43:
404:
2893:
594:
31:
2800:
85:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102,
77:
For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself.
86:
104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150. (sequence
139:. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter
503:
Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are
283:
200:
902:
574:
Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called
546:
363:
321:
607:
2836:
135:
or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a
93:
895:
100:
Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number
1702:
888:
1697:
795:
1712:
1692:
3259:
2882:
770:
113:
3079:
2892:
2405:
1985:
460:
131:
One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a
123:
that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.
62:
that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one
1707:
2491:
2829:
1807:
2157:
1476:
1269:
3033:
2192:
2162:
1837:
1827:
2333:
1747:
1481:
1461:
2023:
2187:
3249:
3069:
2282:
1905:
1662:
1471:
1453:
1347:
1337:
1327:
456:
2167:
787:
3054:
2410:
1955:
1576:
1362:
1357:
1352:
1342:
1319:
425:
2822:
1395:
403:
1652:
219:
3208:
3074:
2998:
2521:
2486:
2272:
2182:
2056:
2031:
1940:
1930:
1542:
1524:
1444:
809:
561:
447:
3244:
3059:
3018:
2781:
2051:
1925:
1556:
1332:
1112:
1039:
779:
622:
434:
145:
2988:
2857:
2036:
1890:
1817:
972:
617:
612:
30:
2745:
2385:
593:
3162:
3064:
2678:
2572:
2536:
2277:
2000:
1980:
1797:
1466:
1254:
1226:
443:
3254:
3223:
3218:
3013:
3008:
2993:
2932:
2400:
2264:
2259:
2227:
1990:
1965:
1960:
1935:
1865:
1861:
1792:
1682:
1514:
1310:
1279:
567:
Composite numbers have also been called "rectangular numbers", but that name can also refer to the
506:
386:
2799:
3147:
3142:
3103:
3023:
3003:
2803:
2557:
2552:
2466:
2440:
2338:
2317:
2089:
1970:
1920:
1842:
1812:
1752:
1519:
1499:
1430:
1143:
599:
71:
1687:
206:
871:
Lists of composites with prime factorization (first 100, 1,000, 10,000, 100,000, and 1,000,000)
333:
291:
3183:
3123:
2697:
2642:
2496:
2471:
2445:
2222:
1900:
1895:
1822:
1802:
1787:
1509:
1491:
1410:
1400:
1385:
1163:
1148:
853:
833:
813:
791:
766:
74: 1, so the composite numbers are exactly the numbers that are not prime and not a unit.
3213:
3188:
3108:
3094:
3028:
2872:
2733:
2526:
2112:
2084:
2074:
2066:
1950:
1915:
1910:
1877:
1571:
1534:
1425:
1420:
1415:
1405:
1377:
1264:
1216:
1211:
1168:
1107:
496:
59:
3198:
3193:
3118:
3112:
3049:
2947:
2937:
2867:
2709:
2598:
2531:
2457:
2380:
2354:
2172:
1885:
1742:
1677:
1647:
1637:
1632:
1298:
1206:
1153:
997:
937:
829:
416:
374:
35:
3203:
3157:
2983:
2967:
2957:
2927:
2714:
2582:
2567:
2431:
2395:
2370:
2246:
2217:
2202:
2079:
1975:
1945:
1672:
1627:
1504:
1102:
1097:
1092:
1064:
1049:
962:
947:
925:
912:
762:
478:
136:
120:
2814:
3238:
3152:
2952:
2942:
2922:
2637:
2621:
2562:
2516:
2212:
2197:
2107:
1832:
1390:
1259:
1221:
1178:
1059:
1044:
1034:
992:
982:
957:
849:
575:
568:
407:
378:
42:
3167:
3084:
2962:
2907:
2877:
2673:
2662:
2577:
2415:
2390:
2307:
2207:
2177:
2152:
2136:
2041:
2008:
1757:
1731:
1642:
1581:
1158:
1054:
987:
967:
942:
579:
469:
105:
101:
67:
17:
2632:
2507:
2312:
1776:
1667:
1622:
1617:
1367:
1274:
1173:
1002:
977:
952:
397:
393:
2769:
2750:
2046:
1657:
589:
47:
880:
2917:
2375:
2302:
2294:
2099:
2013:
1131:
870:
132:
400:= 2 Ă 3 Ă 7, none of the prime factors are repeated, so 42 is squarefree.
