459:
919:
20:
3180:
167:
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 Ă 21 â 32 = 10.
290:
340:
1282:
373:
226:
153:
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... (sequence
444:
862:
160:
1275:
179:
are deficient. More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many
2082:
1268:
2077:
2092:
2072:
908:
782:
2785:
2365:
1105:
918:
515:
2087:
2871:
855:
2187:
3204:
2537:
1856:
1649:
2572:
2542:
2217:
2207:
1059:
2713:
2127:
1861:
1841:
2403:
2567:
3214:
2662:
2285:
2042:
1851:
1833:
1727:
1717:
1707:
1095:
598:
511:
2547:
3209:
2790:
2335:
1956:
1742:
1737:
1732:
1722:
1699:
1080:
480:
1775:
848:
458:
2032:
231:
2901:
2866:
2652:
2562:
2436:
2411:
2320:
2310:
1922:
1904:
1824:
1234:
1100:
1024:
502:
295:
3161:
2431:
2305:
1936:
1712:
1492:
1419:
1085:
1044:
726:
489:
2416:
2270:
2197:
1352:
1014:
883:
615:
3125:
2765:
19:
3058:
2952:
2916:
2657:
2380:
2360:
2177:
1846:
1634:
1606:
1188:
1090:
630:
498:
2780:
2644:
2639:
2607:
2370:
2345:
2340:
2315:
2245:
2241:
2172:
2062:
1894:
1690:
1659:
1249:
1244:
1039:
1034:
1019:
958:
3179:
3183:
2937:
2932:
2846:
2820:
2718:
2697:
2469:
2350:
2300:
2222:
2192:
2132:
1899:
1879:
1810:
1523:
1173:
1168:
1129:
1049:
1029:
757:
2067:
806:
3077:
3022:
2876:
2851:
2825:
2602:
2280:
2275:
2202:
2182:
2167:
1889:
1871:
1790:
1780:
1765:
1543:
1528:
1209:
1149:
813:
778:
345:
3113:
2906:
2492:
2464:
2454:
2446:
2330:
2295:
2257:
1951:
1914:
1805:
1800:
1795:
1785:
1757:
1644:
1596:
1591:
1548:
1487:
1239:
1134:
1120:
1054:
938:
898:
788:
749:
542:
51:
204:
3089:
2978:
2911:
2837:
2760:
2734:
2552:
2265:
2122:
2057:
2027:
2017:
2012:
1678:
1586:
1533:
1377:
1317:
1224:
1219:
1144:
1138:
1075:
973:
963:
893:
792:
774:
625:
620:
575:
471:
24:
740:
Prielipp, Robert W. (1970). "Perfect numbers, abundant numbers, and deficient numbers".
3094:
2962:
2947:
2811:
2775:
2750:
2626:
2597:
2582:
2459:
2355:
2325:
2052:
2007:
1884:
1482:
1477:
1472:
1444:
1429:
1342:
1327:
1305:
1292:
1229:
1183:
1009:
993:
983:
953:
559:
533:
383:
66:
840:
392:
3198:
3017:
3001:
2942:
2896:
2592:
2577:
2487:
2212:
1770:
1639:
1601:
1558:
1439:
1424:
1414:
1372:
1362:
1337:
1178:
978:
968:
948:
816:
730:
462:
32:
3053:
3042:
2957:
2795:
2770:
2687:
2587:
2557:
2532:
2516:
2421:
2388:
2137:
2111:
2022:
1961:
1538:
1434:
1367:
1347:
1322:
1193:
1110:
988:
933:
903:
524:
188:
176:
3012:
2887:
2692:
2156:
2047:
2002:
1997:
1747:
1654:
1553:
1382:
1357:
1332:
603:
596:
was the first to subdivide numbers into deficient, perfect, or abundant, in his
199:
184:
70:
3149:
3130:
2426:
2037:
834:
593:
180:
1260:
2755:
2682:
2674:
2479:
2393:
1511:
943:
821:
769:
SĂĄndor, JĂłzsef; MitrinoviÄ, Dragoslav S.; Crstici, Borislav, eds. (2006).
