1094:
3355:
599:
The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716 (sequence
816:
It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the proportion of the sums of the sociable number cycles divisible by 10 approaches 1 (sequence
290:
212:
362:
1457:
428:
121:
68:
of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
766:
660:
392:
320:
242:
164:
709:
729:
680:
798:
295:
1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636,
217:
1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860,
844:
1037:
824:
607:
1450:
2257:
1443:
865:
2252:
2267:
2247:
90:
is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to
1083:
2960:
2540:
1280:
1093:
444:
The following categorizes all known sociable numbers as of July 2018 by the length of the corresponding aliquot sequence:
2262:
3046:
1030:
2362:
3379:
2712:
2031:
1824:
2747:
2717:
2392:
2382:
1234:
2888:
2302:
2036:
2016:
61:
2578:
613:
These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).
247:
169:
2742:
978:
3389:
2837:
2460:
2217:
2026:
2008:
1902:
1892:
1882:
1270:
2722:
3384:
2965:
2510:
2131:
1917:
1912:
1907:
1897:
1874:
1255:
1950:
1023:
2207:
325:
3076:
3041:
2827:
2737:
2611:
2586:
2495:
2485:
2097:
2079:
1999:
1409:
1275:
1199:
3394:
3336:
2606:
2480:
2111:
1887:
1667:
1594:
1260:
1219:
142:
As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:
2591:
2445:
2372:
1527:
1189:
1058:
3300:
2940:
397:
3233:
3127:
3091:
2832:
2555:
2535:
2352:
2021:
1809:
1781:
1363:
1265:
732:
93:
2955:
2819:
2814:
2782:
2545:
2520:
2515:
2490:
2420:
2416:
2347:
2237:
2069:
1865:
1834:
1424:
1419:
1214:
1209:
1194:
1133:
3354:
75:
of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
3358:
3112:
3107:
3021:
2995:
2893:
2872:
2644:
2525:
2475:
2397:
2367:
2307:
2074:
2054:
1985:
1698:
1348:
1343:
1304:
1224:
1204:
586:
2242:
3252:
3197:
3051:
3026:
3000:
2777:
2455:
2450:
2377:
2357:
2342:
2064:
2046:
1965:
1955:
1940:
1718:
1703:
1384:
1324:
985:
890:
72:
43:
738:
632:
377:
305:
227:
149:
3288:
3081:
2667:
2639:
2629:
2621:
2505:
2470:
2465:
2432:
2126:
2089:
1980:
1975:
1970:
1960:
1932:
1819:
1771:
1766:
1723:
1662:
1414:
1389:
1309:
1295:
1229:
1113:
1073:
940:
880:
622:
131:
3264:
3153:
3086:
3012:
2935:
2909:
2727:
2440:
2297:
2232:
2202:
2192:
2187:
1853:
1761:
1708:
1552:
1492:
1399:
1394:
1313:
1250:
1148:
1138:
1068:
801:
685:
494:
87:
65:
51:
54:. The first two sociable sequences, or sociable chains, were discovered and named by the
3269:
3137:
3122:
2986:
2950:
2925:
2801:
2772:
2757:
2634:
2530:
2500:
2227:
2182:
2059:
1657:
1652:
1647:
1619:
1604:
1517:
1502:
1480:
1467:
1404:
1358:
1184:
1168:
1158:
1128:
805:
714:
665:
626:
474:
83:
79:
47:
1015:
885:
771:
126:
It is an open question whether all numbers end up at either a sociable number or at a
3373:
3192:
3176:
3117:
3071:
2767:
2752:
2662:
2387:
1945:
1814:
1776:
1733:
1614:
1599:
1589:
1547:
1537:
1512:
1353:
1153:
1143:
1123:
988:
78:
If the period of the sequence is 1, the number is a sociable number of order 1, or a
58:
3228:
3217:
3132:
2970:
2945:
2862:
2762:
2732:
2707:
2691:
2596:
2563:
2312:
2286:
2197:
2136:
1713:
1609:
1542:
1522:
1497:
1368:
1285:
1163:
1108:
1078:
851:
127:
367:
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and
3187:
3062:
2867:
2331:
2222:
2177:
2172:
1922:
1829:
1728:
1557:
1532:
1507:
39:
31:
17:
3324:
3305:
2601:
2212:
944:
578:
1435:
894:
433:
1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