2476:
396:= 2 Ă 3, all the prime factors are repeated, so 72 is a powerful number.
875:
108:
can be written as 2 Ă 3 Ă 5; furthermore, this representation is unique
2862:
2481:
2140:
63:
3133:
402:
109:
41:
29:
817:
857:
837:
2818:
2767:
2731:
2695:
2659:
2619:
2244:
2133:
1859:
1774:
1729:
1606:
1296:
1243:
1195:
1129:
1081:
1019:
923:
884:
66:
other than 1 and itself. Every positive integer is composite,
288:
However, for prime numbers, the function also returns â1 and
571:, numbers that are the product of two consecutive integers.
88:
373:
the prime factors of a number are repeated it is called a
213:
is half the total of prime factors), while for the former
876:
Divisor Plot (patterns found in large composite numbers)
509:
336:
294:
222:
148:
652:
650:
104:
can be written as 13 Ă 23, and the composite number
3176:
3132:
3093:
3042:
2976:
2900:
2850:
2591:
2545:
2505:
2456:
2430:
2363:
2347:
2326:
2293:
2258:
2098:
2065:
2022:
1999:
1876:
1564:
1555:
1533:
1490:
1452:
1443:
1376:
1318:
1309:
844:Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970),
540:
357:
315:
277:
194:
112:the order of the factors. This fact is called the
728:
641:
564:(though the first two such numbers are 1 and 2).
385:of its prime factors are repeated, it is called
826:Introduction To Modern Algebra, Revised Edition
608:Canonical representation of a positive integer
2830:
896:
8:
535:
510:
389:. (All prime numbers and 1 are squarefree.)
38:, of the divisors of the composite number 10
2837:
2823:
2815:
2764:
2728:
2692:
2656:
2616:
2290:
2255:
2241:
2130:
1873:
1856:
1771:
1726:
1603:
1561:
1449:
1315:
1306:
1293:
1240:
1197:Possessing a specific set of other numbers
1192:
1126:
1078:
1016:
920:
903:
889:
881:
529:
508:
335:
327:with one or more repeated prime factors,
293:
251:
221:
177:
147:
806:Elementary Introduction to Number Theory
692:
680:
668:
634:
278:{\displaystyle \mu (n)=(-1)^{2x+1}=-1.}
46:Composite numbers can be arranged into
716:
81:The composite numbers up to 150 are:
7:
740:
704:
656:
27:Integer having a non-trivial divisor
2845:Divisibility-based sets of integers
195:{\displaystyle \mu (n)=(-1)^{2x}=1}
759:A First Course In Abstract Algebra
25:
2883:Fundamental theorem of arithmetic
114:fundamental theorem of arithmetic
2891:
2798:
2406:Perfect digit-to-digit invariant
592:
552:that has more divisors than any
346:
340:
304:
298:
248:
238:
232:
226:
174:
164:
158:
152:
1:
1245:Expressible via specific sums
729:Pettofrezzo & Byrkit 1970
642:Pettofrezzo & Byrkit 1970
541:{\displaystyle \{1,p,p^{2}\}}
788:Blaisdell Publishing Company
2334:Multiplicative digital root
808:(2nd ed.), Lexington:
3276:
757:Fraleigh, John B. (1976),
381:are powerful numbers). If
3080:Superior highly composite
2889:
2794:
2777:
2763:
2741:
2727:
2705:
2691:
2669:
2655:
2628:
2615:
2411:Perfect digital invariant
2254:
2240:
2148:
2129:
1986:Superior highly composite
1872:
1855:
1783:
1770:
1738:
1725:
1613:
1602:
1305:
1292:
1250:
1239:
1202:
1191:
1139:
1125:
1088:
1077:
1030:
1015:
933:
919:
846:Elements of Number Theory
761:(2nd ed.), Reading:
461:superior highly composite
358:{\displaystyle \mu (n)=0}
316:{\displaystyle \mu (1)=1}
50:but prime numbers cannot.
