500:
2497:
2487:
2484:
2483:
2480:
2475:
2470:
2466:
2456:
2453:
2451:
2448:
2447:
2444:
2439:
2434:
2430:
2420:
2417:
2415:
2412:
2411:
2408:
2403:
2398:
2394:
2384:
2381:
2379:
2376:
2375:
2372:
2368:
2362:
2358:
2348:
2345:
2343:
2340:
2338:
2335:
2334:
2331:
2327:
2322:
2318:
2304:
2301:
2300:
2298:
2294:
2288:
2285:
2283:
2280:
2278:
2277:Polydivisible
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2254:
2252:
2248:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2218:
2216:
2213:
2208:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2168:
2166:
2163:
2159:
2151:
2148:
2147:
2146:
2143:
2142:
2140:
2137:
2133:
2121:
2118:
2117:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2072:
2070:
2066:
2060:
2057:
2056:
2054:
2050:
2044:
2041:
2039:
2036:
2035:
2033:
2031:Digit product
2029:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2004:
2002:
2000:
1996:
1988:
1985:
1983:
1980:
1979:
1978:
1975:
1974:
1972:
1970:
1965:
1961:
1957:
1952:
1947:
1943:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1867:Erdős–Nicolas
1865:
1863:
1860:
1858:
1855:
1854:
1851:
1846:
1842:
1836:
1832:
1818:
1815:
1813:
1810:
1809:
1807:
1805:
1801:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1776:
1774:
1772:
1768:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1733:
1731:
1729:
1725:
1719:
1716:
1714:
1711:
1710:
1708:
1706:
1702:
1696:
1693:
1691:
1688:
1686:
1685:Superabundant
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
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:
1587:
1585:
1583:
1579:
1575:
1571:
1567:
1562:
1558:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1489:
1486:
1482:
1477:
1473:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1444:
1441:
1437:
1432:
1428:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
1390:
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:
1319:
1316:
1309:
1305:
1287:
1284:
1282:
1279:
1277:
1274:
1273:
1271:
1267:
1264:
1262:
1261:4-dimensional
1258:
1248:
1245:
1244:
1242:
1240:
1236:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1201:
1199:
1197:
1193:
1187:
1184:
1182:
1179:
1177:
1174:
1172:
1171:Centered cube
1169:
1167:
1164:
1163:
1161:
1159:
1155:
1152:
1150:
1149:3-dimensional
1146:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1087:
1085:
1083:
1079:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1029:
1027:
1025:
1021:
1018:
1016:
1015:2-dimensional
1012:
1008:
1004:
999:
995:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
959:Nonhypotenuse
957:
956:
953:
946:
942:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
908:
905:
898:
894:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
845:
842:
837:
832:
828:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
794:
791:
784:
780:
770:
767:
765:
762:
760:
757:
755:
752:
750:
747:
745:
742:
740:
737:
736:
733:
728:
722:
718:
708:
705:
703:
700:
698:
697:Perfect power
695:
693:
690:
688:
687:Seventh power
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
639:
636:
631:
626:
622:
618:
610:
605:
603:
598:
596:
591:
590:
587:
578:
577:
572:
569:
564:
561:
557:
554:
553:
549:
543:
539:
535:
531:
526:
521:
517:
513:
512:
506:
505:
499:
494:
490:
486:
485:
467:
464:
459:
456:
451:
446:
443:
436:
428:
424:
413:
412:
407:
403:
399:
395:
391:
387:
383:
382:
376:
375:
371:
367:
364:
363:
359:
357:
355:
351:
347:
343:
339:
335:
332:
324:
322:
320:
316:
312:
307:
305:
301:
297:
293:
289:
284:
282:
277:
275:
271:
263:
258:
253:
249:
245:
242:
241:
240:
237:
235:
234:pseudoperfect
231:
227:
223:
219:
215:
196:
193:
190:
185:
182:
175:
164:
160:
156:
153:
146:
145:
144:
142:
134:
132:
130:
126:
107:
104:
101:
96:
93:
86:
75:
71:
67:
62:
59:
50:
49:
48:
46:
42:
38:
34:
33:number theory
30:
21:
2241:Transposable