2856:
830:
753:
761:
2861:
2520:
888:
379:
44:
382:
of deficient numbers are deficient. Moreover, all proper divisors of
732:
History of the Theory of
Numbers, Vol. I: Divisibility and Primality
602:(circa 100 CE). However, he applied this classification only to the
1159:
457:
18:
3147:
3111:
3075:
3039:
2999:
2624:
2513:
2239:
2154:
2109:
1986:
1676:
1623:
1575:
1509:
1461:
1399:
1303:
1264:
844:
389:
There exists at least one deficient number in the interval
155:
183:
deficient numbers. There are also an infinite number of
697:
228:
are deficient, because their only proper divisors are
395:
348:
298:
234:
207:
175:
Since the aliquot sums of prime numbers equal 1, all
65:. Equivalently, it is a number for which the sum of
2971:
2925:
2885:
2836:
2810:
2743:
2727:
2706:
2673:
2638:
2478:
2445:
2402:
2379:
2256:
1944:
1935:
1913:
1870:
1832:
1823:
1756:
1698:
1689:
1202:
1158:
1119:
1068:
1002:
926:
876:
81:, and their sum is less than 8, so 8 is deficient.
438:
367:
334:
284:
220:
1276:
856:
77:. For example, the proper divisors of 8 are
8:
16:Number whose divisor sum is less than itself
3144:
3108:
3072:
3036:
2996:
2670:
2635:
2621:
2510:
2253:
2236:
2151:
2106:
1983:
1941:
1829:
1695:
1686:
1673:
1620:
1577:Possessing a specific set of other numbers
1572:
1506:
1458:
1396:
1300:
1283:
1269:
1261:
863:
849:
841:
558:Closely related to deficient numbers are
427:
394:
353:
347:
306:
299:
297:
270:
251:
233:
212:
206:
685:
673:
661:
649:
285:{\displaystyle 1,p,p^{2},\dots ,p^{k-1}}
709:
698:SĂĄndor, MitrinoviÄ & Crstici (2006)
642:
335:{\displaystyle {\frac {p^{k}-1}{p-1}}}
7:
807:The Prime Glossary: Deficient number
149:The first few deficient numbers are
871:Divisibility-based sets of integers
735:. Carnegie Institute of Washington.
27:, of the deficiency of the number 8
14:
909:Fundamental theorem of arithmetic
3178:
2786:Perfect digit-to-digit invariant
917:
193:1 + 2 + 4 + 8 + ... + 2 = 2 - 1
115:. In terms of the aliquot sum
95:the sum of divisors, the value
433:
424:
411:
396:
1:
1625:Expressible via specific sums
2714:Multiplicative digital root
771:Handbook of number theory I
446:for all sufficiently large
3231:
599:Introduction to Arithmetic
3174:
3157:
3143:
3121:
3107:
3085:
3071:
3049:
3035:
3008:
2995:
2791:Perfect digital invariant
2634:
2620:
2528:
2509:
2366:Superior highly composite
2252:
2235:
2163:
2150:
2118:
2105:
1993:
1982:
1685:
1672:
1630:
1619:
1582:
1571:
1519:
1505:
1468:
1457:
1410:
1395:
1313:
1299:
1106:Superior highly composite
915:
652:, Theorem 1, pp. 693â694.