3260:Elementary number theory
2977:Constrained divisor sums
2024:Euler's totient function
1808:EulerâJacobi pseudoprime
1083:Other polynomial numbers
804:Long, Calvin T. (1972),
119:There are several known
1838:SomerâLucas pseudoprime
1828:LucasâCarmichael number
1663:Lazy caterer's sequence
824:McCoy, Neal H. (1968),
810:D. C. Heath and Company
562:highly composite number
1713:WedderburnâEtherington
1113:Lucky numbers of Euler
623:Table of prime factors
542:
500:
410:of numbers under 100:
359:
317:
279:
196:
51:
39:
2858:Integer factorization
2001:Prime omega functions
1818:Frobenius pseudoprime
1608:Combinatorial numbers
1477:Centered dodecahedral
1270:Primary pseudoperfect
618:Sieve of Eratosthenes
613:Integer factorization
543:
406:
360:
318:
280:
197:
45:
33:
2460:-composition related
2260:Arithmetic functions
1862:Arithmetic functions
1798:Elliptic pseudoprime
1482:Centered icosahedral
1462:Centered tetrahedral
848:, Englewood Cliffs:
671:, pp. 198, 266.
507:
334:
292:
220:
146:
34:Demonstration, with
3070:Colossally abundant
2901:Factorization forms
2386:Kaprekar's constant
1906:Colossally abundant
1793:Catalan pseudoprime
1693:SchröderâHipparchus
1472:Centered octahedral
1348:Centered heptagonal
1338:Centered pentagonal
1328:Centered triangular
928:and related numbers
457:Colossally abundant
3055:Primitive abundant
3043:With many divisors
2804:Mathematics portal
2746:Aronson's sequence
2492:SmarandacheâWellin
2249:-dependent numbers
1956:Primitive abundant
1843:Strong pseudoprime
1833:Perrin pseudoprime
1813:Fermat pseudoprime
1753:Wolstenholme prime
1577:Squared triangular
1363:Centered decagonal
1358:Centered nonagonal
1353:Centered octagonal
1343:Centered hexagonal
600:Mathematics portal
538:
501:
426:Primitive abundant
355:
313:
275:
192:
52:
40:
3250:Integer sequences
3232:
3231:
2812:
2811:
2790:
2789:
2759:
2758:
2723:
2722:
2687:
2686:
2651:
2650:
2611:
2610:
2607:
2606:
2426:
2425:
2236:
2235:
2125:
2124:
2121:
2120:
2067:Aliquot sequences
1878:Divisor functions
1851:
1850:
1823:Lucas pseudoprime
1803:Euler pseudoprime
1788:Carmichael number
1766:
1765:
1721:
1720:
1598:
1597:
1594:
1593:
1590:
1589:
1551:
1550:
1439:
1438:
1396:Square triangular
1288:
1287:
1235:
1234:
1187:
1186:
1121:
1120:
1073:
1072:
1011:
1010:
784:Topics In Algebra
644:, pp. 23â24.
18:Composite numbers
16:(Redirected from
3267:
3209:Harmonic divisor
3095:Aliquot sequence
3075:Highly composite
2999:Multiply perfect
2895:
2873:Divisor function
2839:
2832:
2825:
2816:
2802:
2765:
2734:Natural language
2729:
2693:
2661:Generated via a
2657:
2617:
2522:Digit-reassembly
2487:Self-descriptive
2291:
2256:
2242:
2193:LucasâCarmichael
2183:Harmonic divisor
2131:
2057:Sparsely totient
2032:Highly cototient
1941:Multiply perfect
1931:Highly composite
1874:
1857:
1772:
1727:
1708:Telephone number
1604:
1562:
1543:Square pyramidal
1525:Stella octangula
1450:
1316:
1307:
1299:Figurate numbers
1294:
1241:
1193:
1127:
1079:
1017:
921:
905:
898:
891:
882:
860:
840:
820:
800:
775:
744:
738:
732:
726:
720:
714:
708:
702:
696:
690:
684:
678:
672:
666:
660:
654:
645:
639:
602:
597:
596:
582:, respectively.
