56:
2686:
2593:
67:
The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any
229:+ 1 is not an untouchable number. It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 7 is an untouchable number, and
695:
350:
309:
268:
582:
617:
59:
If we draw an arrow pointing from each positive integer to the sum of all its proper divisors, there will be no arrow pointing to untouchable numbers like 2 and 5.
197:
The number 5 is believed to be the only odd untouchable number, but this has not been proven. It would follow from a slightly stronger version of the
2629:
169:
688:
191:
71:
The number 6 is not untouchable, as it is equal to the sum of the proper divisors of 6 itself: 1 + 2 + 3 = 6.
1495:
681:
1490:
1505:
1485:
2675:
652:
525:
The stronger version is obtained by adding to the
Goldbach conjecture the further requirement that the two primes be distinctâsee
2872:
2685:
2198:
1778:
1500:
2284:
2622:
1600:
3037:
1950:
1269:
1062:
2826:
1985:
1955:
1630:
1620:
2126:
1540:
1274:
1254:
64:
The number 4 is not untouchable, as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4.
1816:
1980:
3047:
2862:
2075:
1698:
1455:
1264:
1246:
1140:
1130:
1120:
1960:
3042:
2847:
2203:
1748:
1369:
1155:
1150:
1145:
1135:
1112:
352:, so only 5 can be an odd untouchable number. Thus it appears that besides 2 and 5, all untouchable numbers are
2615:
1188:
669:
sequence A070015 (Least m such that sum of aliquot parts of m equals n or 0 if no such number exists)
1445:
3001:
2867:
2791:
2314:
2279:
2065:
1975:
1849:
1824:
1733:
1723:
1335:
1317:
1237:
2852:
2811:
2574:
1844:
1718:
1349:
1125:
905:
832:
55:
44:
2781:
2650:
1829:
1683:
1610:
765:
2538:
2178:
2955:
2857:
2471:
2365:
2329:
2070:
1793:
1773:
1590:
1259:
1047:
1019:
162:, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence
628:
Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442.
314:
273:
232:
3016:
3011:
2806:
2801:
2786:
2725:
2193:
2057:
2052:
2020:
1783:
1758:
1753:
1728:
1658:
1654:
1585:
1475:
1307:
1103:
1072:
198:
2592:
552:
404: + 1. Also, no untouchable number is three more than a prime number, except 5, since if
2940:
2935:
2816:
2796:
2596:
2350:
2345:
2259:
2233:
2131:
2110:
1763:
1713:
1635:
1605:
1545:
1312:
1292:
1223:
936:
509:
493:
1480:
587:
2976:
2916:
2490:
2435:
2289:
2264:
2238:
2015:
1693:
1688:
1615:
1595:
1580:
1302:
1284:
1203:
1193:
1178:
956:
941:
648:
3006:
2981:
2901:
2887:
2821:
2705:
2665:
2526:
2319:
1905:
1877:
1867:
1859:
1743:
1708:
1703:
1670:
1364:
1327:
1218:
1213:
1208:
1198:
1170:
1057:
1009:
1004:
961:
900:
485:
441:
353:
505:
360:
is untouchable, since, at the very least, it can be expressed as the sum of its own proper
2991:
2986:
2911:
2905:
2842:
2740:
2730:
2660:
2502:
2391:
2324:
2250:
2173:
2147:
1965:
1678:
1535:
1470:
1440:
1430:
1425:
1091:
999:
946:
790:
730:
644:
501:
429:
373:
369:
365:
68:
number's proper divisors (since the list of factors would have to contain both 4 and 2).
2996:
2950:
2776:
2760:
2750:
2720:
2507:
2375:
2360:
2224:
2188:
2163:
2039:
2010:
1995:
1872:
1768:
1738:
1465:
1420:
1297:
895:
890:
885:
857:
842:
755:
740:
718:
705:
636:
529:
357:
36:
2607:
3031:
2945:
2745:
2735:
2715:
2430:
2414:
2355:
2309:
2005:
1990:
1900:
1625:
1183:
1052:
1014:
971:
852:
837:
827:
785:
775:
750:
513:
425:
2960:
2877:
2755:
2700:
2670:
2466:
2455:
2370:
2208:
2183:
2100:
2000:
1970:
1945:
1929:
1834:
1801:
1550:
1524:
1435:
1374:
951:
847:
780:
760:
735:
456:
451:
389:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
2425:
2300:
2105:
1569:
1460:
1415:
1410:
1160:
1067:
966:
795:
770:
745:
95:
91:
87:
40:
20:
2562:
2543:
1839:
1450:
446:
83:
79:
673:
2710:
2168:
2095:
2087:
1892:
1806:
924:
534:
39:
of any positive integer. That is, these numbers are not in the image of the
32:
2269:
424:
There are infinitely many untouchable numbers, a fact that was proven by
497:
2655:
2274:
1933:
489:
361:
180:
28:
476:
Sesiano, J. (1991), "Two problems of number theory in
Islamic times",
2926:
54:
665:
47:(circa 1000 AD), who observed that both 2 and 5 are untouchable.