516:superior highly composite
187:deficient numbers as all
2404:Euler's totient function
2188:EulerâJacobi pseudoprime
1463:Other polynomial numbers
1003:Constrained divisor sums
2218:SomerâLucas pseudoprime
2208:LucasâCarmichael number
2043:Lazy caterer's sequence
742:The Mathematics Teacher
727:Dickson, Leonard Eugene
368:{\displaystyle p^{k}-1}
111:is called the number's
2093:WedderburnâEtherington
1493:Lucky numbers of Euler
555:
465:of numbers under 100:
440:
369:
336:
286:
222:
28:
2381:Prime omega functions
2198:Frobenius pseudoprime
1988:Combinatorial numbers
1857:Centered dodecahedral
1650:Primary pseudoperfect
884:Integer factorization
616:Almost perfect number
461:
441:
370:
337:
287:
223:
221:{\displaystyle p^{k}}
22:
2840:-composition related
2640:Arithmetic functions
2242:Arithmetic functions
2178:Elliptic pseudoprime
1862:Centered icosahedral
1842:Centered tetrahedral
754:10.5951/MT.63.8.0692
688:, Theorem 3, p. 694.
676:, Theorem 7, p. 695.
664:, Theorem 2, p. 694.
631:Superabundant number
393:
346:
296:
232:
205:
198:More generally, all
126:, the deficiency is
23:Demonstration, with
3205:Arithmetic dynamics
2766:Kaprekar's constant
2286:Colossally abundant
2173:Catalan pseudoprime
2073:SchröderâHipparchus
1852:Centered octahedral
1728:Centered heptagonal
1718:Centered pentagonal
1708:Centered triangular
1308:and related numbers
1096:Colossally abundant
927:Factorization forms
512:Colossally abundant
342:, which is at most
3184:Mathematics portal
3126:Aronson's sequence
2872:SmarandacheâWellin
2629:-dependent numbers
2336:Primitive abundant
2223:Strong pseudoprime
2213:Perrin pseudoprime
2193:Fermat pseudoprime
2133:Wolstenholme prime
1957:Squared triangular
1743:Centered decagonal
1738:Centered nonagonal
1733:Centered octagonal
1723:Centered hexagonal
1081:Primitive abundant
1069:With many divisors
817:"Deficient Number"
814:Weisstein, Eric W.
586:) > 2
556:
481:Primitive abundant
436:
365:
332:
282:
218:
29:
3215:Integer sequences
3192:
3191:
3170:
3169:
3139:
3138:
3103:
3102:
3067:
3066:
3031:
3030:
2991:
2990:
2987:
2986:
2806:
2805:
2616:
2615:
2505:
2504:
2501:
2500:
2447:Aliquot sequences
2258:Divisor functions
2231:
2230:
2203:Lucas pseudoprime
2183:Euler pseudoprime
2168:Carmichael number
2146:
2145:
2101:
2100:
1978:
1977:
1974:
1973:
1970:
1969:
1931:
1930:
1819:
1818:
1776:Square triangular
1668:
1667:
1615:
1614:
1567:
1566:
1501:
1500:
1453:
1452:
1391:
1390:
1258:
1257:
330:
3222:
3210:Divisor function
3182:
3145:
3114:Natural language
3109:
3073:
3041:Generated via a
3037:
2997:
2902:Digit-reassembly
2867:Self-descriptive
2671:
2636:
2622:
2573:LucasâCarmichael
2563:Harmonic divisor
2511:
2437:Sparsely totient
2412:Highly cototient
2321:Multiply perfect
2311:Highly composite
2254:
2237:
2152:
2107:
2088:Telephone number
1984:
1942:
1923:Square pyramidal
1905:Stella octangula
1830:
1696:
1687:
1679:Figurate numbers
1674:
1621:
1573:
1507:
1459:
1397:
1301:
1285:
1278:
1271:
1262:
1235:Harmonic divisor
1121:Aliquot sequence
1101:Highly composite
1025:Multiply perfect
921:
899:Divisor function
865:
858:
851:
842:
831:deficient number
827:
826:
796:
765:
736:
713:
707:
701:
695:
689:
683:
677:
671:
665:
659:
653:
647:
576:abundant numbers
549:
540:
531:
522:
509:
503:highly composite
496:
487:
478:
469:
454:Related concepts
445:
443:
442:
439:{\displaystyle }
437:
432:
431:
374:
372:
371:
366:
358:
357:
341:
339:
338:
333:
331:
329:
318:
311:
310:
300:
291:
289:
288:
283:
281:
280:
256:
255:
227:
225:
224:
219:
217:
216:
194:
158:
140:
125:
110:
94:
80:
76:
64:
57:
49:
41:defective number
37:deficient number
3230:
3229:
3225:
3224:
3223:
3221:
3220:
3219:
3195:
3194:
3193:
3188:
3166:
3162:Strobogrammatic
3153:
3135:
3117:
3099:
3081:
3063:
3045:
3027:
3004:
2983:
2967:
2926:Divisor-related
2921:
2881:
2832:
2802:
2739:
2723:
2702:
2669:
2642:
2630:
2612:
2524:
2523:related numbers
2497:
2474:
2441:
2432:Perfect totient
2398:
2375:
2306:Highly abundant
2248:
2227:
2159:
2142:
2114:
2097:
2083:Stirling second
1989:
1966:
1927:
1909:
1866:
1815:
1752:
1713:Centered square
1681:
1664:
1626:
1611:
1578:
1563:
1515:
1514:defined numbers
1497:
1464:
1449:
1420:Double Mersenne
1406:
1387:
1309:
1295:
1293:natural numbers
1289:
1259:
1254:
1198:
1154:
1115:
1086:Highly abundant
1064:
1045:Unitary perfect
998:
922:
913:
894:Unitary divisor
872:
869:
812:
811:
803:
785:
775:Springer-Verlag
768:
739:
725:
722:
717:
716:
708:
704:
696:
692:
686:Prielipp (1970)
684:
680:
674:Prielipp (1970)
672:
668:
662:Prielipp (1970)
660:
656:
650:Prielipp (1970)
648:
644:
639:
626:Sociable number
621:Amicable number
612:
570:) = 2
560:perfect numbers
554:
547:
545:
538:
536:
529:
527:
520:
518:
507:
505:
494:
492:
490:Highly abundant
485:
483:
476:
474:
467:
456:
423:
391:
390:
386:are deficient.
384:perfect numbers
349:
344:
343:
319:
302:
301:
294:
293:
266:
247:
230:
229:
208:
203:
202:
192:
173:
154:
147:
127:
116:
96:
85:
78:
74:
73:) is less than
67:proper divisors
59:
55:
52:sum of divisors
47:
25:Cuisenaire rods
17:
12:
11:
5:
3228:
3226:
3218:
3217:
3212:
3207:
3197:
3196:
3190:
3189:
3187:
3186:
3175:
3172:
3171:
3168:
3167:
3165:
3164:
3158:
3155:
3154:
3148:
3141:
3140:
3137:
3136:
3134:
3133:
3128:
3122:
3119:
3118:
3112:
3105:
3104:
3101:
3100:
3098:
3097:
3095:Sorting number
3092:
3090:Pancake number
3086:
3083:
3082:
3076:
3069:
3068:
3065:
3064:
3062:
3061:
3056:
3050:
3047:
3046:
3040:
3033:
3032:
3029:
3028:
3026:
3025:
3020:
3015:
3009:
3006:
3005:
3002:Binary numbers
3000:
2993:
2992:
2989:
2988:
2985:
2984:
2982:
2981:
2975:
2973:
2969:
2968:
2966:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2929:
2927:
2923:
2922:
2920:
2919:
2914:
2909:
2904:
2899:
2893:
2891:
2883:
2882:
2880:
2879:
2874:
2869:
2864:
2859:
2854:
2849:
2843:
2841:
2834:
2833:
2831:
2830:
2829:
2828:
2817:
2815:
2812:P-adic numbers
2808:
2807:
2804:
2803:
2801:
2800:
2799:
2798:
2788:
2783:
2778:
2773:
2768:
2763:
2758:
2753:
2747:
2745:
2741:
2740:
2738:
2737:
2731:
2729:
2728:Coding-related
2725:
2724:
2722:
2721:
2716:
2710:
2708:
2704:
2703:
2701:
2700:
2695:
2690:
2685:
2679:
2677:
2668:
2667:
2666:
2665:
2663:Multiplicative
2660:
2649:
2647:
2632:
2631:
2627:Numeral system
2625:
2618:
2617:
2614:
2613:
2611:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2535:
2529:
2526:
2525:
2514:
2507:
2506:
2503:
2502:
2499:
2498:
2496:
2495:
2490:
2484:
2482:
2476:
2475:
2473:
2472:
2467:
2462:
2457:
2451:
2449:
2443:
2442:
2440:
2439:
2434:
2429:
2424:
2419:
2417:Highly totient
2414:
2408:
2406:
2400:
2399:
2397:
2396:
2391:
2385:
2383:
2377:
2376:
2374:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2271:Almost perfect
2268:
2262:
2260:
2250:
2249:
2240:
2233:
2232:
2229:
2228:
2226:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2164:
2161:
2160:
2155:
2148:
2147:
2144:
2143:
2141:
2140:
2135:
2130:
2125:
2119:
2116:
2115:
2110:
2103:
2102:
2099:
2098:
2096:
2095:
2090:
2085:
2080:
2078:Stirling first
2075:
2070:
2065:
2060:
2055:
2050:
2045:
2040:
2035:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1994:
1991:
1990:
1987:
1980:
1979:
1976:
1975:
1972:
1971:
1968:
1967:
1965:
1964:
1959:
1954:
1948:
1946:
1939:
1933:
1932:
1929:
1928:
1926:
1925:
1919:
1917:
1911:
1910:
1908:
1907:
1902:
1897:
1892:
1887:
1882:
1876:
1874:
1868:
1867:
1865:
1864:
1859:
1854:
1849:
1844:
1838:
1836:
1827:
1821:
1820:
1817:
1816:
1814:
1813:
1808:
1803:
1798:
1793:
1788:
1783:
1778:
1773:
1768:
1762:
1760:
1754:
1753:
1751:
1750:
1745:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1704:
1702:
1693:
1683:
1682:
1677:
1670:
1669:
1666:
1665:
1663:
1662:
1657:
1652:
1647:
1642:
1637:
1631:
1628:
1627:
1624:
1617:
1616:
1613:
1612:
1610:
1609:
1604:
1599:
1594:
1589:
1583:
1580:
1579:
1576:
1569:
1568:
1565:
1564:
1562:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1526:
1520:
1517:
1516:
1510:
1503:
1502:
1499:
1498:
1496:
1495:
1490:
1485:
1480:
1475:
1469:
1466:
1465:
1462:
1455:
1454:
1451:
1450:
1448:
1447:
1442:
1437:
1432:
1427:
1422:
1417:
1411:
1408:
1407:
1400:
1393:
1392:
1389:
1388:
1386:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1325:
1320:
1314:
1311:
1310:
1304:
1297:
1296:
1290:
1288:
1287:
1280:
1273:
1265:
1256:
1255:
1253:
1252:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1206:
1204:
1200:
1199:
1197:
1196:
1191:
1186:
1181:
1176:
1171:
1165:
1163:
1156:
1155:
1153:
1152:
1147:
1142:
1132:
1126:
1124:
1117:
1116:
1114:
1113:
1108:
1103:
1098:
1093:
1088:
1083:
1078:
1072:
1070:
1066:
1065:
1063:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1015:Almost perfect
1012:
1006:
1004:
1000:
999:
997:
996:
991:
986:
981:
976:
971:
966:
961:
956:
951:
946:
941:
936:
930:
928:
924:
923:
916:
914:
912:
911:
906:
901:
896:
891:
886:
880:
878:
874:
873:
870:
868:
867:
860:
853:
845:
839:
838:
828:
809:
802:
801:External links
799:
798:
797:
783:
766:
748:(8): 692â696.