547:
545:
544:
539:
534:
533:
494:
485:
476:
467:
454:
448:highly composite
441:
432:
423:
414:
364:
362:
361:
356:
322:
320:
319:
314:
284:
282:
281:
276:
265:
264:
205:(where Ό is the
201:
199:
198:
193:
185:
184:
91:
60:positive integer
56:composite number
21:
3275:
3274:
3270:
3269:
3268:
3266:
3265:
3264:
3235:
3234:
3233:
3228:
3172:
3128:
3089:
3060:Highly abundant
3038:
3019:Unitary perfect
2972:
2896:
2887:
2868:Unitary divisor
2846:
2843:
2813:
2808:
2786:
2782:Strobogrammatic
2773:
2755:
2737:
2719:
2701:
2683:
2665:
2647:
2624:
2603:
2587:
2546:Divisor-related
2541:
2501:
2452:
2422:
2359:
2343:
2322:
2289:
2262:
2250:
2232:
2144:
2143:related numbers
2117:
2094:
2061:
2052:Perfect totient
2018:
1995:
1926:Highly abundant
1868:
1847:
1779:
1762:
1734:
1717:
1703:Stirling second
1609:
1586:
1547:
1529:
1486:
1435:
1372:
1333:Centered square
1301:
1284:
1246:
1231:
1198:
1183:
1135:
1134:defined numbers
1117:
1084:
1069:
1040:Double Mersenne
1026:
1007:
929:
915:
913:natural numbers
909:
867:
843:
830:Allyn and Bacon
823:
803:
798:
780:Herstein, I. N.
778:
773:
756:
753:
748:
747:
739:
735:
727:
723:
715:
711:
703:
699:
691:
687:
679:
675:
667:
663:
655:
648:
640:
636:
631:
598:
591:
588:
525:
505:
504:
499:
492:
490:
483:
481:
474:
472:
465:
463:
452:
450:
439:
437:
435:Highly abundant
430:
428:
421:
419:
412:
375:powerful number
332:
331:
323:. For a number
290:
289:
247:
218:
217:
207:Möbius function
173:
144:
143:
129:
121:primality tests
87:
36:Cuisenaire rods
28:
23:
22:
15:
12:
11:
5:
3273:
3271:
3263:
3262:
3257:
3252:
3247:
3237:
3236:
3230:
3229:
3227:
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3180:
3178:
3174:
3173:
3171:
3170:
3165:
3160:
3155:
3150:
3145:
3139:
3137:
3130:
3129:
3127:
3126:
3121:
3116:
3106:
3100:
3098:
3091:
3090:
3088:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3046:
3044:
3040:
3039:
3037:
3036:
3031:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2989:Almost perfect
2986:
2980:
2978:
2974:
2973:
2971:
2970:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2904:
2902:
2898:
2897:
2890:
2888:
2886:
2885:
2880:
2875:
2870:
2865:
2860:
2854:
2852:
2848:
2847:
2844:
2842:
2841:
2834:
2827:
2819:
2810:
2809:
2807:
2806:
2795:
2792:
2791:
2788:
2787:
2785:
2784:
2778:
2775:
2774:
2768:
2761:
2760:
2757:
2756:
2754:
2753:
2748:
2742:
2739:
2738:
2732:
2725:
2724:
2721:
2720:
2718:
2717:
2715:Sorting number
2712:
2710:Pancake number
2706:
2703:
2702:
2696:
2689:
2688:
2685:
2684:
2682:
2681:
2676:
2670:
2667:
2666:
2660:
2653:
2652:
2649:
2648:
2646:
2645:
2640:
2635:
2629:
2626:
2625:
2622:Binary numbers
2620:
2613:
2612:
2609:
2608:
2605:
2604:
2602:
2601:
2595:
2593:
2589:
2588:
2586:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2549:
2547:
2543:
2542:
2540:
2539:
2534:
2529:
2524:
2519:
2513:
2511:
2503:
2502:
2500:
2499:
2494:
2489:
2484:
2479:
2474:
2469:
2463:
2461:
2454:
2453:
2451:
2450:
2449:
2448:
2437:
2435:
2432:P-adic numbers
2428:
2427:
2424:
2423:
2421:
2420:
2419:
2418:
2408:
2403:
2398:
2393:
2388:
2383:
2378:
2373:
2367:
2365:
2361:
2360:
2358:
2357:
2351:
2349:
2348:Coding-related
2345:
2344:
2342:
2341:
2336:
2330:
2328:
2324:
2323:
2321:
2320:
2315:
2310:
2305:
2299:
2297:
2288:
2287:
2286:
2285:
2283:Multiplicative
2280:
2269:
2267:
2252:
2251:
2247:Numeral system
2245:
2238:
2237:
2234:
2233:
2231:
2230:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2149:
2146:
2145:
2134:
2127:
2126:
2123:
2122:
2119:
2118:
2116:
2115:
2110:
2104:
2102:
2096:
2095:
2093:
2092:
2087:
2082:
2077:
2071:
2069:
2063:
2062:
2060:
2059:
2054:
2049:
2044:
2039:
2037:Highly totient
2034:
2028:
2026:
2020:
2019:
2017:
2016:
2011:
2005:
2003:
1997:
1996:
1994:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1903:
1898:
1893:
1891:Almost perfect
1888:
1882:
1880:
1870:
1869:
1860:
1853:
1852:
1849:
1848:
1846:
1845:
1840:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1790:
1784:
1781:
1780:
1775:
1768:
1767:
1764:
1763:
1761:
1760:
1755:
1750:
1745:
1739:
1736:
1735:
1730:
1723:
1722:
1719:
1718:
1716:
1715:
1710:
1705:
1700:
1698:Stirling first
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1614:
1611:
1610:
1607:
1600:
1599:
1596:
1595:
1592:
1591:
1588:
1587:
1585:
1584:
1579:
1574:
1568:
1566:
1559:
1553:
1552:
1549:
1548:
1546:
1545:
1539:
1537:
1531:
1530:
1528:
1527:
1522:
1517:
1512:
1507:
1502:
1496:
1494:
1488:
1487:
1485:
1484:
1479:
1474:
1469:
1464:
1458:
1456:
1447:
1441:
1440:
1437:
1436:
1434:
1433:
1428:
1423:
1418:
1413:
1408:
1403:
1398:
1393:
1388:
1382:
1380:
1374:
1373:
1371:
1370:
1365:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1324:
1322:
1313:
1303:
1302:
1297:
1290:
1289:
1286:
1285:
1283:
1282:
1277:
1272:
1267:
1262:
1257:
1251:
1248:
1247:
1244:
1237:
1236:
1233:
1232:
1230:
1229:
1224:
1219:
1214:
1209:
1203:
1200:
1199:
1196:
1189:
1188:
1185:
1184:
1182:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1140:
1137:
1136:
1130:
1123:
1122:
1119:
1118:
1116:
1115:
1110:
1105:
1100:
1095:
1089:
1086:
1085:
1082:
1075:
1074:
1071:
1070:
1068:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1031:
1028:
1027:
1020:
1013:
1012:
1009:
1008:
1006:
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
934:
931:
930:
924:
917:
916:
910:
908:
907:
900:
893:
885:
879:
878:
873:
866:
865:External links
863:
862:
861:
841:
821:
801:
797:978-1114541016
796:
776:
771:
763:Addison-Wesley
752:
749:
746:
745:
743:, p. 159.
733:
721:
709:
697:
695:, p. 270.
685:
683:, p. 106.