2611:
2560:
2524:
2488:
2452:
2412:
2037:
1926:
1652:
1567:
1522:
1399:
1089:
1036:
988:
922:
874:
812:
716:
677:
668:
164:
385:= 2 â 1 is equal to the sum of the proper divisors of 2.
408:
is an odd prime then the sum of the proper divisors of 2
590:
555:
356:(since except 2, all even numbers are composite). No
317:
276:
235:
225:
can be written as a sum of two distinct primes, then
2969:
2925:
2886:
2835:
2769:
2693:
2643:
2384:
2338:
2298:
2249:
2223:
2156:
2140:
2119:
2086:
2051:
1891:
1858:
1815:
1792:
1669:
1357:
1348:
1326:
1283:
1245:
1236:
1169:
1111:
1102:
188:
Are there any odd untouchable numbers other than 5?
611:
576:
344:
303:
262:
396:is prime, then the sum of the proper divisors of
16:Number that cannot be written as an aliquot sum
528:Adams-Watters, Frank & Weisstein, Eric W.
2623:
689:
8:
43:function. Their study goes back at least to
2630:
2616:
2608:
2557:
2521:
2485:
2449:
2409:
2083:
2048:
2034:
1923:
1666:
1649:
1564:
1519:
1396:
1354:
1242:
1108:
1099:
1086:
1033:
990:Possessing a specific set of other numbers
985:
919:
871:
809:
713:
696:
682:
674:
201:, since the sum of the proper divisors of
589:
554:
388:No untouchable number is one more than a
316:
275:
234:
468:
192:(more unsolved problems in mathematics)
428:. According to Chen & Zhao, their
75:The first few untouchable numbers are
619:. Elemente der Math. 28 (1973), 83-86
478:Archive for History of Exact Sciences
7:
2638:Divisibility-based sets of integers
549:P. Erdos, Ăber die Zahlen der Form
372:are untouchable. Also, none of the
641:Unsolved Problems in Number Theory
14:
2676:Fundamental theorem of arithmetic
2684:
2591:
2199:Perfect digit-to-digit invariant
31:that cannot be expressed as the
183:Unsolved problem in mathematics
606:
600:
565:
559:
345:{\displaystyle 7=\sigma (8)-8}
333:
327:
304:{\displaystyle 3=\sigma (4)-4}
292:
286:
263:{\displaystyle 1=\sigma (2)-2}
251:
245:
1:
1038:Expressible via specific sums
577:{\displaystyle \sigma (n)-n}
2127:Multiplicative digital root
3064:
612:{\displaystyle n-\phi (n)}
2873:Superior highly composite
2682:
2587:
2570:
2556:
2534:
2520:
2498:
2484:
2462:
2448:
2421:
2408:
2204:Perfect digital invariant
2047:
2033:
1941:
1922:
1779:Superior highly composite
1665:
1648:
1576:
1563:
1531:
1518:
1406:
1395:
1098:
1085:
1043:
1032:
995:
984:
932:
918:
881:
870:
823:
808:
726:
712:
432:is at least d > 0.06.
