2424:
108:
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be in increasing order (with the exception of the second number, 10), and â except for the first two digits â all digits must be 7, 8, or 9. There are
49:
Usually, this involves additive or multiplicative persistence of a non-negative integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative
77:
The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit:
361:
The next number in the sequence (the smallest number of additive persistence 5) is 2 Ă 10 â 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is at most proportional to its
90:
10, there is thought to be no number with a multiplicative persistence greater than 11; this is known to be true for numbers up to 2.67Ă10. The smallest numbers with persistence 0, 1, 2, ... are:
526:
261:
208:
341:
321:
301:
228:
182:
159:
129:
above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
389:
For example, the function which takes the minimal digit only allows for persistence 0 or 1, as you either start with or step to a single-digit number.
419:
354:
101:
519:
473:
1326:
512:
1321:
1336:
1316:
2029:
1609:
138:
The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero.
1331:
2115:
1431:
1781:
1100:
893:
1816:
1786:
1461:
1451:
1957:
1371:
1105:
1085:
268:
78:
39 â 27 â 14 â 4. Also, 39 is the smallest number of multiplicative persistence 3.
1647:
1811:
1906:
1529:
1286:
1095:
1077:
971:
961:
951:
1791:
2034:
1579:
1200:
986:
981:
976:
966:
943:
43:
109:
also additional restrictions on the first two digits. Based on these restrictions, the number of candidates for
1019:
1276:
2145:
2110:
1806:
1680:
1655:
1564:
1554:
1166:
1148:
1068:
61:
The single-digit final state reached in the process of calculating an integer's additive persistence is its
2448:
2405:
1675:
1549:
1180:
956:
736:
663:
383:
1660:
1514:
1441:
596:
2369:
2009:
2302:
2196:
2160:
1901:
1624:
1604:
1421:
1090:
878:
850:
2024:
1888:
1883:
1851:
1614:
1589:
1584:
1559:
1489:
1485:
1416:
1306:
1138:
934:
903:
2423:
233:
2427:
2181:
2176:
2090:
2064:
1962:
1941:
1713:
1594:
1544:
1466:
1436:
1376:
1143:
1123:
1054:
767:
367:
280:
1311:
94:
0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. (sequence
2321:
2266:
2120:
2095:
2069:
1846:
1524:
1519:
1446:
1426:
1411:
1133:
1115:
1034:
1024:
1009:
787:
772:
469:
435:
2357:
2150:
1736:
1708:
1698:
1690:
1574:
1539:
1534:
1501:
1195:
1158:
1049:
1044:
1039:
1029:
1001:
888:
840:
835:
792:
731:
479:
187:
126:
114:
2333:
2222:
2155:
2081:
2004:
1978:
1796:
1509:
1366:
1301:
1271:
1261:
1256:
922:
830:
777:
621:
561:
483:
465:
2338:
2206:
2191:
2055:
2019:
1994:
1870:
1841:
1826:
1703:
1599:
1569:
1296:
1251:
1128:
726:
721:
716:
688:
673:
586:
571:
549:
536:
457:
326:
306:
286:
213:
167:
162:
144:
2442:
2261:
2245:
2186:
2140:
1836:
1821:
1731:
1456:
1014:
883:
845:
802:
683:
668:
658:
616:
606:
581:
17:
2297:
2286:
2201:
2039:
2014:
1931:
1831:
1801:
1776:
1760:
1665:
1632:
1381:
1355:
1266:
1205:
782:
678:
611:
591:
566:
62:
65:. Put another way, a number's additive persistence counts how many times we must
2256:
2131:
1936:
1400:
1291:
1246:
1241:
991:
898:
797:
626:
601:
576:
409:
31:
2393:
2374:
1670:
1281:
504:
279:
The additive persistence of a number, however, can become arbitrarily large (
1999:
1926:
1918:
1723:
1637:
755:
371:
363:
113:-digit numbers with record-breaking persistence is only proportional to the
66:
2100:
55:
2105:
1764:
39:
366:; therefore, the additive persistence is at most proportional to the
492:
343:). The smallest numbers of additive persistence 0, 1, 2, ... are:
87:
51:
370:, and the smallest number of a given additive persistence grows
2391:
2355:
2319:
2283:
2243:
1868:
1757:
1483:
1398:
1353:
1230:
920:
867:
819:
753:
705:
643:
547:
508:
38:
is the number of times one must apply a given operation to an
267:
More about the additive persistence of a number can be found
125:-digit numbers. However, any number that is missing from the
413:
349:
96:
347:
0, 10, 19, 199, 19999999999999999999999, ... (sequence
82:
Smallest numbers of a given multiplicative persistence
329:
309:
289:
236:
216:
190:
170:
147:
323:
repetitions of the digit 1 is 1 higher than that of
46:
at which the operation no longer alters the number.