673:
661:
646:
633:
632:
630:
627:
626:
625:
620:
615:
610:
604:
603:
587:
584:
576:smooth numbers
569:pronic numbers
537:
532:
528:
524:
521:
518:
515:
512:
491:
482:
473:
464:
451:
438:
429:
420:
411:
379:perfect powers
367:
366:
354:
351:
348:
345:
342:
339:
312:
309:
306:
303:
300:
297:
286:
285:
274:
271:
268:
263:
260:
257:
254:
250:
246:
243:
240:
237:
234:
231:
228:
225:
203:
202:
191:
188:
183:
180:
176:
172:
169:
166:
163:
160:
157:
154:
151:
137:sphenic number
128:
125:
98:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3272:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3245:Prime numbers
3243:
3242:
3240:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3181:
3179:
3175:
3169:
3166:
3164:
3163:Polydivisible
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3140:
3138:
3135:
3131:
3125:
3122:
3120:
3117:
3114:
3110:
3107:
3105:
3102:
3101:
3099:
3096:
3092:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3065:Superabundant
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3047:
3045:
3041:
3035:
3034:ErdĆsâNicolas
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2981:
2979:
2975:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2943:Perfect power
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2905:
2903:
2899:
2894:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2853:
2849:
2840:
2835:
2833:
2828:
2826:
2821:
2820:
2817:
2805:
2801:
2797:
2796:
2793:
2783:
2780:
2779:
2776:
2771:
2766:
2762:
2752:
2749:
2747:
2744:
2743:
2740:
2735:
2730:
2726:
2716:
2713:
2711:
2708:
2707:
2704:
2699:
2694:
2690:
2680:
2677:
2675:
2672:
2671:
2668:
2664:
2658:
2654:
2644:
2641:
2639:
2636:
2634:
2631:
2630:
2627:
2623:
2618:
2614:
2600:
2597:
2596:
2594:
2590:
2584:
2581:
2579:
2576:
2574:
2573:Polydivisible
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2550:
2548:
2544:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2514:
2512:
2509:
2504:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2464:
2462:
2459:
2455:
2447:
2444:
2443:
2442:
2439:
2438:
2436:
2433:
2429:
2417:
2414:
2413:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2368:
2366:
2362:
2356:
2353:
2352:
2350:
2346:
2340:
2337:
2335:
2332:
2331:
2329:
2327:Digit product
2325:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2300:
2298:
2296:
2292:
2284:
2281:
2279:
2276:
2275:
2274:
2271:
2270:
2268:
2266:
2261:
2257:
2253:
2248:
2243:
2239:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2163:ErdĆsâNicolas
2161:
2159:
2156:
2154:
2151:
2150:
2147:
2142:
2138:
2132:
2128:
2114:
2111:
2109:
2106:
2105:
2103:
2101:
2097:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2072:
2070:
2068:
2064:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2029:
2027:
2025:
2021:
2015:
2012:
2010:
2007:
2006:
2004:
2002:
1998:
1992:
1989:
1987:
1984:
1982:
1981:Superabundant
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1883:
1881:
1879:
1875:
1871:
1867:
1863:
1858:
1854:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1785:
1782:
1778:
1773:
1769:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1740:
1737:
1733:
1728:
1724:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1615:
1612:
1605:
1601:
1583:
1580:
1578:
1575:
1573:
1570:
1569:
1567:
1563:
1560:
1558:
1557:4-dimensional
1554:
1544:
1541:
1540:
1538:
1536:
1532:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1497:
1495:
1493:
1489:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1467:Centered cube
1465:
1463:
1460:
1459:
1457:
1455:
1451:
1448:
1446:
1445:3-dimensional
1442:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1383:
1381:
1379:
1375:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1325:
1323:
1321:
1317:
1314:
1312:
1311:2-dimensional
1308:
1304:
1300:
1295:
1291:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1255:Nonhypotenuse
1253:
1252:
1249:
1242:
1238:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1204:
1201:
1194:
1190:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1141:
1138:
1133:
1128:
1124:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1090:
1087:
1080:
1076:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1032:
1029:
1024:
1018:
1014:
1004:
1001:
999:
996:
994:
993:Perfect power
991:
989:
986:
984:
983:Seventh power
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
935:
932:
927:
922:
918:
914:
906:
901:
899:
894:
892:
887:
886:
883:
877:
874:
872:
869:
868:
864:
859:
855:
851:
850:Prentice Hall
847:
842:
839:
835:
831:
827:
822:
819:
815:
811:
807:
802:
799:
793:
789:
785:
781:
777:
774:
772:0-201-01984-1
768:
764:
760:
755:
754:
750:
742:
737:
734:
731:, p. 53.