364:. Similarly, none of the
2770:Constrained divisor sums
1817:Euler's totient function
1601:EulerâJacobi pseudoprime
876:Other polynomial numbers
213:distinct primes) is 1 +
1631:SomerâLucas pseudoprime
1621:LucasâCarmichael number
1456:Lazy caterer's sequence
376:are untouchable, since
1506:WedderburnâEtherington
906:Lucky numbers of Euler
613:
578:
346:
305:
264:
60:
45:Abu Mansur al-Baghdadi
2651:Integer factorization
1794:Prime omega functions
1611:Frobenius pseudoprime
1401:Combinatorial numbers
1270:Centered dodecahedral
1063:Primary pseudoperfect
614:
579:
347:
306:
265:
58:
2253:-composition related
2053:Arithmetic functions
1655:Arithmetic functions
1591:Elliptic pseudoprime
1275:Centered icosahedral
1255:Centered tetrahedral
588:
553:
530:"Untouchable Number"
315:
274:
233:
221:. Thus, if a number
3038:Arithmetic dynamics
2863:Colossally abundant
2694:Factorization forms
2179:Kaprekar's constant
1699:Colossally abundant
1586:Catalan pseudoprime
1486:SchröderâHipparchus
1265:Centered octahedral
1141:Centered heptagonal
1131:Centered pentagonal
1121:Centered triangular
721:and related numbers
199:Goldbach conjecture
2848:Primitive abundant
2836:With many divisors
2597:Mathematics portal
2539:Aronson's sequence
2285:SmarandacheâWellin
2042:-dependent numbers
1749:Primitive abundant
1636:Strong pseudoprime
1626:Perrin pseudoprime
1606:Fermat pseudoprime
1546:Wolstenholme prime
1370:Squared triangular
1156:Centered decagonal
1151:Centered nonagonal
1146:Centered octagonal
1136:Centered hexagonal
609:
574:
490:10.1007/BF00348408
342:
301:
260:
61:
25:untouchable number
3048:Integer sequences
3025:
3024:
2605:
2604:
2583:
2582:
2552:
2551:
2516:
2515:
2480:
2479:
2444:
2443:
2404:
2403:
2400:
2399:
2219:
2218:
2029:
2028:
1918:
1917:
1914:
1913:
1860:Aliquot sequences
1671:Divisor functions
1644:
1643:
1616:Lucas pseudoprime
1596:Euler pseudoprime
1581:Carmichael number
1559:
1558:
1514:
1513:
1391:
1390:
1387:
1386:
1383:
1382:
1344:
1343:
1232:
1231:
1189:Square triangular
1081:
1080:
1028:
1027:
980:
979:
914:
913:
866:
865:
804:
803:
354:composite numbers
3055:
3043:Divisor function
3002:Harmonic divisor
2888:Aliquot sequence
2868:Highly composite
2792:Multiply perfect
2688:
2666:Divisor function
2632:
2625:
2618:
2609:
2595:
2558:
2527:Natural language
2522:
2486:
2454:Generated via a
2450:
2410:
2315:Digit-reassembly
2280:Self-descriptive
2084:
2049:
2035:
1986:LucasâCarmichael
1976:Harmonic divisor
1924:
1850:Sparsely totient
1825:Highly cototient
1734:Multiply perfect
1724:Highly composite
1667:
1650:
1565:
1520:
1501:Telephone number
1397:
1355:
1336:Square pyramidal
1318:Stella octangula
1243:
1109:
1100:
1092:Figurate numbers
1087:
1034:
986:
920:
872:
810:
714:
698:
691:
684:
675:
667:
629:
626:
620:
618:
616:
615:
610:
583:
581:
580:
575:
547:
541:
540:
539:
523:
517:
516:
473:
442:Aliquot sequence
416: + 3.
374:Mersenne numbers
370:sociable numbers
366:amicable numbers
351:
349:
348:
343:
310:
308:
307:
302:
269:
267:
266:
261:
184:
167:
3063:
3062:
3058:
3057:
3056:
3054:
3053:
3052:
3028:
3027:
3026:
3021:
2965:
2921:
2882:
2853:Highly abundant
2831:
2812:Unitary perfect
2765:
2689:
2680:
2661:Unitary divisor
2639:
2636:
2606:
2601:
2579:
2575:Strobogrammatic
2566:
2548:
2530:
2512:
2494:
2476:
2458:
2440:
2417:
2396:
2380:
2339:Divisor-related
2334:
2294:
2245:
2215:
2152:
2136:
2115:
2082:
2055:
2043:
2025:
1937:
1936:related numbers
1910:
1887:
1854:
1845:Perfect totient
1811:
1788:
1719:Highly abundant
1661:
1640:
1572:
1555:
1527:
1510:
1496:Stirling second
1402:
1379:
1340:
1322:
1279:
1228:
1165:
1126:Centered square
1094:
1077:
1039:
1024:
991:
976:
928:
927:defined numbers
910:
877:
862:
833:Double Mersenne
819:
800:
722:
708:
706:natural numbers
702:
662:
645:Springer Verlag
633:
632:
627:
623:
586:
585:
551:
550:
548:
544:
527:
526:
524:
520:
475:
474:
470:
465:
438:
430:natural density
422:
384:
313:
312:
272:
271:
231:
230:
195:
194:
189:
186:
182:
179:
163:
53:
37:proper divisors
17:
12:
11:
5:
3061:
3059:
3051:
3050:
3045:
3040:
3030:
3029:
3023:
3022:
3020:
3019:
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2973:
2971:
2967:
2966:
2964:
2963:
2958:
2953:
2948:
2943:
2938:
2932:
2930:
2923:
2922:
2920:
2919:
2914:
2909:
2899:
2893:
2891:
2884:
2883:
2881:
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2839:
2837:
2833:
2832:
2830:
2829:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2784:
2782:Almost perfect
2779:
2773:
2771:
2767:
2766:
2764:
2763:
2758:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2697:
2695:
2691:
2690:
2683:
2681:
2679:
2678:
2673:
2668:
2663:
2658:
2653:
2647:
2645:
2641:
2640:
2637:
2635:
2634:
2627:
2620:
2612:
2603:
2602:
2600:
2599:
2588:
2585:
2584:
2581:
2580:
2578:
2577:
2571:
2568:
2567:
2561:
2554:
2553:
2550:
2549:
2547:
2546:
2541:
2535:
2532:
2531:
2525:
2518:
2517:
2514:
2513:
2511:
2510:
2508:Sorting number
2505:
2503:Pancake number
2499:
2496:
2495:
2489:
2482:
2481:
2478:
2477:
2475:
2474:
2469:
2463:
2460:
2459:
2453:
2446:
2445:
2442:
2441:
2439:
2438:
2433:
2428:
2422:
2419:
2418:
2415:Binary numbers
2413:
2406:
2405:
2402:
2401:
2398:
2397:
2395:
2394:
2388:
2386:
2382:
2381:
2379:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2342:
2340:
2336:
2335:
2333:
2332:
2327:
2322:
2317:
2312:
2306:
2304:
2296:
2295:
2293:
2292:
2287:
2282:
2277:
2272:
2267:
2262:
2256:
2254:
2247:
2246:
2244:
2243:
2242:
2241:
2230:
2228:
2225:P-adic numbers
2221:
2220:
2217:
2216:
2214:
2213:
2212:
2211:
2201:
2196:
2191:
2186:
2181:
2176:
2171:
2166:
2160:
2158:
2154:
2153:
2151:
2150:
2144:
2142:
2141:Coding-related
2138:
2137:
2135:
2134:
2129:
2123:
2121:
2117:
2116:
2114:
2113:
2108:
2103:
2098:
2092:
2090:
2081:
2080:
2079:
2078:
2076:Multiplicative
2073:
2062:
2060:
2045:
2044:
2040:Numeral system
2038:
2031:
2030:
2027:
2026:
2024:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1942:
1939:
1938:
1927:
1920:
1919:
1916:
1915:
1912:
1911:
1909:
1908:
1903:
1897:
1895:
1889:
1888:
1886:
1885:
1880:
1875:
1870:
1864:
1862:
1856:
1855:
1853:
1852:
1847:
1842:
1837:
1832:
1830:Highly totient
1827:
1821:
1819:
1813:
1812:
1810:
1809:
1804:
1798:
1796:
1790:
1789:
1787:
1786:
1781:
1776:
1771:
1766:
1761:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1701:
1696:
1691:
1686:
1684:Almost perfect
1681:
1675:
1673:
1663:
1662:
1653:
1646:
1645:
1642:
1641:
1639:
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1583:
1577:
1574:
1573:
1568:
1561:
1560:
1557:
1556:
1554:
1553:
1548:
1543:
1538:
1532:
1529:
1528:
1523:
1516:
1515:
1512:
1511:
1509:
1508:
1503:
1498:
1493:
1491:Stirling first
1488:
1483:
1478:
1473:
1468:
1463:
1458:
1453:
1448:
1443:
1438:
1433:
1428:
1423:
1418:
1413:
1407:
1404:
1403:
1400:
1393:
1392:
1389:
1388:
1385:
1384:
1381:
1380:
1378:
1377:
1372:
1367:
1361:
1359:
1352:
1346:
1345:
1342:
1341:
1339:
1338:
1332:
1330:
1324:
1323:
1321:
1320:
1315:
1310:
1305:
1300:
1295:
1289:
1287:
1281:
1280:
1278:
1277:
1272:
1267:
1262:
1257:
1251:
1249:
1240:
1234:
1233:
1230:
1229:
1227:
1226:
1221:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1181:
1175:
1173:
1167:
1166:
1164:
1163:
1158:
1153:
1148:
1143:
1138:
1133:
1128:
1123:
1117:
1115:
1106:
1096:
1095:
1090:
1083:
1082:
1079:
1078:
1076:
1075:
1070:
1065:
1060:
1055:
1050:
1044:
1041:
1040:
1037:
1030:
1029:
1026:
1025:
1023:
1022:
1017:
1012:
1007:
1002:
996:
993:
992:
989:
982:
981:
978:
977:
975:
974:
969:
964:
959:
954:
949:
944:
939:
933:
930:
929:
923:
916:
915:
912:
911:
909:
908:
903:
898:
893:
888:
882:
879:
878:
875:
868:
867:
864:
863:
861:
860:
855:
850:
845:
840:
835:
830:
824:
821:
820:
813:
806:
805:
802:
801:
799:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
727:
724:
723:
717:
710:
709:
703:
701:
700:
693:
686:
678:
672:
671:
661:
660:External links
658:
657:
656:
655:; section B10.