737:
721:
718:
715:
714:
710:Dickson (1919)
702:
700:, p. 108.
690:
678:
666:
654:
641:
640:
638:
635:
634:
633:
628:
623:
618:
611:
608:
546:
537:
528:
519:
506:
493:
484:
475:
466:
455:
452:
435:
430:
426:
422:
419:
416:
413:
410:
407:
404:
401:
398:
364:
361:
356:
352:
328:
325:
322:
317:
314:
309:
305:
279:
276:
273:
269:
265:
262:
259:
254:
250:
246:
243:
240:
237:
215:
211:
191:have the sum (
172:
169:
165:
164:
146:
143:
50:for which the
43:is a positive
15:
13:
10:
9:
6:
4:
3:
2:
3227:
3216:
3213:
3211:
3208:
3206:
3203:
3202:
3200:
3185:
3181:
3177:
3176:
3173:
3163:
3160:
3159:
3156:
3151:
3146:
3142:
3132:
3129:
3127:
3124:
3123:
3120:
3115:
3110:
3106:
3096:
3093:
3091:
3088:
3087:
3084:
3079:
3074:
3070:
3060:
3057:
3055:
3052:
3051:
3048:
3044:
3038:
3034:
3024:
3021:
3019:
3016:
3014:
3011:
3010:
3007:
3003:
2998:
2994:
2980:
2977:
2976:
2974:
2970:
2964:
2961:
2959:
2956:
2954:
2953:Polydivisible
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2928:
2924:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2894:
2892:
2889:
2884:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2844:
2842:
2839:
2835:
2827:
2824:
2823:
2822:
2819:
2818:
2816:
2813:
2809:
2797:
2794:
2793:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2748:
2746:
2742:
2736:
2733:
2732:
2730:
2726:
2720:
2717:
2715:
2712:
2711:
2709:
2707:Digit product
2705:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2680:
2678:
2676:
2672:
2664:
2661:
2659:
2656:
2655:
2654:
2651:
2650:
2648:
2646:
2641:
2637:
2633:
2628:
2623:
2619:
2609:
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2543:ErdĆsâNicolas
2541:
2539:
2536:
2534:
2531:
2530:
2527:
2522:
2518:
2512:
2508:
2494:
2491:
2489:
2486:
2485:
2483:
2481:
2477:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2452:
2450:
2448:
2444:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2409:
2407:
2405:
2401:
2395:
2392:
2390:
2387:
2386:
2384:
2382:
2378:
2372:
2369:
2367:
2364:
2362:
2361:Superabundant
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2263:
2261:
2259:
2255:
2251:
2247:
2243:
2238:
2234:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2165:
2162:
2158:
2153:
2149:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2120:
2117:
2113:
2108:
2104:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1996:
1995:
1992:
1985:
1981:
1963:
1960:
1958:
1955:
1953:
1950:
1949:
1947:
1943:
1940:
1938:
1937:4-dimensional
1934:
1924:
1921:
1920:
1918:
1916:
1912:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1877:
1875:
1873:
1869:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1847:Centered cube
1845:
1843:
1840:
1839:
1837:
1835:
1831:
1828:
1826:
1825:3-dimensional
1822:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1763:
1761:
1759:
1755:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1705:
1703:
1701:
1697:
1694:
1692:
1691:2-dimensional
1688:
1684:
1680:
1675:
1671:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1635:Nonhypotenuse