2930:
2857:
2849:
2654:
2568:
1686:
1118:
993:
3031:
3036:
2695:
1063:
55:
130:(and hence 1), or, equivalently, whether there exist numbers whose
1334:
1010:
3322:
3286:
3250:
3214:
3174:
2799:
2688:
2414:
2329:
2284:
2161:
1851:
1798:
1750:
1684:
1636:
1574:
1478:
1439:
1019:
64:
in 1918. In a sociable sequence, each number is the sum of the
973:
1006:
1002:
911:
907:
819:
602:
596:
The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264
979:
Extensive tables of perfect, amicable and sociable numbers
446:
850:(1918), pp. 100–101. (The full text can be found at
589:
to 3 modulo 4 then there is no such sequence with length
937:
Distributed cycle detection in large-scale sparse graphs
939:, SimpĂłsio Brasileiro de Pesquisa Operacional (SBPO),
923:
86:
of 6 are 1, 2, and 3, whose sum is again 6. A pair of
774:
741:
717:
688:
668:
635:
400:
380:
328:
308:
250:
230:
172:
152:
96:
935:
Rocha, Rodrigo
Caetano; Thatte, Bhalchandra (2015),
3146:
3100:
3060:
3011:
2985:
2918:
2902:
2881:
2848:
2813:
2653:
2620:
2577:
2554:
2431:
2119:
2110:
2088:
2045:
2007:
1998:
1931:
1873:
1864:
1377:
1333:
1294:
1243:
1177:
1101:
1051:
1003:A003416 (smallest sociable number from each cycle)
792:
760:
723:
703:
674:
654:
422:
386:
356:
314:
284:
236:
206:
158:
115:
864:Bratley, Paul; Lunnon, Fred; McKay, John (1970).
134:never terminates, and hence grows without bound.
27:Numbers whose aliquot sums form a cyclic sequence
812:Conjecture of the sum of sociable number cycles
768:represent sociable numbers within the interval
285:{\displaystyle =2^{2}\cdot 5\cdot 193\cdot 401}
207:{\displaystyle =2^{2}\cdot 5\cdot 17\cdot 3719}
448:
46:. They are generalizations of the concepts of
1451:
1031:
800:. Two special cases are loops that represent
8:
3319:
3283:
3247:
3211:
3171:
2845:
2810:
2796:
2685:
2428:
2411:
2326:
2281:
2158:
2116:
2004:
1870:
1861:
1848:
1795:
1752:Possessing a specific set of other numbers
1747:
1681:
1633:
1571:
1475:
1458:
1444:
1436:
1038:
1024:
1016:
711:denotes the sum of the proper divisors of
884:
866:"Amicable numbers and their distribution"
773:
746:
740:
716:
687:
667:
640:
634:
408:
399:
379:
336:
327:
307:
258:
249:
229:
180:
171:
151:
107:
95:
804:and cycles of length two that represent
357:{\displaystyle =2^{2}\cdot 521\cdot 829}
836:
852:ProofWiki: Catalan-Dickson Conjecture
7:
1046:Divisibility-based sets of integers
845:L'Intermédiaire des Mathématiciens
374:the sum of the proper divisors of
302:the sum of the proper divisors of
224:the sum of the proper divisors of
146:The sum of the proper divisors of
25:
1084:Fundamental theorem of arithmetic
957:On amicable and sociable numbers,
886:10.1090/S0025-5718-1970-0271005-8
423:{\displaystyle =2^{5}\cdot 40787}
3353:
2961:Perfect digit-to-digit invariant
1092:
974:A list of known sociable numbers
1007:A122726 (all sociable numbers)
787:
775:
698:
692:
617:Searching for sociable numbers
440:List of known sociable numbers
116:{\displaystyle 5\times 10^{7}}
1:
1800:Expressible via specific sums
563:
552:
541:
530:
519:
508:
488:
468:
2889:Multiplicative digital root
3411:
873:Mathematics of Computation
3349:
3332:
3318:
3296:
3282:
3260:
3246:
3224:
3210:
3183:
3170:
2966:Perfect digital invariant
2809:
2795:
2703:
2684:
2541:Superior highly composite
2427:
2410:
2338:
2325:
2293:
2280:
2168:
2157:
1860:
1847:
1805:
1794:
1757:
1746:
1694:
1680:
1643:
1632:
1585:
1570:
1488:
1474:
1281:Superior highly composite