637:Richard K. Guy
631:
630:
621:
608:
605:
602:
599:
596:
593:
573:
570:
567:
564:
561:
558:
542:
518:
484:(3): 235â238,
467:
466:
464:
461:
460:
459:
454:
449:
444:
437:
434:
421:
418:
380:
358:perfect number
341:
338:
335:
332:
329:
326:
323:
320:
300:
297:
294:
291:
288:
285:
282:
279:
259:
256:
253:
250:
247:
244:
241:
238:
190:
187:
181:
178:
175:
174:
173:
73:
72:
69:
65:
52:
49:
27:is a positive
15:
13:
10:
9:
6:
4:
3:
2:
3060:
3049:
3046:
3044:
3041:
3039:
3036:
3035:
3033:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2972:
2968:
2962:
2959:
2957:
2956:Polydivisible
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2933:
2931:
2928:
2924:
2918:
2915:
2913:
2910:
2907:
2903:
2900:
2898:
2895:
2894:
2892:
2889:
2885:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2858:Superabundant
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2840:
2838:
2834:
2828:
2827:ErdĆsâNicolas
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2774:
2772:
2768:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2736:Perfect power
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2698:
2696:
2692:
2687:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2648:
2646:
2642:
2633:
2628:
2626:
2621:
2619:
2614:
2613:
2610:
2598:
2594:
2590:
2589:
2586:
2576:
2573:
2572:
2569:
2564:
2559:
2555:
2545:
2542:
2540:
2537:
2536:
2533:
2528:
2523:
2519:
2509:
2506:
2504:
2501:
2500:
2497:
2492:
2487:
2483:
2473:
2470:
2468:
2465:
2464:
2461:
2457:
2451:
2447:
2437:
2434:
2432:
2429:
2427:
2424:
2423:
2420:
2416:
2411:
2407:
2393:
2390:
2389:
2387:
2383:
2377:
2374:
2372:
2369:
2367:
2366:Polydivisible
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2343:
2341:
2337:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2307:
2305:
2302:
2297:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2257:
2255:
2252:
2248:
2240:
2237:
2236:
2235:
2232:
2231:
2229:
2226:
2222:
2210:
2207:
2206:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2161:
2159:
2155:
2149:
2146:
2145:
2143:
2139:
2133:
2130:
2128:
2125:
2124:
2122:
2120:Digit product
2118:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2093:
2091:
2089:
2085:
2077:
2074:
2072:
2069:
2068:
2067:
2064:
2063:
2061:
2059:
2054:
2050:
2046:
2041:
2036:
2032:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1956:ErdĆsâNicolas
1954:
1952:
1949:
1947:
1944:
1943:
1940:
1935:
1931:
1925:
1921:
1907:
1904:
1902:
1899:
1898:
1896:
1894:
1890:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1865:
1863:
1861:
1857:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1822:
1820:
1818:
1814:
1808:
1805:
1803:
1800:
1799:
1797:
1795:
1791:
1785:
1782:
1780:
1777:
1775:
1774:Superabundant
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1676:
1674:
1672:
1668:
1664:
1660:
1656:
1651:
1647:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1578:
1575:
1571:
1566:
1562:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1533:
1530:
1526:
1521:
1517:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1408:
1405:
1398:
1394:
1376:
1373:
1371:
1368:
1366:
1363:
1362:
1360:
1356:
1353:
1351:
1350:4-dimensional
1347:
1337:
1334:
1333:
1331:
1329:
1325:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1290:
1288:
1286:
1282:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1260:Centered cube
1258:
1256:
1253:
1252:
1250:
1248:
1244:
1241:
1239:
1238:3-dimensional
1235:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1176:
1174:
1172:
1168:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1118:
1116:
1114:
1110:
1107:
1105:
1104:2-dimensional
1101:
1097:
1093:
1088:
1084:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1048:Nonhypotenuse
1046:
1045:
1042:
1035:
1031:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
997:
994:
987:
983:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
934:
931:
926:
921:
917:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
883:
880:
873:
869:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
825:
822:
817:
811:
807:
797:
794:
792:
789:
787:
786:Perfect power
784:
782:
779:
777:
776:Seventh power
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
728:
725:
720:
715:
711:
707:
699:
694:
692:
687:
685:
680:
679:
676:
670:
664:
663:
659:
654:
653:0-387-20860-7
650:
646:
642:
638:
635:
634:
625:
622:
603:
597:
594:
591:
571:
568:
562:
556:
546:
543:
537:
536:
531:
522:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
479:
472:
469:
462:
458:
455:
453:
450:
448:
445:
443:
440:
439:
435:
433:
431:
427:
419:
417:
415:
411:
407:
403:
399:
395:
391:
386:
383:
379:
375:
371:
367:
363:
359:
355:
339:
336:
330:
324:
321:
318:
298:
295:
289:
283:
280:
277:
257:
254:
248:
242:
239:
236:
228:
224:
220:
216:
212:
208:
204:
200:
193:
176:
171:
166:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
78:
77:
76:
70:
66:
63:
62:
57:
50:
48:
46:
42:
38:
34:
30:
26:
22:
3017:Superperfect
3012:Refactorable
2896:
2807:Superperfect
2802:Hyperperfect
2787:Quasiperfect
2671:Prime factor
2330:Transposable
2194:Narcissistic
2101:Digital root
2021:Super-Poulet
1981:JordanâPĂłlya
1930:prime factor
1882:
1835:Noncototient
1802:Almost prime
1784:Superperfect
1759:Refactorable
1754:Quasiperfect
1729:Hyperperfect
1570:Pseudoprimes
1541:WallâSunâSun
1476:Ordered Bell
1446:FussâCatalan
1358:non-centered
1308:Dodecahedral
1285:non-centered
1171:non-centered
1073:Wolstenholme
818:× 2 ± 1
815:
814:Of the form
781:Eighth power
761:Fourth power
640:
624:
545:
533:
521:
481:
477:
471:
457:Weird number
452:Noncototient
423:
413:
409:
405:
401:
397:
393:
390:prime number
387:
381:
377:
226:
222:
218:
214:
210:
206:
202:
196:
74:
24:
18:
2941:Extravagant
2936:Equidigital
2897:Untouchable
2817:Semiperfect
2797:Hemiperfect
2726:Square-free
2351:Extravagant
2346:Equidigital
2301:permutation
2260:Palindromic
2234:Automorphic
2132:Sum-product
2111:Sum-product
2066:Persistence
1961:ErdĆsâWoods
1883:Untouchable
1764:Semiperfect
1714:Hemiperfect
1375:Tesseractic
1313:Icosahedral
1293:Tetrahedral
1224:Dodecagonal
925:Recursively
796:Prime power
771:Sixth power
766:Fifth power
746:Power of 10
704:Classes of
392:, since if
41:aliquot sum
35:of all the
21:mathematics
3032:Categories
2977:Arithmetic
2970:Other sets
2929:-dependent
2563:Graphemics
2436:Pernicious
2290:Undulating
2265:Pandigital
2239:Trimorphic
1840:Nontotient
1689:Arithmetic
1303:Octahedral
1204:Heptagonal
1194:Pentagonal