2215:
2169:
2129:
2080:
2054:
1987:
1971:
1950:
1917:
1882:
1722:
1689:
1646:
1623:
1500:
1188:
1179:
1157:
1114:
1076:
1067:
1000:
942:
933:
335:
315:
295:
255:
222:
202:
176:
153:
494:On the additive persistence of a number in base p
275:Smallest numbers of a given additive persistence
386:only allow persistence up to a certain degree.
303:, the persistence of the number consisting of
520:
8:
404:
402:
2388:
2352:
2316:
2280:
2240:
1914:
1879:
1865:
1754:
1497:
1480:
1395:
1350:
1227:
1185:
1073:
939:
930:
917:
864:
821:Possessing a specific set of other numbers
816:
750:
702:
640:
544:
527:
513:
505:
420:On-Line Encyclopedia of Integer Sequences
328:
308:
288:
247:
235:
215:
189:
169:
146:
398:
7:
54:. In the remainder of this article,
263:have the same additive persistence.
462:Unsolved problems in number theory
378:Functions with limited persistence
133:Properties of additive persistence
121:, a tiny fraction of all possible
25:
2422:
2030:Perfect digit-to-digit invariant
69:to arrive at its digital root.
1:
869:Expressible via specific sums
436:"Multiplicative Persistence"
256:{\displaystyle n\cdot b^{k}}
1958:Multiplicative digital root
491:Meimaris, Antonios (2015).
50:persistence depends on the
2465:
410:Sloane, N. J. A.
2418:
2401:
2387:
2365:
2351:
2329:
2315:
2293:
2279:
2252:
2239:
2035:Perfect digital invariant
1878:
1864:
1772:
1753:
1610:Superior highly composite
1496:
1479:
1407:
1394:
1362:
1349:
1237:
1226:
929:
916:
874:
863:
826:
815:
763:
749:
712:
701:
654:
639:
557:
543:
1648:Euler's totient function
1432:EulerâJacobi pseudoprime
707:Other polynomial numbers
1462:SomerâLucas pseudoprime
1452:LucasâCarmichael number
1287:Lazy caterer's sequence
414:"Sequence A003001"
36:persistence of a number
1337:WedderburnâEtherington
737:Lucky numbers of Euler
337:
317:
297:
257:
224:
204:
203:{\displaystyle n>9}
178:
155:
1625:Prime omega functions
1442:Frobenius pseudoprime
1232:Combinatorial numbers
1101:Centered dodecahedral
894:Primary pseudoperfect
440:mathworld.wolfram.com
338:
318:
298:
283:: for a given number
258:
225:
205:
179:
156:
2084:-composition related
1884:Arithmetic functions
1486:Arithmetic functions
1422:Elliptic pseudoprime
1106:Centered icosahedral
1086:Centered tetrahedral
468:. pp. 398â399.
327:
307:
287:
234:
214:
188:
168:
145:
27:Property of a number
18:Additive persistence
2010:Kaprekar's constant
1530:Colossally abundant
1417:Catalan pseudoprime
1317:SchröderâHipparchus
1096:Centered octahedral
972:Centered heptagonal
962:Centered pentagonal
952:Centered triangular
552:and related numbers
434:Eric W. Weisstein.
