3118:
39:, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors contain precisely all the digits of the original number, in any order, counting multiplicity. The two factors cannot both have trailing zeroes. The first vampire number is 1260 = 21 Ă— 60.
924:
However, 126000 (which can be expressed as 21 Ă— 6000 or 210 Ă— 600) is not a vampire number, since although 126000 = 21 Ă— 6000 and the digits (216000) are a permutation of the original number, the two factors 21 and 6000 do not have the correct number of digits. Furthermore, although 126000 = 210 Ă—
1038:
is 10392BA45768 = 105628 Ă— BA3974, where A means ten and B means eleven. Another example in the same base is a vampire number with three fangs, 572164B9A830 = 8752 Ă— 9346 Ă— A0B1. An example with four fangs is 3715A6B89420 = 763 Ă— 824 Ă— 905 Ă— B1A. In these examples, all 12 digits are used exactly
710:
967:
1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125500, ... (sequence
921:
1260 is a vampire number, with 21 and 60 as fangs, since 21 Ă— 60 = 1260 and the digits of the concatenation of the two factors (2160) are a permutation of the digits of the original number (1260).
171:
406:
331:
1220:
441:
495:
468:
737:
518:
88:
821:
797:
777:
757:
562:
542:
254:
234:
214:
194:
65:
566:
3142:
974:
1213:
2020:
1206:
2015:
2030:
2010:
1172:
1128:
2723:
2303:
997:
A vampire number can have multiple distinct pairs of fangs. The first of infinitely many vampire numbers with 2 pairs of fangs:
2025:
2809:
2125:
2475:
1794:
1587:
2510:
2480:
2155:
2145:
2651:
2065:
1799:
1779:
2341:
2505:
96:
2600:
2223:
1980:
1789:
1771:
1665:
1655:
1645:
2485:
1025:
24959017348650 = 2947050 Ă— 8469153 = 2949705 Ă— 8461530 = 4125870 Ă— 6049395 = 4129587 Ă— 6043950 = 4230765 Ă— 5899410
337:
262:
2728:
2273:
1894:
1680:
1675:
1670:
1660:
1637:
1107:
1713:
17:
1970:
2839:
2804:
2590:
2500:
2374:
2349:
2258:
2248:
1860:
1842:
1762:
3099:
2369:
2243:
1874:
1650:
1430:
1357:
2354:
2208:
2135:
1290:
3063:
2703:
2996:
2890:
2854:
2595:
2318:
2298:
2115:
1784:
1572:
1544:
827:
2718:
2582:
2577:
2545:
2308:
2283:
2278:
2253:
2183:
2179:
2110:
2000:
1832:
1628:
1597:
3117:
3121:
2875:
2870:
2784:
2758:
2656:
2635:
2407:
2288:
2238:
2160:
2130:
2070:
1837:
1817:
1748:
1461:
414:
2005:
1163:
1145:
981:
There are many known sequences of infinitely many vampire numbers following a pattern, such as:
3015:
2960:
2814:
2789:
2763:
2540:
2218:
2213:
2140:
2120:
2105:
1827:
1809:
1728:
1718:
1703:
1481:
1466:
1124:
1069:
1017:
16758243290880 = 1982736 Ă— 8452080 = 2123856 Ă— 7890480 = 2751840 Ă— 6089832 = 2817360 Ă— 5948208
3051:
2844:
2430:
2402:
2392:
2384:
2268:
2233:
2228:
2195:
1889:
1852:
1743:
1738:
1733:
1723:
1695:
1582:
1534:
1529:
1486:
1425:
29:
473:
446:
3027:
2916:
2849:
2775:
2698:
2672:
2490:
2203:
2060:
1995:
1965:
1955:
1950:
1616:
1524:
1471:
1315:
1255:
1048:
1034:
Vampire numbers also exist for bases other than base 10. For example, a vampire number in
36:
1184:
719:
500:
70:
3032:
2885:
2749:
2713:
2688:
2564:
2535:
2520:
2397:
2293:
2263:
1990:
1945:
1822:
1420:
1415:
1410:
1382:
1367:
1280:
1265:
1243:
1230:
834:
group sci.math, and the article he later wrote was published in chapter 30 of his book
806:
782:
762:
742:
547:
527:
239:
219:
199:
179:
50:
32:
3136:
2955:
2939:
2880:
2834:
2530:
2515:
2425:
2150:
1708:
1577:
1539:
1496:
1377:
1362:
1352:
1310:
1300:
1275:
521:
705:{\displaystyle ({a_{k}}{a_{k-1}}...{a_{2}}{a_{1}}{b_{k}}{b_{k-1}}...{b_{2}}{b_{1}})}
2991:
2980:
2895:
2733:
2708:
2625:
2525:
2495:
2470:
2454:
2359:
2326:
2075:
2049:
1960:
1899:
1476:
1372:
1305:
1285:
1260:
1072:
2950:
2825:
2630:
2094:
1985:
1940:
1935:
1685:
1592:
1491:
1320:
1295:
1270:
1180:
713:
3087:
3068:
2364:
1975:
1035:
1198:
2693:
2620:
2612:
2417:
2331:
1449:
1077:
989:
Al
Sweigart calculated all the vampire numbers that have at most 10 digits.
2794:
2799:
2458:
831:
1094:
196:
is a vampire number if and only if there exist two natural numbers
3085:
3049:
3013:
2977:
2937:
2562:
2451:
2177:
2092:
2047:
1924:
1614:
1561:
1513:
1447:
1399:
1337:
1241:
1202:
985:
1530 = 30 Ă— 51, 150300 = 300 Ă— 501, 15003000 = 3000 Ă— 5001, ...
969:
826:
Vampire numbers were first described in a 1994 post by
809:
785:
765:
745:
722:
569:
550:
530:
503:
476:
449:
417:
340:
265:
242:
222:
202:
182:
99:
73:
53:
925:
600, both factors 210 and 600 have trailing zeroes.
2909:
2863:
2823:
2774:
2748:
2681:
2665:
2644:
2611:
2576:
2416:
2383:
2340:
2317:
2194:
1882:
1873:
1851:
1808:
1770:
1761:
1694:
1636:
1627:
1108:
Pickover's original post describing vampire numbers
1009:13078260 = 1620 Ă— 8073 = 1863 Ă— 7020 = 2070 Ă— 6318
815:
791:
771:
751:
731:
704:
556:
536:
512:
489:
462:
435:
400:
325:
248:
228:
208:
188:
165:
82:
59:
166:{\displaystyle N={n_{2k}}{n_{2k-1}}...{n_{1}}}
1214:
8:
401:{\displaystyle B={b_{k}}{b_{k-1}}...{b_{1}}}
326:{\displaystyle A={a_{k}}{a_{k-1}}...{a_{1}}}
3082:
3046:
3010:
2974:
2934:
2608:
2573:
2559:
2448:
2191:
2174:
2089:
2044:
1921:
1879:
1767:
1633:
1624:
1611:
1558:
1515:Possessing a specific set of other numbers
1510:
1444:
1396:
1334:
1238:
1221:
1207:
1199:
808:
784:
764:
744:
721:
692:
687:
680:
675:
653:
648:
641:
636:
629:
624:
617:
612:
590:
585:
578:
573:
568:
549:
529:
502:
481:
475:
454:
448:
416:
391:
386:
364:
359:
352:
347:
339:
316:
311:
289:
284:
277:
272:
264:
241:
221:
201:
181:
156:
151:
126:
121:
111:
106:
98:
72:
52:
845:
1060:
7:
963:The sequence of vampire numbers is:
928:The first few vampire numbers are:
856:Count of vampire numbers of length
14:
1021:The first with 5 pairs of fangs:
1013:The first with 4 pairs of fangs:
1005:The first with 3 pairs of fangs:
3143:Base-dependent integer sequences
3116:
2724:Perfect digit-to-digit invariant
1119:Pickover, Clifford A. (1995).