730:
725:
722:
719:, p. 85.
718:
713:
710:
707:, p. 44.
706:
701:
698:
694:
693:Fraleigh 1976
689:
686:
682:
681:Herstein 1964
677:
674:
670:
669:Fraleigh 1976
665:
662:
659:, p. 16.
658:
653:
651:
647:
643:
638:
635:
628:
624:
621:
619:
616:
614:
611:
609:
606:
605:
601:
595:
590:
585:
583:
581:
580:rough numbers
577:
572:
570:
565:
563:
559:
555:
551:
530:
526:
522:
519:
516:
513:
498:
489:
480:
471:
462:
458:
449:
445:
444:Superabundant
436:
427:
418:
409:
408:Euler diagram
405:
401:
399:
395:
392:For example,
390:
388:
384:
380:
376:
372:
352:
349:
343:
337:
330:
329:
328:
326:
310:
307:
301:
295:
272:
269:
266:
261:
258:
255:
252:
244:
241:
235:
229:
223:
216:
215:
214:
212:
208:
189:
186:
181:
178:
170:
167:
161:
155:
149:
142:
141:
140:
138:
134:
126:
124:
122:
117:
115:
111:
107:
103:
95:
90:
84:
83:
82:
79:
75:
73:
69:
65:
61:
57:
49:
44:
37:
32:
19:
3224:Superperfect
3219:Refactorable
3014:Superperfect
3009:Hyperperfect
2994:Quasiperfect
2912:
2878:Prime factor
2537:Transposable
2401:Narcissistic
2308:Digital root
2228:Super-Poulet
2188:JordanâPĂłlya
2137:prime factor
2042:Noncototient
2009:Almost prime
1991:Superperfect
1966:Refactorable
1961:Quasiperfect
1936:Hyperperfect
1777:Pseudoprimes
1748:WallâSunâSun
1683:Ordered Bell
1653:FussâCatalan
1565:non-centered
1515:Dodecahedral
1492:non-centered
1378:non-centered
1280:Wolstenholme
1025:× 2 ± 1
1022:
1021:Of the form
988:Eighth power
968:Fourth power
845:
825:
805:
783:
758:
736:
724:
712:
700:
688:
676:
664:
637:
573:
566:
557:
553:
549:
502:
487:
391:
382:
370:
368:
324:
287:
210:
204:
130:
118:
99:
80:
76:
55:
53:
3148:Extravagant
3143:Equidigital
3104:Untouchable
3024:Semiperfect
3004:Hemiperfect
2933:Square-free
2558:Extravagant
2553:Equidigital
2508:permutation
2467:Palindromic
2441:Automorphic
2339:Sum-product
2318:Sum-product
2273:Persistence
2168:ErdĆsâWoods
2090:Untouchable
1971:Semiperfect
1921:Hemiperfect
1582:Tesseractic
1520:Icosahedral
1500:Tetrahedral
1431:Dodecagonal
1132:Recursively
1003:Prime power
978:Sixth power
973:Fifth power
953:Power of 10
911:Classes of
786:, Waltham:
548:. A number
3255:Arithmetic
3239:Categories
3184:Arithmetic
3177:Other sets
3136:-dependent
2770:Graphemics
2643:Pernicious
2497:Undulating
2472:Pandigital
2446:Trimorphic
2047:Nontotient
1896:Arithmetic
1510:Octahedral
1411:Heptagonal
1401:Pentagonal
1386:Triangular
1227:SierpiĆski
1149:Jacobsthal
948:Power of 3
943:Power of 2
828:, Boston:
751:References
717:McCoy 1968
387:squarefree
48:rectangles
3214:Descartes
3189:Deficient
3124:Betrothed
3029:Practical
2918:Semiprime
2913:Composite
2527:Parasitic
2376:Factorion
2303:Digit sum
2295:Digit sum
2113:Fortunate
2100:Primorial
2014:Semiprime
1951:Practical
1916:Descartes
1911:Deficient
1901:Betrothed
1743:Wieferich