2428:Mathematics portal
2370:Aronson's sequence
2116:SmarandacheâWellin
1873:-dependent numbers
1580:Primitive abundant
1467:Strong pseudoprime
1457:Perrin pseudoprime
1437:Fermat pseudoprime
1377:Wolstenholme prime
1201:Squared triangular
987:Centered decagonal
982:Centered nonagonal
977:Centered octagonal
967:Centered hexagonal
423:. OEIS Foundation.
368:iterated logarithm
333:
313:
293:
253:
220:
200:
174:
151:
42:before reaching a
2436:
2435:
2414:
2413:
2383:
2382:
2347:
2346:
2311:
2310:
2275:
2274:
2235:
2234:
2231:
2230:
2050:
2049:
1860:
1859:
1749:
1748:
1745:
1744:
1691:Aliquot sequences
1502:Divisor functions
1475:
1474:
1447:Lucas pseudoprime
1427:Euler pseudoprime
1412:Carmichael number
1390:
1389:
1345:
1344:
1222:
1221:
1218:
1217:
1214:
1213:
1175:
1174:
1063:
1062:
1020:Square triangular
912:
911:
859:
858:
811:
810:
745:
744:
697:
696:
635:
634:
475:978-0-387-20860-2
336:{\displaystyle n}
316:{\displaystyle n}
296:{\displaystyle n}
223:{\displaystyle n}
177:{\displaystyle k}
154:{\displaystyle b}
16:(Redirected from
2456:
2426:
2389:
2358:Natural language
2353:
2317:
2285:Generated via a
2281:
2241:
2146:Digit-reassembly
2111:Self-descriptive
1915:
1880:
1866:
1817:LucasâCarmichael
1807:Harmonic divisor
1755:
1681:Sparsely totient
1656:Highly cototient
1565:Multiply perfect
1555:Highly composite
1498:
1481:
1396:
1351:
1332:Telephone number
1228:
1186:
1167:Square pyramidal
1149:Stella octangula
1074:
940:
931:
923:Figurate numbers
918:
865:
817:
751:
703:
641:
545:
529:
522:
515:
506:
498:
487:
464:(3rd ed.).
444:
443:
431:
425:
424:
406:
352:
342:
340:
339:
334:
322:
320:
319:
314:
302:
300:
299:
294:
262:
260:
259:
254:
252:
251:
229:
227:
226:
221:
209:
207:
206:
201:
183:
181:
180:
175:
160:
158:
157:
152:
99:
21:
2464:
2463:
2459:
2458:
2457:
2455:
2454:
2453:
2439:
2438:
2437:
2432:
2410:
2406:Strobogrammatic
2397:
2379:
2361:
2343:
2325:
2307:
2289:
2271:
2248:
2227:
2211:
2170:Divisor-related
2165:
2125:
2076:
2046:
1983:
1967:
1946:
1913:
1886:
1874:
1856:
1768:
1767:related numbers
1741:
1718:
1685:
1676:Perfect totient
1642:
1619:
1550:Highly abundant
1492:
1471:
1403:
1386:
1358:
1341:
1327:Stirling second
1233:
1210:
1171:
1153:
1110:
1059:
996:
957:Centered square
925:
908:
870:
855:
822:
807:
759:
758:defined numbers
741:
708:
693:
664:Double Mersenne
650:
631:
553:
539:
537:natural numbers
533:
502:
490:
476:
466:Springer-Verlag
458:Guy, Richard K.
