29:
3197:
571:
601:. That is, if one forms a sequence of concentric shells in three dimensions, where the first shell consists of a single point, the second shell consists of the six vertices of a pentagonal pyramid, and each successive shell forms a larger pentagonal pyramid with a
279:, formed by accreting concentric layers of cubes onto a central cube. The centered octahedral numbers count the cubes used by this construction. HaĂĽy proposed this construction, and several related constructions of other polyhedra, as a model for the structure of
378:
171:
472:
553:
1299:
934:
842:
208:
1292:
578:
The octahedron in the three-dimensional integer lattice, whose number of lattice points is counted by the centered octahedral number, is a
2099:
1285:
2094:
2109:
2089:
772:
668:
2802:
2382:
301:
94:
736:
Fathauer, Robert W. (2013), "Iterative arrangements of polyhedra – Relationships to classical fractals and Haüy constructions",
2104:
927:
2888:
755:
Maitte, Bernard (2013), "The
Construction of Group Theory in Crystallography", in Barbin, Evelyne; Pisano, Raffaele (eds.),
2204:
609:
of points on the pentagonal face, then the total number of points in this configuration is a centered octahedral number.
2554:
1873:
1666:
1112:
2589:
2559:
2234:
2224:
597:
The same numbers can be viewed as figurate numbers in a different way, as the centered figurate numbers generated by a
2730:
2144:
1878:
1858:
1117:
1097:
624:). As for Delannoy numbers more generally, these numbers count the paths from the southwest corner of a 3 Ă—
2420:
405:
2584:
2679:
2302:
2059:
1850:
1744:
1734:
1724:
1089:
988:
978:
968:
2564:
3221:
2807:
2352:
1973:
1759:
1754:
1749:
1739:
1716:
1212:
1003:
998:
993:
983:
960:
920:
1792:
2049:
2918:
2883:
2669:
2579:
2453:
2428:
2337:
2327:
1939:
1921:
1841:
1178:
1155:
1080:
486:
3178:
2448:
2322:
1953:
1729:
1509:
1436:
1192:
973:
275:
active in the late 18th and early 19th centuries. His "HaĂĽy construction" approximates an octahedron as a
2433:
2287:
2214:
1369:
1247:
3142:
2782:
3075:
2969:
2933:
2674:
2397:
2377:
2194:
1863:
1651:
1623:
1102:
883:
703:
586:, a geometry in which distance is measured by the sum of the coordinatewise distances rather than by
268:
256:
42:
837:"Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))"
2797:
2661:
2656:
2624:
2387:
2362:
2357:
2332:
2262:
2258:
2189:
2079:
1911:
1707:
1676:
1145:
951:
478:
396:
3196:
3200:
2954:
2949:
2863:
2837:
2735:
2714:
2486:
2367:
2317:
2239:
2209:
2149:
1916:
1896:
1827:
1540:
1173:
1150:
1066:
899:
873:
598:
587:
579:
255:, which count certain two-dimensional lattice paths. The HaĂĽy octahedral numbers are named after
2084:
3094:
3039:
2893:
2868:
2842:
2619:
2297:
2292:
2219:
2199:
2184:
1906:
1888:
1807:
1797:
1782:
1560:
1545:
1140:
1127:
1046:
1036:
1026:
805:
768:
664:
658:
606:
602:
559:
77:
799:
779:
737:
3130:
2923:
2509:
2481:
2471:
2463:
2347:
2312:
2307:
2274:
1968:
1931:
1822:
1817:
1812:
1802:
1774:
1661:
1613:
1608:
1565:
1504:
1207:
1165:
1061:
1056:
1051:
1041:
1018:
891:
760:
688:
583:
28:
895:
715:
3106:
2995:
2928:
2854:
2777:
2751:
2569:
2282:
2139:
2074:
2044:
2034:
2029:
1695:
1603:
1550:
1394:
1334:
943:
711:
613:
252:
244:
240:
82:
887:
707:
3111:
2979:
2964:
2828:
2792:
2767:
2643:
2614:
2599:
2476:
2372:
2342:
2069:
2024:
1901:
1499:
1494:
1489:
1461:
1446:
1359:
1344:
1322:
1309:
1135:
808:
628:
grid to the northeast corner, using steps that go one unit east, north, or northeast.