1633:
1632:
1629:
1622:
1618:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1584:
1581:
1574:
1570:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1521:
1518:
1513:
1508:
1504:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1470:
1467:
1460:
1456:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1412:
1409:
1404:
1398:
1394:
1384:
1381:
1379:
1376:
1374:
1373:Perfect power
1371:
1369:
1366:
1364:
1363:Seventh power
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1315:
1312:
1307:
1302:
1298:
1294:
1286:
1281:
1279:
1274:
1272:
1267:
1266:
1263:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1207:
1205:
1201:
1195:
1192:
1190:
1189:Polydivisible
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1166:
1164:
1161:
1157:
1151:
1148:
1146:
1143:
1140:
1136:
1133:
1131:
1128:
1127:
1125:
1122:
1118:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1091:Superabundant
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1073:
1071:
1067:
1061:
1060:ErdĆsâNicolas
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1007:
1005:
1001:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
969:Perfect power
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
931:
929:
925:
920:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
881:
879:
875:
866:
861:
859:
854:
852:
847:
846:
843:
836:
832:
829:
824:
823:
818:
815:
810:
808:
805:
804:
800:
794:
790:
786:
784:1-4020-4215-9
780:
776:
773:. Dordrecht:
772:
767:
763:
759:
755:
751:
747:
743:
738:
734:
733:
728:
724:
723:
719:
711:
706:
703:
699:
694:
691:
687:
682:
679:
675:
670:
667:
663:
658:
655:
651:
646:
643:
636:
632:
629:
627:
624:
622:
619:
617:
614:
613:
609:
607:
605:
601:
600:
595:
591:
589:
585:
581:
577:
573:
569:
565:
561:
553:
544:
535:
526:
517:
513:
504:
500:
499:Superabundant
491:
482:
473:
464:
463:Euler diagram
460:
453:
451:
449:
428:
420:
417:
414:
408:
405:
402:
399:
387:
385:
381:
376:
362:
359:
354:
350:
326:
323:
320:
315:
312:
307:
303:
292:which sum to
277:
274:
271:
267:
263:
260:
257:
252:
248:
244:
241:
238:
235:
213:
209:
201:
196:
190:
189:powers of two
186:
182:
178:
177:prime numbers
170:
168:
162:
157:
152:
151:
150:
144:
142:
138:
134:
130:
123:
119:
114:
108:
104:
100:
92:
88:
82:
72:
68:
63:
58:is less than
53:
46:
42:
38:
34:
33:number theory
26:
21:
2917:Transposable
2781:Narcissistic
2688:Digital root
2608:Super-Poulet
2568:JordanâPĂłlya
2517:prime factor
2422:Noncototient
2389:Almost prime
2371:Superperfect
2346:Refactorable
2341:Quasiperfect
2316:Hyperperfect
2290:
2157:Pseudoprimes
2128:WallâSunâSun
2063:Ordered Bell
2033:FussâCatalan
1945:non-centered
1895:Dodecahedral
1872:non-centered
1758:non-centered
1660:Wolstenholme
1405:× 2 ± 1
1402:
1401:Of the form
1368:Eighth power
1348:Fourth power
1250:Superperfect
1245:Refactorable
1214:
1040:Superperfect
1035:Hyperperfect
1020:Quasiperfect
904:Prime factor
820:
770:
745:
741:
731:
712:, p. 3.