1090:
945:10.13140/RG.2.1.1233.8640
2579:Euler's totient function
2363:Euler–Jacobi pseudoprime
1638:Other polynomial numbers
1178:Constrained divisor sums
963:(1970), pp. 423–429
912:https://oeis.org/A052470
908:https://oeis.org/A003416
625:can be represented as a
2393:Somer–Lucas pseudoprime
2383:Lucas–Carmichael number
2218:Lazy caterer's sequence
761:{\displaystyle G_{n,s}}
655:{\displaystyle G_{n,s}}
387:{\displaystyle 1305184}
315:{\displaystyle 1727636}
237:{\displaystyle 1547860}
159:{\displaystyle 1264460}
2268:Wedderburn–Etherington
1668:Lucky numbers of Euler
910:cross referenced with
794:
762:
725:
705:
676:
662:, for a given integer
656:
424:
388:
358:
316:
286:
238:
208:
160:
117:
2556:Prime omega functions
2373:Frobenius pseudoprime
2163:Combinatorial numbers
2032:Centered dodecahedral
1825:Primary pseudoperfect
1059:Integer factorization
795:
763:
726:
706:
677:
657:
425:
389:
359:
317:
287:
239:
209:
161:
118:
3015:-composition related
2815:Arithmetic functions
2417:Arithmetic functions
2353:Elliptic pseudoprime
2037:Centered icosahedral
2017:Centered tetrahedral
772:
739:
715:
704:{\displaystyle s(k)}
686:
666:
633:
398:
378:
326:
306:
248:
228:
170:
150:
94:
3380:Arithmetic dynamics
2941:Kaprekar's constant
2461:Colossally abundant
2348:Catalan pseudoprime
2248:Schröder–Hipparchus
2027:Centered octahedral
1903:Centered heptagonal
1893:Centered pentagonal
1883:Centered triangular
1483:and related numbers
1271:Colossally abundant
1102:Factorization forms
924:Amicable pairs list
3359:Mathematics portal
3301:Aronson's sequence
3047:Smarandache–Wellin
2804:-dependent numbers
2511:Primitive abundant
2398:Strong pseudoprime
2388:Perrin pseudoprime
2368:Fermat pseudoprime
2308:Wolstenholme prime
2132:Squared triangular
1918:Centered decagonal
1913:Centered nonagonal
1908:Centered octagonal
1898:Centered hexagonal
1256:Primitive abundant
1244:With many divisors
989:"Sociable numbers"
986:Weisstein, Eric W.
843:P. Poulet, #4865,
790:
758:
721:
701:
672:
652:
420:
384:
354:
312:
282:
234:
204:
156:
113:
82:—for example, the
38:are numbers whose
3390:Integer sequences
3367:
3366:
3345:
3344:
3314:
3313:
3278:
3277:
3242:
3241:
3206:
3205:
3166:
3165:
3162:
3161:
2981:
2980:
2791:
2790:
2680:
2679:
2676:
2675:
2622:Aliquot sequences
2433:Divisor functions
2406:
2405:
2378:Lucas pseudoprime
2358:Euler pseudoprime
2343:Carmichael number
2321:
2320:
2276:
2275:
2153:
2152:
2149:
2148:
2145:
2144:
2106:
2105:
1994:
1993:
1951:Square triangular
1843:
1842:
1790:
1789:
1742:
1741:
1676:
1675:
1628:
1627:
1566:
1565:
1433:
1432:
724:{\displaystyle k}
675:{\displaystyle n}
575:
574:
44:periodic sequence
16:(Redirected from
3402:
3385:Divisor function
3357:
3320:
3289:Natural language
3284:
3248:
3216:Generated via a
3212:
3172:
3077:Digit-reassembly
3042:Self-descriptive
2846:
2811:
2797:
2748:Lucas–Carmichael
2738:Harmonic divisor
2686:
2612:Sparsely totient
2587:Highly cototient
2496:Multiply perfect
2486:Highly composite
2429:
2412:
2327:
2282:
2263:Telephone number
2159:
2117:
2098:Square pyramidal
2080:Stella octangula
2005:
1871:
1862:
1854:Figurate numbers
1849:
1796:
1748:
1682:
1634:
1572:
1476:
1460:
1453:
1446:
1437:
1410:Harmonic divisor
1296:Aliquot sequence
1276:Highly composite
1200:Multiply perfect
1096:
1074:Divisor function
1040:
1033:
1026:
1017:
999:
998:
948:
947:
932:
926:
922:Sergei Chernykh
920:
914:
905:
899:
898:
888:
879:(110): 431–432.