1572:Pentatope
1535:pyramidal
1426:Decagonal
1421:Nonagonal
1416:Octagonal
1406:Hexagonal
1265:Practical
1212:Congruent
1144:Fibonacci
1108:Loeschian
818:77-171950
741:Long 1972
705:Long 1972
657:Long 1972
497:Deficient
488:Composite
338:μ
296:μ
270:−
242:−
224:μ
168:−
150:μ
133:semiprime
70:, or the
3199:Solitary
3194:Friendly
3119:Sociable
3109:Amicable
3097:-related
3050:Abundant
2948:Achilles
2938:Powerful
2851:Overview
2599:Friedman
2532:Primeval
2477:Repdigit
2434:-related
2381:Kaprekar
2355:Meertens
2278:Additive
2265:dynamics
2173:Friendly
2085:Sociable
2075:Amicable
1886:Abundant
1866:dynamics
1688:Schröder
1678:Narayana
1648:Eulerian
1638:Delannoy
1633:Dedekind
1454:centered
1320:centered
1207:Amenable
1164:Narayana
1154:Leonardo
1050:Mersenne
998:Powerful
938:Achilles
858:77-81766
838:68-15225
782:(1964),
586:See also
417:Abundant
3204:Sublime
3158:Harshad
2984:Perfect
2968:Unusual
2958:Regular
2928:Sphenic
2863:Divisor
2772:related
2736:related
2700:related
2698:Sorting
2583:Vampire
2568:Harshad
2510:related
2482:Repunit
2396:Lychrel
2371:Dudeney
2223:StĂžrmer
2218:Sphenic
2203:Regular
2141:divisor
2080:Perfect
1976:Sublime
1946:Perfect
1673:Motzkin
1628:Catalan
1169:Padovan
1103:Leyland
1098:Idoneal
1093:Hilbert
1065:Woodall
495:
486:
479:Perfect
477:
468:
455:
442:
433:
424:
415:
92:in the
89:A002808
64:divisor
3153:Frugal
3113:Triple
2953:Smooth
2923:Pronic
2638:Odious
2563:Frugal
2517:Cyclic
2506:Digit-
2213:Smooth
2198:Pronic
2158:Cyclic
2135:Other
2108:Euclid
1758:Wilson
1732:Primes
1391:Square
1260:Polite
1222:Riesel
1217:Knödel
1179:Perrin
1060:Thabit
1045:Fermat
1035:Cullen
958:Square
926:Powers
856:
836:
816:
794:
769:
493:
484:
475:
466:
453:
440:
431:
422:
413:
3168:Smith
3085:Weird
2963:Rough
2908:Prime
2679:Prime
2674:Lucky
2663:sieve
2592:Other
2578:Smith
2458:Digit
2416:Happy
2391:Keith
2364:Other
2208:Rough
2178:Giuga
1643:Euler
1505:Cubic
1159:Lucas
1055:Proth
629:Notes
560:is a
556:<
470:Weird
459:and
446:and
377:(All
127:Types
110:up to
68:prime
58:is a
3134:Base
2633:Evil
2313:Self
2263:and
2153:Blum
1864:and
1668:Lobb
1623:Cake
1618:Bell
1368:Star
1275:Ulam
1174:Pell
963:Cube
854:LCCN
834:LCCN
814:LCCN
792:ISBN
767:ISBN
578:and
383:none
209:and
94:OEIS
72:unit
2751:Ban
2139:or
1658:Lah
371:all
369:If
106:360
102:299
3241::
852:,
832:,
812:,
790:,
765:,
649:^
398:42
394:72
273:1.
116:.
54:A
3115:)
3111:(
2838:e
2831:t
2824:v
1023:a
904:e
897:t
890:v
558:n
554:x
550:n
536:}
531:2
527:p
523:,
520:p
517:,
514:1
511:{
365:.
353:0
350:=
347:)
344:n
341:(
325:n
311:1
308:=
305:)
302:1
299:(
267:=
262:1
259:+
256:x
253:2
249:)
245:1
239:(
236:=
233:)
230:n
227:(
211:x
190:1
187:=
182:x
179:2
175:)
171:1
165:(
162:=
159:)
156:n
153:(
96:)
20:)
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