705:
693:
681:
669:
657:
645:
604:even numbers
597:
592:
587:
583:
579:
571:
567:
563:
557:
551:
447:
388:
377:
200:prime powers
197:
174:
166:
148:
136:
132:
128:
121:
117:
112:
106:
102:
98:
90:
86:
84:Denoting by
83:
61:
40:
36:
30:
2938:Extravagant
2933:Equidigital
2888:permutation
2847:Palindromic
2821:Automorphic
2719:Sum-product
2698:Sum-product
2653:Persistence
2548:ErdĆsâWoods
2470:Untouchable
2351:Semiperfect
2301:Hemiperfect
1962:Tesseractic
1900:Icosahedral
1880:Tetrahedral
1811:Dodecagonal
1512:Recursively
1383:Prime power
1358:Sixth power
1353:Fifth power
1333:Power of 10
1291:Classes of
1174:Extravagant
1169:Equidigital
1130:Untouchable
1050:Semiperfect
1030:Hemiperfect
959:Square-free
378:All proper
79:1, 2, and 4
71:aliquot sum
3199:Categories
3150:Graphemics
3023:Pernicious
2877:Undulating
2852:Pandigital
2826:Trimorphic
2427:Nontotient
2276:Arithmetic
1890:Octahedral
1791:Heptagonal
1781:Pentagonal
1766:Triangular
1607:SierpiĆski
1529:Jacobsthal
1328:Power of 3
1323:Power of 2
1210:Arithmetic
1203:Other sets
1162:-dependent
835:PlanetMath
793:1151.11300
720:References
594:Nicomachus
171:Properties
113:deficiency
2907:Parasitic
2756:Factorion
2683:Digit sum
2675:Digit sum
2493:Fortunate
2480:Primorial
2394:Semiprime
2331:Practical
2296:Descartes
2291:Deficient
2281:Betrothed
2123:Wieferich
1952:Pentatope
1915:pyramidal
1806:Decagonal
1801:Nonagonal
1796:Octagonal
1786:Hexagonal
1645:Practical
1592:Congruent
1524:Fibonacci
1488:Loeschian
1240:Descartes
1215:Deficient
1150:Betrothed
1055:Practical
944:Semiprime
939:Composite
822:MathWorld
552:Deficient
543:Composite
418:
360:−
324:−
313:−
275:−
261:…
2979:Friedman
2912:Primeval
2857:Repdigit
2814:-related
2761:Kaprekar
2735:Meertens
2658:Additive
2645:dynamics
2553:Friendly
2465:Sociable
2455:Amicable
2266:Abundant
2246:dynamics
2068:Schröder
2058:Narayana
2028:Eulerian
2018:Delannoy
2013:Dedekind
1834:centered
1700:centered
1587:Amenable
1544:Narayana
1534:Leonardo
1430:Mersenne
1378:Powerful
1318:Achilles
1225:Solitary
1220:Friendly
1145:Sociable
1135:Amicable
1123:-related
1076:Abundant
974:Achilles
964:Powerful
877:Overview
762:27958492
729:(1919).
610:See also
472:Abundant
380:divisors
145:Examples
3152:related
3116:related
3080:related
3078:Sorting
2963:Vampire
2948:Harshad
2890:related
2862:Repunit
2776:Lychrel
2751:Dudeney
2603:StĂžrmer
2598:Sphenic
2583:Regular
2521:divisor
2460:Perfect
2356:Sublime
2326:Perfect
2053:Motzkin
2008:Catalan
1549:Padovan
1483:Leyland
1478:Idoneal
1473:Hilbert
1445:Woodall
1230:Sublime
1184:Harshad
1010:Perfect
994:Unusual
984:Regular
954:Sphenic
889:Divisor
550:
541:
534:Perfect
532:
523:
510:
497:
488:
479:
470:
159:in the
156:A005100
45:integer
3018:Odious
2943:Frugal
2897:Cyclic
2886:Digit-
2593:Smooth
2578:Pronic
2538:Cyclic
2515:Other
2488:Euclid
2138:Wilson
2112:Primes
1771:Square
1640:Polite
1602:Riesel
1597:Knödel
1559:Perrin
1440:Thabit
1425:Fermat
1415:Cullen
1338:Square
1306:Powers
1179:Frugal
1139:Triple
979:Smooth
949:Pronic
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3054:Lucky
3043:sieve
2972:Other
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2796:Happy
2771:Keith
2744:Other
2588:Rough
2558:Giuga
2023:Euler
1885:Cubic
1539:Lucas
1435:Proth
1194:Smith
1111:Weird
989:Rough
934:Prime
758:JSTOR
637:Notes
578:with
562:with
525:Weird
514:and
501:and
3013:Evil
2693:Self
2643:and
2533:Blum
2244:and
2048:Lobb
2003:Cake
1998:Bell
1748:Star
1655:Ulam
1554:Pell
1343:Cube
1160:Base
779:ISBN
185:even
161:OEIS
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3131:Ban
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