1179:Triangular
1020:SierpiĆski
942:Jacobsthal
741:Power of 3
736:Power of 2
643:(3rd ed),
463:References
447:Nontotient
426:Paul ErdĆs
420:Infinitude
177:Properties
3007:Descartes
2982:Deficient
2917:Betrothed
2822:Practical
2711:Semiprime
2706:Composite
2320:Parasitic
2169:Factorion
2096:Digit sum
2088:Digit sum
1906:Fortunate
1893:Primorial
1807:Semiprime
1744:Practical
1709:Descartes
1704:Deficient
1694:Betrothed
1536:Wieferich
1365:Pentatope
1328:pyramidal
1219:Decagonal
1214:Nonagonal
1209:Octagonal
1199:Hexagonal
1058:Practical
1005:Congruent
937:Fibonacci
901:Loeschian
598:ϕ
595:−
569:−
557:σ
535:MathWorld
514:115235810
337:−
325:σ
296:−
284:σ
255:−
243:σ
2992:Solitary
2987:Friendly
2912:Sociable
2902:Amicable
2890:-related
2843:Abundant
2741:Achilles
2731:Powerful
2644:Overview
2392:Friedman
2325:Primeval
2270:Repdigit
2227:-related
2174:Kaprekar
2148:Meertens
2071:Additive
2058:dynamics
1966:Friendly
1878:Sociable
1868:Amicable
1679:Abundant
1659:dynamics
1481:Schröder
1471:Narayana
1441:Eulerian
1431:Delannoy
1426:Dedekind
1247:centered
1113:centered
1000:Amenable
957:Narayana
947:Leonardo
843:Mersenne
791:Powerful
731:Achilles
498:41133889
436:See also
412:is
400:is
362:divisors
51:Examples
2997:Sublime
2951:Harshad
2777:Perfect
2761:Unusual
2751:Regular
2721:Sphenic
2656:Divisor
2565:related
2529:related
2493:related
2491:Sorting
2376:Vampire
2361:Harshad
2303:related
2275:Repunit
2189:Lychrel
2164:Dudeney
2016:StĂžrmer
2011:Sphenic
1996:Regular
1934:divisor
1873:Perfect
1769:Sublime
1739:Perfect
1466:Motzkin
1421:Catalan
962:Padovan
896:Leyland
891:Idoneal
886:Hilbert
858:Woodall
647:, 2004
506:1107382
168:in the
165:A005114
29:integer
2946:Frugal
2906:Triple
2746:Smooth
2716:Pronic
2431:Odious
2356:Frugal
2310:Cyclic
2299:Digit-
2006:Smooth
1991:Pronic
1951:Cyclic
1928:Other
1901:Euclid
1551:Wilson
1525:Primes
1184:Square
1053:Polite
1015:Riesel
1010:Knödel
972:Perrin
853:Thabit
838:Fermat
828:Cullen
751:Square
719:Powers
651:
512:
504:
496:
205:(with
2961:Smith
2878:Weird
2756:Rough
2701:Prime
2472:Prime
2467:Lucky
2456:sieve
2385:Other
2371:Smith
2251:Digit
2209:Happy
2184:Keith
2157:Other
2001:Rough
1971:Giuga
1436:Euler
1298:Cubic
952:Lucas
848:Proth
510:S2CID
494:JSTOR
23:, an
2927:Base
2426:Evil
2106:Self
2056:and
1946:Blum
1657:and
1461:Lobb
1416:Cake
1411:Bell
1161:Star
1068:Ulam
967:Pell
756:Cube
666:OEIS
649:ISBN
584:und
170:OEIS
2544:Ban
1932:or
1451:Lah
486:doi
368:or
160:290
156:288
152:276
148:268
144:262
140:248
136:246
132:238
128:216
124:210
120:206
116:188
112:162
108:146
104:124
100:120
33:sum
19:In
3034::
639:,
532:.
508:,
502:MR
500:,
492:,
482:41
480:,
311:,
270:,
217:+
209:,
203:pq
172:).
158:,
154:,
150:,
146:,
142:,
138:,
134:,
130:,
126:,
122:,
118:,
114:,
110:,
106:,
102:,
98:,
96:96
94:,
92:88
90:,
88:52
86:,
82:,
2908:)
2904:(
2631:e
2624:t
2617:v
816:a
697:e
690:t
683:v
607:)
604:n
601:(
592:n
572:n
566:)
563:n
560:(
538:.
488::
414:p
410:p
406:p
402:p
398:p
394:p
382:n
378:M
340:8
334:)
331:8
328:(
322:=
319:7
299:4
293:)
290:4
287:(
281:=
278:3
258:2
252:)
249:2
246:(
240:=
237:1
227:n
223:n
219:q
215:p
211:q
207:p
185::
84:5
80:2
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