870:
861:
855:
841:
822:
799:
797:
796:
793:{\displaystyle }
791:
767:
765:
764:
759:
757:
756:
730:
728:
727:
722:
710:
708:
707:
702:
681:
679:
678:
673:
661:
659:
658:
653:
651:
650:
623:aliquot sequence
605:
456:Number of known
447:
429:
427:
426:
421:
413:
412:
393:
391:
390:
385:
363:
361:
360:
355:
341:
340:
321:
319:
318:
313:
291:
289:
288:
283:
263:
262:
243:
241:
240:
235:
213:
211:
210:
205:
185:
184:
165:
163:
162:
157:
132:aliquot sequence
122:
120:
119:
114:
112:
111:
88:amicable numbers
52:amicable numbers
36:sociable numbers
21:
18:Sociable numbers
3410:
3409:
3405:
3404:
3403:
3401:
3400:
3399:
3370:
3369:
3368:
3363:
3341:
3337:Strobogrammatic
3328:
3310:
3292:
3274:
3256:
3238:
3220:
3202:
3179:
3158:
3142:
3101:Divisor-related
3096:
3056:
3007:
2977:
2914:
2898:
2877:
2844:
2817:
2805:
2787:
2699:
2698:related numbers
2672:
2649:
2616:
2607:Perfect totient
2573:
2550:
2481:Highly abundant
2423:
2402:
2334:
2317:
2289:
2272:
2258:Stirling second
2164:
2141:
2102:
2084:
2041:
1990:
1927:
1888:Centered square
1856:
1839:
1801:
1786:
1753:
1738:
1690:
1689:defined numbers
1672:
1639:
1624:
1595:Double Mersenne
1581:
1562:
1484:
1470:
1468:natural numbers
1464:
1434:
1429:
1373:
1329:
1290:
1261:Highly abundant
1239:
1220:Unitary perfect
1173:
1097:
1088:
1069:Unitary divisor
1047:
1044:
984:
983:
970:
952:
951:
934:
933:
929:
921:
917:
906:
902:
868:
863:
862:
858:
842:
838:
833:
818:
814:
802:perfect numbers
770:
769:
742:
737:
736:
713:
712:
684:
683:
664:
663:
636:
631:
630:
619:
601:
538:21,548,919,483
495:Amicable number
442:
404:
396:
395:
376:
375:
332:
324:
323:
304:
303:
254:
246:
245:
226:
225:
176:
168:
167:
148:
147:
140:
103:
92:
91:
84:proper divisors
66:proper divisors
48:perfect numbers
28:
23:
22:
15:
12:
11:
5:
3408:
3406:
3398:
3397:
3392:
3387:
3382:
3372:
3371:
3365:
3364:
3362:
3361:
3350:
3347:
3346:
3343:
3342:
3340:
3339:
3333:
3330:
3329:
3323:
3316:
3315:
3312:
3311:
3309:
3308:
3303:
3297:
3294:
3293:
3287:
3280:
3279:
3276:
3275:
3273:
3272:
3270:Sorting number
3267:
3265:Pancake number
3261:
3258:
3257:
3251:
3244:
3243:
3240:
3239:
3237:
3236:
3231:
3225:
3222:
3221:
3215:
3208:
3207:
3204:
3203:
3201:
3200:
3195:
3190:
3184:
3181:
3180:
3177:Binary numbers
3175:
3168:
3167:
3164:
3163:
3160:
3159:
3157:
3156:
3150:
3148:
3144:
3143:
3141:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3104:
3102:
3098:
3097:
3095:
3094:
3089:
3084:
3079:
3074:
3068:
3066:
3058:
3057:
3055:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3018:
3016:
3009:
3008:
3006:
3005:
3004:
3003:
2992:
2990:
2987:P-adic numbers
2983:
2982:
2979:
2978:
2976:
2975:
2974:
2973:
2963:
2958:
2953:
2948:
2943:
2938:
2933:
2928:
2922:
2920:
2916:
2915:
2913:
2912:
2906:
2904:
2903:Coding-related
2900:
2899:
2897:
2896:
2891:
2885:
2883:
2879:
2878:
2876:
2875:
2870:
2865:
2860:
2854:
2852:
2843:
2842:
2841:
2840:
2838:Multiplicative
2835:
2824:
2822:
2807:
2806:
2802:Numeral system
2800:
2793:
2792:
2789:
2788:
2786:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2745:
2740:
2735:
2730:
2725:
2720:
2715:
2710:
2704:
2701:
2700:
2689:
2682:
2681:
2678:
2677:
2674:
2673:
2671:
2670:
2665:
2659:
2657:
2651:
2650:
2648:
2647:
2642:
2637:
2632:
2626:
2624:
2618:
2617:
2615:
2614:
2609:
2604:
2599:
2594:
2592:Highly totient
2589:
2583:
2581:
2575:
2574:
2572:
2571:
2566:
2560:
2558:
2552:
2551:
2549:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2446:Almost perfect
2443:
2437:
2435:
2425:
2424:
2415:
2408:
2407:
2404:
2403:
2401:
2400:
2395:
2390:
2385:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2339:
2336:
2335:
2330:
2323:
2322:
2319:
2318:
2316:
2315:
2310:
2305:
2300:
2294:
2291:
2290:
2285:
2278:
2277:
2274:
2273:
2271:
2270:
2265:
2260:
2255:
2253:Stirling first
2250:
2245:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2169:
2166:
2165:
2162:
2155:
2154:
2151:
2150:
2147:
2146:
2143:
2142:
2140:
2139:
2134:
2129:
2123:
2121:
2114:
2108:
2107:
2104:
2103:
2101:
2100:
2094:
2092:
2086:
2085:
2083:
2082:
2077:
2072:
2067:
2062:
2057:
2051:
2049:
2043:
2042:
2040:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2002:
1996:
1995:
1992:
1991:
1989:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1937:
1935:
1929:
1928:
1926:
1925:
1920:
1915:
1910:
1905:
1900:
1895:
1890:
1885:
1879:
1877:
1868:
1858:
1857:
1852:
1845:
1844:
1841:
1840:
1838:
1837:
1832:
1827:
1822:
1817:
1812:
1806:
1803:
1802:
1799:
1792:
1791:
1788:
1787:
1785:
1784:
1779:
1774:
1769:
1764:
1758:
1755:
1754:
1751:
1744:
1743:
1740:
1739:
1737:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1701:
1695:
1692:
1691:
1685:
1678:
1677:
1674:
1673:
1671:
1670:
1665:
1660:
1655:
1650:
1644:
1641:
1640:
1637:
1630:
1629:
1626:
1625:
1623:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1586:
1583:
1582:
1575:
1568:
1567:
1564:
1563:
1561:
1560:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1489:
1486:
1485:
1479:
1472:
1471:
1465:
1463:
1462:
1455:
1448:
1440:
1431:
1430:
1428:
1427:
1422:
1417:
1412:
1407:
1402:
1397:
1392:
1387:
1381:
1379:
1375:
1374:
1372:
1371:
1366:
1361:
1356:
1351:
1346:
1340:
1338:
1331:
1330:
1328:
1327:
1322:
1317:
1307:
1301:
1299:
1292:
1291:
1289:
1288:
1283:
1278:
1273:
1268:
1263:
1258:
1253:
1247:
1245:
1241:
1240:
1238:
1237:
1232:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1190:Almost perfect
1187:
1181:
1179:
1175:
1174:
1172:
1171:
1166:
1161:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1121:
1116:
1111:
1105:
1103:
1099:
1098:
1091:
1089:
1087:
1086:
1081:
1076:
1071:
1066:
1061:
1055:
1053:
1049:
1048:
1045:
1043:
1042:
1035:
1028:
1020:
1014:
1013:
1000:
981:
976:
969:
968:External links
966:
965:
964:
950:
949:
927:
915:
900:
856:
835:
834:
832:
829:
813:
810:
806:amicable pairs
789:
786:
783:
780:
777:
755:
752:
749:
745:
720:
700:
697:
694:
691:
671:
649:
646:
643:
639:
627:directed graph
618:
615:
573:
572:
569:
566:
562:
561:
558:
555:
551:
550:
549:1,095,447,416
547:
544:
540:
539:
536:
533:
529:
528:
525:
522:
518:
517:
514:
511:
507:
506:
503:
500:
487:
486:
483:
480:
475:Perfect number
467:
466:
462:lowest number
460:
454:
441:
438:
437:
436:
435:
434:
419:
416:
411:
407:
403:
383:
371:
370:
369:
368:
353:
350:
347:
344:
339:
335:
331:
311:
299:
298:
297:
296:
281:
278:
275:
272:
269:
266:
261:
257:
253:
233:
221:
220:
219:
218:
203:
200:
197:
194:
191:
188:
183:
179:
175:
155:
139:
136:
110:
106:
102:
99:
80:perfect number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3407:
3396:
3395:Number theory
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3377:
3375:
3360:
3356:
3352:
3351:
3348:
3338:
3335:
3334:
3331:
3326:
3321:
3317:
3307:
3304:
3302:
3299:
3298:
3295:
3290:
3285:
3281:
3271:
3268:
3266:
3263:
3262:
3259:
3254:
3249:
3245:
3235:
3232:
3230:
3227:
3226:
3223:
3219:
3213:
3209:
3199:
3196:
3194:
3191:
3189:
3186:
3185:
3182:
3178:
3173:
3169:
3155:
3152:
3151:
3149:
3145:
3139:
3136:
3134:
3131:
3129:
3128:Polydivisible
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3105:
3103:
3099:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3069:
3067:
3064:
3059:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3019:
3017:
3014:
3010:
3002:
2999:
2998:
2997:
2994:
2993:
2991:
2988:
2984:
2972:
2969:
2968:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2923:
2921:
2917:
2911:
2908:
2907:
2905:
2901:
2895:
2892:
2890:
2887:
2886:
2884:
2882:Digit product
2880:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2853:
2851:
2847:
2839:
2836:
2834:
2831:
2830:
2829:
2826:
2825:
2823:
2821:
2816:
2812:
2808:
2803:
2798:
2794:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2718:Erdős–Nicolas
2716:
2714:
2711:
2709:
2706:
2705:
2702:
2697:
2693:
2687:
2683:
2669:
2666:
2664:
2661:
2660:
2658:
2656:
2652:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2627:
2625:
2623:
2619:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2584:
2582:
2580:
2576:
2570:
2567:
2565:
2562:
2561:
2559:
2557:
2553:
2547:
2544:
2542:
2539:
2537:
2536:Superabundant
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2438:
2436:
2434:
2430:
2426:
2422:
2418:
2413:
2409:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2340:
2337:
2333:
2328:
2324:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2292:
2288:
2283:
2279:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2170:
2167:
2160:
2156:
2138:
2135:
2133:
2130:
2128:
2125:
2124:
2122:
2118:
2115:
2113:
2112:4-dimensional
2109:
2099:
2096:
2095:
2093:
2091:
2087:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2052:
2050:
2048:
2044:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2022:Centered cube
2020:
2018:
2015:
2014:
2012:
2010:
2006:
2003:
2001:
2000:3-dimensional
1997:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1938:
1936:
1934:
1930:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1880:
1878:
1876:
1872:
1869:
1867:
1866:2-dimensional
1863:
1859:
1855:
1850:
1846:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1810:Nonhypotenuse
1808:
1807:
1804:
1797:
1793:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1759:
1756:
1749:
1745:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1693:
1688:
1683:
1679:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1642:
1635:
1631:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1584:
1579:
1573:
1569:
1559:
1556:
1554:
1551:
1549:
1548:Perfect power
1546:
1544:
1541:
1539:
1538:Seventh power
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1490:
1487:
1482:
1477:
1473:
1469:
1461:
1456:
1454:
1449:
1447:
1442:
1441:
1438:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1382:
1380:
1376:
1370:
1367:
1365:
1364:Polydivisible
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1341:
1339:
1336:
1332:
1326:
1323:
1321:
1318:
1315:
1311:
1308:
1306:
1303:
1302:
1300:
1297:
1293:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1266:Superabundant
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1248:
1246:
1242:
1236:
1235:Erdős–Nicolas
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1182:
1180:
1176:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1144:Perfect power
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1106:
1104:
1100:
1095:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1056:
1054:
1050:
1041:
1036:
1034:
1029:
1027:
1022:
1021:
1018:
1012:
1008:
1004:
1001:
996:
995:
990:
987:
982:
980:
977:
975:
972:
971:
967:
962:
958:
954:
953:
946:
942:
938:
931:
928:
925:
919:
916:
913:
909:
904:
901:
896:
892:
887:
882:
878:
874:
867:
860:
857:
853:
849:
846:
840:
837:
830:
828:
826:
821:
811:
809:
807:
803:
784:
781:
778:
753:
750:
747:
743:
734:
718:
695:
689:
669:
647:
644:
641:
637:
628:
624:
616:
614:
611:
609:
604:
597:
594:
592:
588:
584:
580:
570:
567:
564:
559:
556:
553:
548:
545:
542:
537:
534:
531:
526:
523:
520:
515:
512:
509:
504:
501:
499:
497:
496:
489:
484:
481:
479:
477:
476:
469:
465:
461:
459:
455:
453:
449:
445:
439:
432:
431:
417:
414:
409:
405:
401:
381:
373:
372:
366:
365:
351:
348:
345:
342:
337:
333:
329:
309:
301:
300:
294:
293:
279:
276:
273:
270:
267:
264:
259:
255:
251:
231:
223:
222:
216:
215:
201:
198:
195:
192:
189:
186:
181:
177:
173:
153:
145:
144:
143:
137:
135:
133:
129:
124:
108:
104:
100:
97:
89:
85:
81:
76:
74:
69:
67:
63:
60:
59:mathematician
57:
53:
49:
45:
41:
37:
33:
19:
3092:Transposable
2956:Narcissistic
2863:Digital root
2783:Super-Poulet
2743:Jordan–Pólya
2692:prime factor
2597:Noncototient
2564:Almost prime
2546:Superperfect
2521:Refactorable
2516:Quasiperfect
2491:Hyperperfect
2332:Pseudoprimes
2303:Wall–Sun–Sun
2238:Ordered Bell
2208:Fuss–Catalan
2120:non-centered
2070:Dodecahedral
2047:non-centered
1933:non-centered
1835:Wolstenholme
1580:× 2 ± 1
1577:
1576:Of the form
1543:Eighth power
1523:Fourth power
1425:Superperfect
1420:Refactorable
1319:
1215:Superperfect
1210:Hyperperfect
1195:Quasiperfect
1079:Prime factor
992:
960:
959:Math. Comp.
956:
936:
930:
918:
903:
876:
872:
859:
847:
839:
815:
620:
612:
598:
595:
590:
582:
576:
560:805,984,760
493:
491:
473:
471:
464:in sequence
463:
457:
451:
443:
141:
125:
123:as of 1970.
77:
70:
40:aliquot sums
35:
29:
3113:Extravagant
3108:Equidigital
3063:permutation
3022:Palindromic
2996:Automorphic
2894:Sum-product
2873:Sum-product
2828:Persistence
2723:Erdős–Woods
2645:Untouchable
2526:Semiperfect
2476:Hemiperfect
2137:Tesseractic
2075:Icosahedral
2055:Tetrahedral
1986:Dodecagonal
1687:Recursively
1558:Prime power
1533:Sixth power
1528:Fifth power
1508:Power of 10
1466:Classes of
1349:Extravagant
1344:Equidigital
1305:Untouchable
1225:Semiperfect
1205:Hemiperfect
1134:Square-free
579:conjectured
502:1225736919
458:sequences
62:Paul Poulet
32:mathematics
3374:Categories
3325:Graphemics
3198:Pernicious
3052:Undulating
3027:Pandigital
3001:Trimorphic
2602:Nontotient
2451:Arithmetic
2065:Octahedral
1966:Heptagonal
1956:Pentagonal
1941:Triangular
1782:Sierpiński
1704:Jacobsthal
1503:Power of 3
1498:Power of 2
1385:Arithmetic