456:
453:
448:
447:
433:
432:
428:
408:
407:
400:
395:
380:
348:
325:
324:
305:
304:
285:
284:
277:
243:
232:
231:
212:
211:
186:
185:
166:
165:
163:natural numbers
143:
142:
135:
95:
84:
75:
28:
23:
22:
15:
12:
11:
5:
2462:
2460:
2452:
2451:
2441:
2440:
2434:
2433:
2431:
2430:
2419:
2416:
2415:
2412:
2411:
2409:
2408:
2402:
2399:
2398:
2392:
2385:
2384:
2381:
2380:
2378:
2377:
2372:
2366:
2363:
2362:
2356:
2349:
2348:
2345:
2344:
2342:
2341:
2339:Sorting number
2336:
2334:Pancake number
2330:
2327:
2326:
2320:
2313:
2312:
2309:
2308:
2306:
2305:
2300:
2294:
2291:
2290:
2284:
2277:
2276:
2273:
2272:
2270:
2269:
2264:
2259:
2253:
2250:
2249:
2246:Binary numbers
2244:
2237:
2236:
2233:
2232:
2229:
2228:
2226:
2225:
2219:
2217:
2213:
2212:
2210:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2173:
2171:
2167:
2166:
2164:
2163:
2158:
2153:
2148:
2143:
2137:
2135:
2127:
2126:
2124:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2087:
2085:
2078:
2077:
2075:
2074:
2073:
2072:
2061:
2059:
2056:P-adic numbers
2052:
2051:
2048:
2047:
2045:
2044:
2043:
2042:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1991:
1989:
1985:
1984:
1982:
1981:
1975:
1973:
1972:Coding-related
1969:
1968:
1966:
1965:
1960:
1954:
1952:
1948:
1947:
1945:
1944:
1939:
1934:
1929:
1923:
1921:
1912:
1911:
1910:
1909:
1907:Multiplicative
1904:
1893:
1891:
1876:
1875:
1871:Numeral system
1869:
1862:
1861:
1858:
1857:
1855:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1773:
1770:
1769:
1758:
1751:
1750:
1747:
1746:
1743:
1742:
1740:
1739:
1734:
1728:
1726:
1720:
1719:
1717:
1716:
1711:
1706:
1701:
1695:
1693:
1687:
1686:
1684:
1683:
1678:
1673:
1668:
1663:
1661:Highly totient
1658:
1652:
1650:
1644:
1643:
1641:
1640:
1635:
1629:
1627:
1621:
1620:
1618:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1515:Almost perfect
1512:
1506:
1504:
1494:
1493:
1484:
1477:
1476:
1473:
1472:
1470:
1469:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1408:
1405:
1404:
1399:
1392:
1391:
1388:
1387:
1385:
1384:
1379:
1374:
1369:
1363:
1360:
1359:
1354:
1347:
1346:
1343:
1342:
1340:
1339:
1334:
1329:
1324:
1322:Stirling first
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1238:
1235:
1234:
1231:
1224:
1223:
1220:
1219:
1216:
1215:
1212:
1211:
1209:
1208:
1203:
1198:
1192:
1190:
1183:
1177:
1176:
1173:
1172:
1170:
1169:
1163:
1161:
1155:
1154:
1152:
1151:
1146:
1141:
1136:
1131:
1126:
1120:
1118:
1112:
1111:
1109:
1108:
1103:
1098:
1093:
1088:
1082:
1080:
1071:
1065:
1064:
1061:
1060:
1058:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1006:
1004:
998:
997:
995:
994:
989:
984:
979:
974:
969:
964:
959:
954:
948:
946:
937:
927:
926:
921:
914:
913:
910:
909:
907:
906:
901:
896:
891:
886:
881:
875:
872:
871:
868:
861:
860:
857:
856:
854:
853:
848:
843:
838:
833:
827:
824:
823:
820:
813:
812:
809:
808:
806:
805:
800:
795:
790:
785:
780:
775:
770:
764:
761:
760:
754:
747:
746:
743:
742:
740:
739:
734:
729:
724:
719:
713:
710:
709:
706:
699:
698:
695:
694:
692:
691:
686:
681:
676:
671:
666:
661:
655:
652:
651:
644:
637:
636:
633:
632:
630:
629:
624:
619:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
564:
558:
555:
554:
548:
541:
540:
534:
532:
531:
524:
517:
509:
500:
499:
488:
474:
452:
449:
446:
445:
426:
397:
396:
394:
391:
379:
376:
359:
358:
332:
312:
292:
276:
273:
265:
264:
250:
246:
242:
239:
219:
199:
196:
193:
173:
150:
139:
134:
131:
106:
105:
83:
80:
74:
71:
67:sum its digits
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2461:
2450:
2449:Number theory
2447:
2446:
2444:
2429:
2425:
2421:
2420:
2417:
2407:
2404:
2403:
2400:
2395:
2390:
2386:
2376:
2373:
2371:
2368:
2367:
2364:
2359:
2354:
2350:
2340:
2337:
2335:
2332:
2331:
2328:
2323:
2318:
2314:
2304:
2301:
2299:
2296:
2295:
2292:
2288:
2282:
2278:
2268:
2265:
2263:
2260:
2258:
2255:
2254:
2251:
2247:
2242:
2238:
2224:
2221:
2220:
2218:
2214:
2208:
2205:
2203:
2200:
2198:
2197:Polydivisible
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2172:
2168:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2138:
2136:
2133:
2128:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2088:
2086:
2083:
2079:
2071:
2068:
2067:
2066:
2063:
2062:
2060:
2057:
2053:
2041:
2038:
2037:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1992:
1990:
1986:
1980:
1977:
1976:
1974:
1970:
1964:
1961:
1959:
1956:
1955:
1953:
1951:Digit product
1949:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1924:
1922:
1920:
1916:
1908:
1905:
1903:
1900:
1899:
1898:
1895:
1894:
1892:
1890:
1885:
1881:
1877:
1872:
1867:
1863:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1795:
1793:
1790:
1788:
1787:ErdĆsâNicolas
1785:
1783:
1780:
1778:
1775:
1774:
1771:
1766:
1762:
1756:
1752:
1738:
1735:
1733:
1730:
1729:
1727:
1725:
1721:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1694:
1692:
1688:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1653:
1651:
1649:
1645:
1639:
1636:
1634:
1631:
1630:
1628:
1626:
1622:
1616:
1613:
1611:
1608:
1606:
1605:Superabundant
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1507:
1505:
1503:
1499:
1495:
1491:
1487:
1482:
1478:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1409:
1406:
1402:
1397:
1393:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1364:
1361:
1357:
1352:
1348:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1239:
1236:
1229:
1225:
1207:
1204:
1202:
1199:
1197:
1194:
1193:
1191:
1187:
1184:
1182:
1181:4-dimensional
1178:
1168:
1165:
1164:
1162:
1160:
1156:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1121:
1119:
1117:
1113:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1091:Centered cube
1089:
1087:
1084:
1083:
1081:
1079:
1075:
1072:
1070:
1069:3-dimensional
1066:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1007:
1005:
1003:
999:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
949:
947:
945:
941:
938:
936:
935:2-dimensional
932:
928:
924:
919:
915:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
879:Nonhypotenuse
877:
876:
873:
866:
862:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
828:
825:
818:
814:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
765:
762:
757:
752:
748:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
714:
711:
704:
700:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
656:
653:
648:
642:
638:
628:
625:
623:
620:
618:
617:Perfect power
615:
613:
610:
608:
607:Seventh power
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
559:
556:
551:
546:
542:
538:
530:
525:
523:
518:
516:
511:
510:
507:
503:
496:
495:
489:
485:
481:
477:
471:
467:
463:
459:
455:
454:
450:
441:
437:
430:
427:
422:
421:
415:
411:
405:
403:
399:
392:
390:
387:
385:
377:
375:
373:
372:tetrationally
369:
365:
356:
351:
346:
345:
344:
330:
310:
290:
282:
274:
272:
270:
248:
244:
240:
237:
217:
197:
194:
191:
171:
164:
148:
140:
137:
136:
132:
130:
128:
124:
120:
116:
112:
103:
98:
93:
92:
91:
89:
81:
79:
72:
70:
68:
64:
59:
57:
53:
47:
45:
41:
37:
33:
19:
2161:Transposable
2025:Narcissistic
1932:Digital root
1896:
1852:Super-Poulet
1812:JordanâPĂłlya
1761:prime factor
1666:Noncototient
1633:Almost prime
1615:Superperfect
1590:Refactorable
1585:Quasiperfect
1560:Hyperperfect
1401:Pseudoprimes
1372:WallâSunâSun
1307:Ordered Bell
1277:FussâCatalan
1189:non-centered
1139:Dodecahedral
1116:non-centered
1002:non-centered
904:Wolstenholme
649:× 2 ± 1
646:
645:Of the form
612:Eighth power
592:Fourth power
501:
493:
461:
439:
429:
417:
388:
381:
360:
278:
266:
210:the numbers
122:
118:
110:
107:
85:
76:
63:digital root
60:
58:is assumed.