2105:Narcissistic
2012:Digital root
1932:Super-Poulet
1892:Jordan–Pólya
1841:prime factor
1746:Noncototient
1713:Almost prime
1695:Superperfect
1670:Refactorable
1665:Quasiperfect
1640:Hyperperfect
1481:Pseudoprimes
1452:Wall–Sun–Sun
1387:Ordered Bell
1357:Fuss–Catalan
1269:non-centered
1219:Dodecahedral
1196:non-centered
1082:non-centered
984:Wolstenholme
973:
729:× 2 ± 1
726:
725:Of the form
692:Eighth power
672:Fourth power
574:
515:
509:
488:
482:
385:
379:
366:Giuga number
345:
341:
337:
333:
328:
318:
314:
310:
308:
303:
295:
291:
285:
278:
267:
238:
229:
225:
221:
217:
213:
211:
140:
138:
128:
122:
40:
36:
26:
2262:Extravagant
2257:Equidigital
2212:permutation
2171:Palindromic
2145:Automorphic
2043:Sum-product
2022:Sum-product
1977:Persistence
1872:Erdős–Woods
1794:Untouchable
1675:Semiperfect
1625:Hemiperfect
1286:Tesseractic
1224:Icosahedral
1204:Tetrahedral
1135:Dodecagonal
836:Recursively
707:Prime power
682:Sixth power
677:Fifth power
657:Power of 10
615:Classes of
491:: 407–420,
392:: 296–300,
300:conjectures
29:mathematics
2523:Categories
2474:Graphemics
2347:Pernicious
2201:Undulating
2176:Pandigital
2150:Trimorphic
1751:Nontotient
1600:Arithmetic
1214:Octahedral
1115:Heptagonal
1105:Pentagonal
1090:Triangular
931:Sierpiński
853:Jacobsthal
652:Power of 3
647:Power of 2
560:PlanetMath
525:1812.06566
372:References
135:Properties
2231:Parasitic
2080:Factorion
2007:Digit sum
1999:Digit sum
1817:Fortunate
1804:Primorial
1718:Semiprime
1655:Practical
1620:Descartes
1615:Deficient
1605:Betrothed
1447:Wieferich
1276:Pentatope
1239:pyramidal
1130:Decagonal
1125:Nonagonal
1120:Octagonal
1110:Hexagonal
969:Practical
916:Congruent
848:Fibonacci
812:Loeschian
576:MathWorld
542:119618783
425:∑
298:≤ 8, and
276:diverge.
274:sequences
161:∑
72:∑
47:equation
2303:Friedman
2236:Primeval
2181:Repdigit
2138:-related
2085:Kaprekar
2059:Meertens
1982:Additive
1969:dynamics
1877:Friendly
1789:Sociable
1779:Amicable
1590:Abundant
1570:dynamics
1392:Schröder
1382:Narayana
1352:Eulerian
1342:Delannoy
1337:Dedekind
1158:centered
1024:centered
911:Amenable
868:Narayana
858:Leonardo
754:Mersenne
702:Powerful
642:Achilles
360:See also
228:(except
2476:related
2440:related
2404:related
2402:Sorting
2287:Vampire
2272:Harshad
2214:related
2186:Repunit
2100:Lychrel
2075:Dudeney
1927:Størmer
1922:Sphenic
1907:Regular
1845:divisor
1784:Perfect
1680:Sublime
1650:Perfect
1377:Motzkin
1332:Catalan
873:Padovan
807:Leyland
802:Idoneal
797:Hilbert
769:Woodall
406:2687685
331:integer
325:History
260:in the
257:A054377
2342:Odious
2267:Frugal
2221:Cyclic
2210:Digit-
1917:Smooth
1902:Pronic
1862:Cyclic
1839:Other
1812:Euclid
1462:Wilson
1436:Primes
1095:Square
964:Polite
926:Riesel
921:Knödel
883:Perrin
764:Thabit
749:Fermat
739:Cullen
662:Square
630:Powers
540:
404:
350:modulo
2383:Prime
2378:Lucky
2367:sieve
2296:Other
2282:Smith
2162:Digit
2120:Happy
2095:Keith
2068:Other
1912:Rough
1882:Giuga
1347:Euler
1209:Cubic
863:Lucas
759:Proth
538:S2CID
520:arXiv
402:JSTOR
388:(4),
39:is a
2337:Evil
2017:Self
1967:and
1857:Blum
1568:and
1372:Lobb
1327:Cake
1322:Bell
1072:Star
979:Ulam
878:Pell
667:Cube
262:OEIS
2455:Ban
1843:or
1362:Lah
558:at
530:doi
516:124
493:doi
394:doi
281:odd
127:of
27:In
2525::
573:.
536:,
528:,
514:,
489:69
487:,
400:,
386:29
384:,
306:.
264:).
252:42
250:,
246:,
236:.
131:.
35:,
727:a
608:e
601:t
594:v
579:.
562:.
545:.
532::
522::
502:.
495::
468:1
465:=
460:N
457:1
452:+
447:p
444:1
437:N
433:|
429:p
409:.
396::
346:r
342:r
338:r
334:r
319:N
315:N
311:N
304:k
296:k
292:k
248:6
244:2
230:N
226:N
222:N
218:N
214:N
197:.
194:N
191:=
186:p
183:N
176:N
170:|
165:p
157:+
154:1
141:N
129:N
108:,
105:1
102:=
97:p
94:1
87:N
81:|
76:p
68:+
63:N
60:1
37:N
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