1378:Other sets
1337:-dependent
955:H. Cohen,
831:References
516:1,264,460
3082:Parasitic
2931:Factorion
2858:Digit sum
2850:Digit sum
2668:Fortunate
2655:Primorial
2569:Semiprime
2506:Practical
2471:Descartes
2466:Deficient
2456:Betrothed
2298:Wieferich
2127:Pentatope
2090:pyramidal
1981:Decagonal
1976:Nonagonal
1971:Octagonal
1961:Hexagonal
1820:Practical
1767:Congruent
1699:Fibonacci
1663:Loeschian
1415:Descartes
1390:Deficient
1325:Betrothed
1230:Practical
1119:Semiprime
1114:Composite
994:MathWorld
895:0025-5718
587:congruent
450:Sequence
415:⋅
349:⋅
343:⋅
277:⋅
271:⋅
265:⋅
199:⋅
193:⋅
187:⋅
101:×
3154:Friedman
3087:Primeval
3032:Repdigit
2989:-related
2936:Kaprekar
2910:Meertens
2833:Additive
2820:dynamics
2728:Friendly
2640:Sociable
2630:Amicable
2441:Abundant
2421:dynamics
2243:Schröder
2233:Narayana
2203:Eulerian
2193:Delannoy
2188:Dedekind
2009:centered
1875:centered
1762:Amenable
1719:Narayana
1709:Leonardo
1605:Mersenne
1553:Powerful
1493:Achilles
1400:Solitary
1395:Friendly
1320:Sociable
1310:Amicable
1298:-related
1251:Abundant
1149:Achilles
1139:Powerful
1052:Overview
682:, where
581:that if
3327:related
3291:related
3255:related
3253:Sorting
3138:Vampire
3123:Harshad
3065:related
3037:Repunit
2951:Lychrel
2926:Dudeney
2778:Størmer
2773:Sphenic
2758:Regular
2696:divisor
2635:Perfect
2531:Sublime
2501:Perfect
2228:Motzkin
2183:Catalan
1724:Padovan
1658:Leyland
1653:Idoneal
1648:Hilbert
1620:Woodall
1405:Sublime
1359:Harshad
1185:Perfect
1169:Unusual
1159:Regular
1129:Sphenic
1064:Divisor
823:in the
820:A292217
606:in the
603:A072890
571:14,316
527:12,496
452:length
382:1305184
310:1727636
232:1547860
154:1264460
138:Example
56:Belgian
42:form a
3193:Odious
3118:Frugal
3072:Cyclic
3061:Digit-
2768:Smooth
2753:Pronic
2713:Cyclic
2690:Other
2663:Euclid
2313:Wilson
2287:Primes
1946:Square
1815:Polite
1777:Riesel
1772:Knödel
1734:Perrin
1615:Thabit
1600:Fermat
1590:Cullen
1513:Square
1481:Powers
1354:Frugal
1314:Triple
1154:Smooth
1124:Pronic
893:
733:Cycles
577:It is
73:period
3234:Prime
3229:Lucky
3218:sieve
3147:Other
3133:Smith
3013:Digit
2971:Happy
2946:Keith
2919:Other
2763:Rough
2733:Giuga
2198:Euler
2060:Cubic
1714:Lucas
1610:Proth
1369:Smith
1286:Weird
1164:Rough
1109:Prime
869:(PDF)
513:5398
430:) is
418:40787
364:) is
292:) is
214:) is
128:prime
3188:Evil
2868:Self
2818:and
2708:Blum
2419:and
2223:Lobb
2178:Cake
2173:Bell
1923:Star
1830:Ulam
1729:Pell
1518:Cube
1335:Base
1011:OEIS
1005:and
891:ISSN
825:OEIS
621:The
610:).
608:OEIS
505:220
202:3719
71:The
50:and
3306:Ban
2694:or
2213:Lah
1009:in
941:doi
881:doi
827:).
735:in
585:is
565:28
482:51
352:829
346:521
280:401
274:193
30:In
3376::
991:.
961:24
889:.
877:24
875:.
871:.
854:.)
848:25
808:.
731:.
629:,
593:.
568:1
557:1
554:9
546:4
543:8
535:5
532:6
524:1
521:5
510:4
498:)
490:2
485:6
478:)
470:1
196:17
105:10
34:,
1578:a
1459:e
1452:t
1445:v
1316:)
1312:(
1039:e
1032:t
1025:v
997:.
943::
897:.
883::
788:]
785:n
782:,
779:1
776:[
754:s
751:,
748:n
744:G
719:k
699:)
696:k
693:(
690:s
670:n
648:s
645:,
642:n
638:G
591:n
583:n
492:(
472:(
410:5
406:2
402:=
394:(
338:2
334:2
330:=
322:(
268:5
260:2
256:2
252:=
244:(
190:5
182:2
178:2
174:=
166:(
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