48:
35:
29:
2182:Extravagant
2177:Equidigital
2132:permutation
2091:Palindromic
2065:Automorphic
1963:Sum-product
1942:Sum-product
1897:Persistence
1792:ErdĆsâWoods
1714:Untouchable
1595:Semiperfect
1545:Hemiperfect
1206:Tesseractic
1144:Icosahedral
1124:Tetrahedral
1055:Dodecagonal
756:Recursively
627:Prime power
602:Sixth power
597:Fifth power
577:Power of 10
535:Classes of
497:. Preprint.
44:fixed point
32:mathematics
2394:Graphemics
2267:Pernicious
2121:Undulating
2096:Pandigital
2070:Trimorphic
1671:Nontotient
1520:Arithmetic
1134:Octahedral
1035:Heptagonal
1025:Pentagonal
1010:Triangular
851:SierpiĆski
773:Jacobsthal
572:Power of 3
567:Power of 2
484:1058.11001
451:Literature
393:References
2151:Parasitic
2000:Factorion
1927:Digit sum
1919:Digit sum
1737:Fortunate
1724:Primorial
1638:Semiprime
1575:Practical
1540:Descartes
1535:Deficient
1525:Betrothed
1367:Wieferich
1196:Pentatope
1159:pyramidal
1050:Decagonal
1045:Nonagonal
1040:Octagonal
1030:Hexagonal
889:Practical
836:Congruent
768:Fibonacci
732:Loeschian
384:functions
364:logarithm
241:⋅
141:For base
2443:Category
2223:Friedman
2156:Primeval
2101:Repdigit
2058:-related
2005:Kaprekar
1979:Meertens
1902:Additive
1889:dynamics
1797:Friendly
1709:Sociable
1699:Amicable
1510:Abundant
1490:dynamics
1312:Schröder
1302:Narayana
1272:Eulerian
1262:Delannoy
1257:Dedekind
1078:centered
944:centered
831:Amenable
788:Narayana
778:Leonardo
674:Mersenne
622:Powerful
562:Achilles
460:(2004).
127:sequence
73:Examples
56:base ten
2396:related
2360:related
2324:related
2322:Sorting
2207:Vampire
2192:Harshad
2134:related
2106:Repunit
2020:Lychrel
1995:Dudeney
1847:StĂžrmer
1842:Sphenic
1827:Regular
1765:divisor
1704:Perfect
1600:Sublime
1570:Perfect
1297:Motzkin
1252:Catalan
793:Padovan
727:Leyland
722:Idoneal
717:Hilbert
689:Woodall
412:(ed.).
353:in the
350:A006050
100:in the
97:A003001
40:integer
2262:Odious
2187:Frugal
2141:Cyclic
2130:Digit-
1837:Smooth
1822:Pronic
1782:Cyclic
1759:Other
1732:Euclid
1382:Wilson
1356:Primes
1015:Square
884:Polite
846:Riesel
841:Knödel
803:Perrin
684:Thabit
669:Fermat
659:Cullen
582:Square
550:Powers
482:
472:
115:square
34:, the
2303:Prime
2298:Lucky
2287:sieve
2216:Other
2202:Smith
2082:Digit
2040:Happy
2015:Keith
1988:Other
1832:Rough
1802:Giuga
1267:Euler
1129:Cubic
783:Lucas
679:Proth
382:Some
281:proof
52:radix
2257:Evil
1937:Self
1887:and
1777:Blum
1488:and
1292:Lobb
1247:Cake
1242:Bell
992:Star
899:Ulam
798:Pell
587:Cube
470:ISBN
418:The
355:OEIS
269:here
230:and
195:>
184:and
161:and
102:OEIS
88:base
2375:Ban
1763:or
1282:Lah
480:Zbl
117:of
86:In
30:In
2445::
478:.
438:.
416:.
401:^
374:.
271:.
647:a
528:e
521:t
514:v
486:.
442:.
357:)
331:n
311:n
291:n
249:k
245:b
238:n
218:n
198:9
192:n
172:k
149:b
123:n
119:n
111:n
104:)
20:)
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