1001:125460 = 204 Ă— 615 = 246 Ă— 510
699:
570:
1:
1563:Expressible via specific sums
1171:Grime, James; Copeland, Ed.
1146:"Vampire Numbers Visualized"
2652:Multiplicative digital root
497:are not both zero, and the
436:{\displaystyle A\times B=N}
3159:
1165:Vampire Numbers Visualized
3112:
3095:
3081:
3059:
3045:
3023:
3009:
2987:
2973:
2946:
2933:
2729:Perfect digital invariant
2572:
2558:
2466:
2447:
2304:Superior highly composite
2190:
2173:
2101:
2088:
2056:
2043:
1931:
1920:
1623:
1610:
1568:
1557:
1520:
1509:
1457:
1443:
1406:
1395:
1348:
1333:
1251:
1237:
67:be a natural number with
2342:Euler's totient function
2126:Euler–Jacobi pseudoprime
1401:Other polynomial numbers
18:recreational mathematics
2156:Somer–Lucas pseudoprime
2146:Lucas–Carmichael number
1981:Lazy caterer's sequence
35:with an even number of
2031:Wedderburn–Etherington
1431:Lucky numbers of Euler
817:
793:
773:
753:
733:
706:
558:
538:
514:
491:
464:
437:
402:
327:
250:
230:
210:
190:
167:
84:
61:
2319:Prime omega functions
2136:Frobenius pseudoprime
1926:Combinatorial numbers
1795:Centered dodecahedral
1588:Primary pseudoperfect
818:
794:
774:
754:
734:
707:
559:
539:
515:
492:
490:{\displaystyle b_{1}}
465:
463:{\displaystyle a_{1}}
438:
403:
328:
251:
231:
211:
191:
168:
85:
62:
2778:-composition related
2578:Arithmetic functions
2180:Arithmetic functions
2116:Elliptic pseudoprime
1800:Centered icosahedral
1780:Centered tetrahedral
828:Clifford A. Pickover
807:
783:
763:
743:
720:
567:
548:
528:
501:
474:
447:
415:
338:
263:
240:
220:
200:
180:
97:
71:
51:
2704:Kaprekar's constant
2224:Colossally abundant
2111:Catalan pseudoprime
2011:Schröder–Hipparchus
1790:Centered octahedral
1666:Centered heptagonal
1656:Centered pentagonal
1646:Centered triangular
1246:and related numbers
993:Multiple fang pairs
847:
26:true vampire number
3122:Mathematics portal
3064:Aronson's sequence
2810:Smarandache–Wellin
2567:-dependent numbers
2274:Primitive abundant
2161:Strong pseudoprime
2151:Perrin pseudoprime
2131:Fermat pseudoprime
2071:Wolstenholme prime
1895:Squared triangular
1681:Centered decagonal
1676:Centered nonagonal
1671:Centered octagonal
1661:Centered hexagonal
1093:Andersen, Jens K.
1070:Weisstein, Eric W.
959:105210 = 210 Ă— 501
956:104260 = 260 Ă— 401
953:102510 = 201 Ă— 510
846:
813:
789:
769:
759:. The two numbers
749:
732:{\displaystyle 2k}
729:
702:
554:
534:
513:{\displaystyle 2k}
510:
487:
460:
433:
398:
323:
246:
226:
206:
186:
163:
83:{\displaystyle 2k}
80:
57:
3130:
3129:
3108:
3107:
3077:
3076:
3041:
3040:
3005:
3004:
2969:
2968:
2929:
2928:
2925:
2924:
2744:
2743:
2554:
2553:
2443:
2442:
2439:
2438:
2385:Aliquot sequences
2196:Divisor functions
2169:
2168:
2141:Lucas pseudoprime
2121:Euler pseudoprime
2106:Carmichael number
2084:
2083:
2039:
2038:
1916:
1915:
1912:
1911:
1908:
1907:
1869:
1868:
1757:
1756:
1714:Square triangular
1606:
1605:
1553:
1552:
1505:
1504:
1439:
1438:
1391:
1390:
1329:
1328:
1173:"Vampire numbers"
1095:"Vampire numbers"
1073:"Vampire Numbers"
919:
918:
816:{\displaystyle N}
792:{\displaystyle B}
772:{\displaystyle A}
752:{\displaystyle N}
557:{\displaystyle B}
537:{\displaystyle A}
249:{\displaystyle k}
229:{\displaystyle B}
209:{\displaystyle A}
189:{\displaystyle N}
60:{\displaystyle N}
3150:
3120:
3083:
3052:Natural language
3047:
3011:
2979:Generated via a
2975:
2935:
2840:Digit-reassembly
2805:Self-descriptive
2609:
2574:
2560:
2511:Lucas–Carmichael
2501:Harmonic divisor
2449:
2375:Sparsely totient
2350:Highly cototient
2259:Multiply perfect
2249:Highly composite
2192:
2175:
2090:
2045:
2026:Telephone number
1922:
1880:
1861:Square pyramidal
1843:Stella octangula
1768:
1634:
1625:
1617:Figurate numbers
1612:
1559:
1511:
1445:
1397:
1335:
1239:
1223:
1216:
1209:
1200:
1195:
1193:
1192:
1183:. Archived from
1150:
1149:
1141:
1135:
1134:
1121:Keys to Infinity
1116:
1110:
1105:
1099:
1098:
1090:
1084:
1083:
1082:
1065:
972:
848:
836:Keys to Infinity
822:
820:
819:
814:
798:
796:
795:
790:
778:
776:
775:
770:
758:
756:
755:
750:
738:
736:
735:
730:
711:
709:
708:
703:
698:
697:
696:
686:
685:
684:
665:
664:
663:
647:
646:
645:
635:
634:
633:
623:
622:
621:
602:
601:
600:
584:
583:
582:
563:
561:
560:
555:
543:
541:
540:
535:
519:
517:
516:
511:
496:
494:
493:
488:
486:
485:
469:
467:
466:
461:
459:
458:
442:
440:
439:
434:
407:
405:
404:
399:
397:
396:
395:
376:
375:
374:
358:
357:
356:
332:
330:
329:
324:
322:
321:
320:
301:
300:
299:
283:
282:
281:
255:
253:
252:
247:
235:
233:
232:
227:
215:
213:
212:
207:
195:
193:
192:
187:
172:
170:
169:
164:
162:
161:
160:
141:
140:
139:
120:
119:
118:
89:
87:
86:
81:
66:
64:
63:
58:
3158:
3157:
3153:
3152:
3151:
3149:
3148:
3147:
3133:
3132:
3131:
3126:
3104:
3100:Strobogrammatic
3091:
3073:
3055:
3037:
3019:
3001:
2983:
2965:
2942:
2921:
2905:
2864:Divisor-related
2859:
2819:
2770:
2740:
2677:
2661:
2640:
2607:
2580:
2568:
2550:
2462:
2461:related numbers
2435:
2412:
2379:
2370:Perfect totient
2336:
2313:
2244:Highly abundant
2186:
2165:
2097:
2080:
2052:
2035:
2021:Stirling second
1927:
1904:
1865:
1847:
1804:
1753:
1690:
1651:Centered square
1619:
1602:
1564:
1549:
1516:
1501:
1453:
1452:defined numbers
1435:
1402:
1387:
1358:Double Mersenne
1344:
1325:
1247:
1233:
1231:natural numbers
1227:
1190:
1188:
1170:
1159:
1154:
1153:
1143:
1142:
1138:
1131:
1118:
1117:
1113:
1106:
1102:
1092:
1091:
1087:
1068:
1067:
1066:
1062:
1057:
1049:Friedman number
1045:
1032:
995:
968:
844:
805:
804:
799:are called the
781:
780:
761:
760:
741:
740:
718:
717:
688:
676:
649:
637:
625:
613:
586:
574:
565:
564:
546:
545:
526:
525:
499:
498:
477:
472:
471:
450:
445:
444:
413:
412:
387:
360:
348:
336:
335:
312:
285:
273:
261:
260:
238:
237:
218:
217:
198:
197:
178:
177:
152:
122:
107:
95:
94:
69:
68:
49:
48:
45:
12:
11:
5:
3156:
3154:
3146:
3145:
3135:
3134:
3128:
3127:
3125:
3124:
3113:
3110:
3109:
3106:
3105:
3103:
3102:
3096:
3093:
3092:
3086:
3079:
3078:
3075:
3074:
3072:
3071:
3066:
3060:
3057:
3056:
3050:
3043:
3042:
3039:
3038:
3036:
3035:
3033:Sorting number
3030:
3028:Pancake number
3024:
3021:
3020:
3014:
3007:
3006:
3003:
3002:
3000:
2999:
2994:
2988:
2985:
2984:
2978:
2971:
2970:
2967:
2966:
2964:
2963:
2958:
2953:
2947:
2944:
2943:
2940:Binary numbers
2938:
2931:
2930:
2927:
2926:
2923:
2922:
2920:
2919:
2913:
2911:
2907:
2906:
2904:
2903:
2898:
2893:
2888:
2883:
2878:
2873:
2867:
2865:
2861:
2860:
2858:
2857:
2852:
2847:
2842:
2837:
2831:
2829:
2821:
2820:
2818:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2781:
2779:
2772:
2771:
2769:
2768:
2767:
2766:
2755:
2753:
2750:P-adic numbers
2746:
2745:
2742:
2741:
2739:
2738:
2737:
2736:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2691:
2685:
2683:
2679:
2678:
2676:
2675:
2669:
2667:
2666:Coding-related
2663:
2662:
2660:
2659:
2654:
2648:
2646:
2642:
2641:
2639:
2638:
2633:
2628:
2623:
2617:
2615:
2606:
2605:
2604:
2603:
2601:Multiplicative
2598:
2587:
2585:
2570:
2569:
2565:Numeral system
2563:
2556:
2555:
2552:
2551:
2549:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2467:
2464:
2463:
2452:
2445:
2444:
2441:
2440:
2437:
2436:
2434:
2433:
2428:
2422:
2420:
2414:
2413:
2411:
2410:
2405:
2400:
2395:
2389:
2387:
2381:
2380:
2378:
2377:
2372:
2367:
2362:
2357:
2355:Highly totient
2352:
2346:
2344:
2338:
2337:
2335:
2334:
2329:
2323:
2321:
2315:
2314:
2312:
2311:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2266:
2261:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2209:Almost perfect
2206:
2200:
2198:
2188:
2187:
2178:
2171:
2170:
2167:
2166:
2164:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2102:
2099:
2098:
2093:
2086:
2085:
2082:
2081:
2079:
2078:
2073:
2068:
2063:
2057:
2054:
2053:
2048:
2041:
2040:
2037:
2036:
2034:
2033:
2028:
2023:
2018:
2016:Stirling first
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1932:
1929:
1928:
1925:
1918:
1917:
1914:
1913:
1910:
1909:
1906:
1905:
1903:
1902:
1897:
1892:
1886:
1884:
1877:
1871:
1870:
1867:
1866:
1864:
1863:
1857:
1855:
1849:
1848:
1846:
1845:
1840:
1835:
1830:
1825:
1820:
1814:
1812:
1806:
1805:
1803:
1802:
1797:
1792:
1787:
1782:
1776:
1774:
1765:
1759:
1758:
1755:
1754:
1752:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1700:
1698:
1692:
1691:
1689:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1642:
1640:
1631:
1621:
1620:
1615:
1608:
1607:
1604:
1603:
1601:
1600:
1595:
1590:
1585:
1580:
1575:
1569:
1566:
1565:
1562:
1555:
1554:
1551:
1550:
1548:
1547:
1542:
1537:
1532:
1527:
1521:
1518:
1517:
1514:
1507:
1506:
1503:
1502:
1500:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1458:
1455:
1454:
1448:
1441:
1440:
1437:
1436:
1434:
1433:
1428:
1423:
1418:
1413:
1407:
1404:
1403:
1400:
1393:
1392:
1389:
1388:
1386:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1349:
1346:
1345:
1338:
1331:
1330:
1327:
1326:
1324:
1323:
1318:
1313:
1308:
1303:
1298:
1293:
1288:
1283:
1278:
1273:
1268:
1263:
1258:
1252:
1249:
1248:
1242:
1235:
1234:
1228:
1226:
1225:
1218:
1211:
1203:
1197:
1196:
1168:
1162:Sweigart, Al.
1158:
1157:External links
1155:
1152:
1151:
1144:Sweigart, Al.
1136:
1129:
1111:
1100:
1085:
1059:
1058:
1056:
1053:
1052:
1051:
1044:
1041:
1031:
1028:
1027:
1026:
1019:
1018:
1011:
1010:
1003:
1002:
994:
991:
987:
986:
979:
978:
961:
960:
957:
954:
951:
950:6880 = 80 Ă— 86
948:
947:2187 = 27 Ă— 81
945:
944:1827 = 21 Ă— 87
942:
941:1530 = 30 Ă— 51
939:
938:1435 = 35 Ă— 41
936:
935:1395 = 15 Ă— 93
933:
932:1260 = 21 Ă— 60
917:
916:
913:
909:
908:
905:
901:
900:
897:
893:
892:
889:
885:
884:
881:
877:
876:
873:
869:
868:
865:
861:
860:
854:
843:
840:
812:
788:
768:
748:
728:
725:
701:
695:
691:
683:
679:
674:
671:
668:
662:
659:
656:
652:
644:
640:
632:
628:
620:
616:
611:
608:
605:
599:
596:
593:
589:
581:
577:
572:
553:
533:
520:digits of the
509:
506:
484:
480:
457:
453:
432:
429:
426:
423:
420:
409:
408:
394:
390:
385:
382:
379:
373:
370:
367:
363:
355:
351:
346:
343:
333:
319:
315:
310:
307:
304:
298:
295:
292:
288:
280:
276:
271:
268:
245:
225:
205:
185:
174:
173:
159:
155:
150:
147:
144:
138:
135:
132:
129:
125:
117:
114:
110:
105:
102:
79:
76:
56:
44:
41:
33:natural number
22:vampire number
13:
10:
9:
6:
4:
3:
2:
3155:
3144:
3141:
3140:
3138:
3123:
3119:
3115:
3114:
3111:
3101:
3098:
3097:
3094:
3089:
3084:
3080:
3070:
3067:
3065:
3062:
3061:
3058:
3053:
3048:
3044:
3034:
3031:
3029:
3026:
3025:
3022:
3017:
3012:
3008:
2998:
2995:
2993:
2990:
2989:
2986:
2982:
2976:
2972:
2962:
2959:
2957:
2954:
2952:
2949:
2948:
2945:
2941:
2936:
2932:
2918:
2915:
2914:
2912:
2908:
2902:
2899:
2897:
2894:
2892:
2891:Polydivisible
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2868:
2866:
2862:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2832:
2830:
2827:
2822:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2782:
2780:
2777:
2773:
2765:
2762:
2761:
2760:
2757:
2756:
2754:
2751:
2747:
2735:
2732:
2731:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2686:
2684:
2680:
2674:
2671:
2670:
2668:
2664:
2658:
2655:
2653:
2650:
2649:
2647:
2645:Digit product
2643:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2618:
2616:
2614:
2610:
2602:
2599:
2597:
2594:
2593:
2592:
2589:
2588:
2586:
2584:
2579:
2575:
2571:
2566:
2561:
2557:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2481:Erdős–Nicolas
2479:
2477:
2474:
2472:
2469:
2468:
2465:
2460:
2456:
2450:
2446:
2432:
2429:
2427:
2424:
2423:
2421:
2419:
2415:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2390:
2388:
2386:
2382:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2347:
2345:
2343:
2339:
2333:
2330:
2328:
2325:
2324:
2322:
2320:
2316:
2310:
2307:
2305:
2302:
2300:
2299:Superabundant
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2201:
2199:
2197:
2193:
2189:
2185:
2181:
2176:
2172:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2103:
2100:
2096:
2091:
2087:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2058:
2055:
2051:
2046:
2042:
2032:
2029:
2027:
2024:
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:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1933:
1930:
1923:
1919:
1901:
1898:
1896:
1893:
1891:
1888:
1887:
1885:
1881:
1878:
1876:
1875:4-dimensional
1872:
1862:
1859:
1858:
1856:
1854:
1850:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1815:
1813:
1811:
1807:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1785:Centered cube
1783:
1781:
1778:
1777:
1775:
1773:
1769:
1766:
1764:
1763:3-dimensional
1760:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1701:
1699:
1697:
1693:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1643:
1641:
1639:
1635:
1632:
1630:
1629:2-dimensional
1626:
1622:
1618:
1613:
1609:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1573:Nonhypotenuse
1571:
1570:
1567:
1560:
1556:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1522:
1519:
1512:
1508:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1459:
1456:
1451:
1446:
1442:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1408:
1405:
1398:
1394:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1350:
1347:
1342:
1336:
1332:
1322:
1319:
1317:
1314:
1312:
1311:Perfect power
1309:
1307:
1304:
1302:
1301:Seventh power
1299:
1297:
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1253:
1250:
1245:
1240:
1236:
1232:
1224:
1219:
1217:
1212:
1210:
1205:
1204:
1201:
1187:on 2017-10-14
1186:
1182:
1178:
1174:
1169:
1167:
1166:
1161:
1160:
1156:
1147:
1140:
1137:
1132:
1130:0-471-19334-8
1126:
1122:
1115:
1112:
1109:
1104:
1101:
1096:
1089:
1086:
1080:
1079:
1074:
1071:
1064:
1061:
1054:
1050:
1047:
1046:
1042:
1040:
1037:
1029:
1024:
1023:
1022:
1016:
1015:
1014:
1008:
1007:
1006:
1000:
999:
998:
992:
990:
984:
983:
982:
976:
971:
966:
965:
964:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
930:
929:
926:
922:
914:
911:
910:
906:
903:
902:
898:
895:
894:
890:
887:
886:
882:
879:
878:
874:
871:
870:
866:
863:
862:
859:
855:
853:
850:
849:
841:
839:
837:
833:
829:
824:
810:
802:
786:
766:
746:
726:
723:
715:
693:
689:
681:
677:
672:
669:
666:
660:
657:
654:
650:
642:
638:
630:
626:
618:
614:
609:
606:
603:
597:
594:
591:
587:
579:
575:
551:
531:
523:
522:concatenation
507:
504:
482:
478:
455:
451:
430:
427:
424:
421:
418:
392:
388:
383:
380:
377:
371:
368:
365:
361:
353:
349:
344:
341:
334:
317:
313:
308:
305:
302:
296:
293:
290:
286:
278:
274:
269:
266:
259:
258:
257:
243:
223:
203:
183:
157:
153:
148:
145:
142:
136:
133:
130:
127:
123:
115:
112:
108:
103:
100:
93:
92:
91:
77:
74:
54:
42:
40:
38:
34:
31:
27:
23:
19:
2900:
2855:Transposable
2719:Narcissistic
2626:Digital root
2546:Super-Poulet
2506:Jordan–Pólya
2455:prime factor
2360:Noncototient
2327:Almost prime
2309:Superperfect
2284:Refactorable
2279:Quasiperfect
2254:Hyperperfect
2095:Pseudoprimes
2066:Wall–Sun–Sun
2001:Ordered Bell
1971:Fuss–Catalan
1883:non-centered
1833:Dodecahedral
1810:non-centered
1696:non-centered
1598:Wolstenholme
1343:× 2 ± 1
1340:
1339:Of the form
1306:Eighth power
1286:Fourth power
1189:. Retrieved
1185:the original
1176:
1164:
1139:
1120:
1114:
1103:
1088:
1076:
1063:
1033:
1020:
1012:
1004:
996:
988:
980:
962:
927:
923:
920:
915:11039126154
857:
851:
835:
825:
800:
410:
236:, each with
175:
46:
25:
21:
15:
2876:Extravagant
2871:Equidigital
2826:permutation
2785:Palindromic
2759:Automorphic
2657:Sum-product
2636:Sum-product
2591:Persistence
2486:Erdős–Woods
2408:Untouchable
2289:Semiperfect
2239:Hemiperfect
1900:Tesseractic
1838:Icosahedral
1818:Tetrahedral
1749:Dodecagonal
1450:Recursively
1321:Prime power
1296:Sixth power
1291:Fifth power
1271:Power of 10
1229:Classes of
1181:Brady Haran
1177:Numberphile
1030:Other bases
714:permutation
3088:Graphemics
2961:Pernicious
2815:Undulating
2790:Pandigital
2764:Trimorphic
2365:Nontotient
2214:Arithmetic
1828:Octahedral
1729:Heptagonal
1719:Pentagonal
1704:Triangular
1545:Sierpiński
1467:Jacobsthal
1266:Power of 3
1261:Power of 2
1191:2013-04-08
1055:References
907:208423682
739:digits of
411:such that
43:Definition
2845:Parasitic
2694:Factorion
2621:Digit sum
2613:Digit sum
2431:Fortunate
2418:Primorial
2332:Semiprime
2269:Practical
2234:Descartes
2229:Deficient
2219:Betrothed
2061:Wieferich
1890:Pentatope
1853:pyramidal
1744:Decagonal
1739:Nonagonal
1734:Octagonal
1724:Hexagonal
1583:Practical
1530:Congruent
1462:Fibonacci
1426:Loeschian
1123:. Wiley.
1078:MathWorld
658:−
595:−
422:×
369:−
294:−
134:−
30:composite
3137:Category
2917:Friedman
2850:Primeval
2795:Repdigit
2752:-related
2699:Kaprekar
2673:Meertens
2596:Additive
2583:dynamics
2491:Friendly
2403:Sociable
2393:Amicable
2204:Abundant
2184:dynamics
2006:Schröder
1996:Narayana
1966:Eulerian
1956:Delannoy
1951:Dedekind
1772:centered
1638:centered
1525:Amenable
1482:Narayana
1472:Leonardo
1368:Mersenne
1316:Powerful
1256:Achilles
1043:See also
899:4390670
842:Examples
256:digits:
90:digits:
3090:related
3054:related
3018:related
3016:Sorting
2901:Vampire
2886:Harshad
2828:related
2800:Repunit
2714:Lychrel
2689:Dudeney
2541:Størmer
2536:Sphenic
2521:Regular
2459:divisor
2398:Perfect
2294:Sublime
2264:Perfect
1991:Motzkin
1946:Catalan
1487:Padovan
1421:Leyland
1416:Idoneal
1411:Hilbert
1383:Woodall
1036:base 12
973:in the
970:A014575
891:108454
830:to the
716:of the
28:) is a
2956:Odious
2881:Frugal
2835:Cyclic
2824:Digit-
2531:Smooth
2516:Pronic
2476:Cyclic
2453:Other
2426:Euclid
2076:Wilson
2050:Primes
1709:Square
1578:Polite
1540:Riesel
1535:Knödel
1497:Perrin
1378:Thabit
1363:Fermat
1353:Cullen
1276:Square
1244:Powers
1127:
1039:once.
832:Usenet
712:are a
37:digits
2997:Prime
2992:Lucky
2981:sieve
2910:Other
2896:Smith
2776:Digit
2734:Happy
2709:Keith
2682:Other
2526:Rough
2496:Giuga
1961:Euler
1823:Cubic
1477:Lucas
1373:Proth
883:3228
801:fangs
176:Then
2951:Evil
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