3215:
3034:
3018:
2959:
2913:
2609:
2594:
2504:
2229:
1787:
1656:
1618:
1575:
1456:
1441:
1431:
1389:
1379:
1354:
1257:
1031:
903:
861:
3070:
3059:
2974:
2812:
2787:
2704:
2604:
2574:
2549:
2533:
2438:
2405:
2154:
2128:
2039:
1978:
1555:
1451:
1384:
1364:
1339:
1262:
1217:
759:, History of Mechanism and Machine Science, vol. 16, Springer, pp. 1–30,
272:
200:
196:
3029:
2904:
2709:
2173:
2064:
2019:
2014:
1764:
1671:
1570:
1399:
1374:
1349:
1252:
1008:
832:
764:
654:
192:
188:
71:
3166:
3147:
2443:
2054:
650:
248:
1277:
2772:
2699:
2691:
2496:
2410:
1528:
1232:
813:
757:
The
Dialectic Relation Between Physics and Mathematics in the XIXth Century
739:
Proceedings of
Bridges 2013: Mathematics, Music, Art, Architecture, Culture
2873:
276:
64:
570:
2878:
2537:
594:
call the centered octahedral numbers "the volume of the crystal ball".
280:
912:
878:
251:
centered at the origin. The same numbers are special cases of the
3164:
3128:
3092:
3056:
3016:
2641:
2530:
2256:
2171:
2126:
2003:
1693:
1640:
1592:
1526:
1478:
1416:
1320:
1281:
916:
558:
They may also be computed as the sums of pairs of consecutive
836:
373:{\displaystyle {\frac {(2n+1)\left(2n^{2}+2n+3\right)}{3}}}
166:{\displaystyle {\frac {(2n+1)\left(2n^{2}+2n+3\right)}{3}}}
267:
The name "HaĂĽy octahedral number" comes from the work of
184:
180:
866:
Journal of
Statistical Mechanics: Theory and Experiment
291:
The number of three-dimensional lattice points within
489:
408:
304:
217:
97:
2988:
2942:
2902:
2853:
2827:
2760:
2744:
2723:
2690:
2655:
2495:
2462:
2419:
2396:
2273:
1961:
1952:
1930:
1887:
1849:
1840:
1773:
1715:
1706:
1240:
1230:
1200:
1191:
1164:
1126:
1088:
1079:
1017:
959:
950:
207:
176:
88:
70:
56:
48:
38:
547:
466:
372:
165:
794:Essai d'une théorie sur la structure des crystaux
689:"Objects counted by the central Delannoy numbers"
574:63 Delannoy paths through a 3 Ă— 3 grid
391:1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ...
467:{\displaystyle {\frac {(1+x)^{3}}{(1-x)^{4}}}.}
33:HaĂĽy construction of an octahedron by 129 cubes
243:that counts the points of a three-dimensional
1293:
928:
862:"Counting lattice animals in high dimensions"
612:The centered octahedral numbers are also the
591:
8:
860:Luther, Sebastian; Mertens, Stephan (2011),
295:steps of the origin is given by the formula
21:
663:, World Scientific, pp. 107–109, 132,
3161:
3125:
3089:
3053:
3013:
2687:
2652:
2638:
2527:
2270:
2253:
2168:
2123:
2000:
1958:
1846:
1712:
1703:
1690:
1637:
1594:Possessing a specific set of other numbers
1589:
1523:
1475:
1413:
1317:
1300:
1286:
1278:
1237:
1197:
1085:
956:
935:
921:
913:
20:
877:
843:On-Line Encyclopedia of Integer Sequences
533:
488:
477:The centered octahedral numbers obey the
452:
428:
409:
407:
338:
305:
303:
131:
98:
96:
605:of points on each triangular face and a
569:
637:
855:
853:
827:
825:
399:of the centered octahedral numbers is
682:
680:
548:{\displaystyle C(n)=C(n-1)+4n^{2}+2.}
7:
645:
643:
641:
383:The first few of these numbers (for
14:
809:"Haűy [sic] Construction"
3195:
2803:Perfect digit-to-digit invariant
896:10.1088/1742-5468/2011/09/P09026
27:
520:
508:
499:
493:
449:
436:
425:
412:
323:
308:
116:
101:
58:
1:
1642:Expressible via specific sums
1113:Centered dodecahedral numbers
1118:Centered icosahedral numbers
1098:Centered tetrahedral numbers
696:Journal of Integer Sequences
2731:Multiplicative digital root
1108:Centered octahedral numbers
989:Centered heptagonal numbers
979:Centered pentagonal numbers
969:Centered triangular numbers
765:10.1007/978-94-007-5380-8_1
687:Sulanke, Robert A. (2003),
592:Luther & Mertens (2011)
566:Alternative interpretations
3238:
1213:Squared triangular numbers
1004:Centered decagonal numbers
999:Centered nonagonal numbers
994:Centered octagonal numbers
984:Centered hexagonal numbers
833:Sloane, N. J. A.
233:centered octahedral number
22:Centered octahedral number
3191:
3174:
3160:
3138:
3124:
3102:
3088:
3066:
3052:
3025:
3012:
2808:Perfect digital invariant
2651:
2637:
2545:
2526:
2383:Superior highly composite
2269:
2252:
2180:
2167:
2135:
2122:
2010:
1999:
1702:
1689:
1647:
1636:
1599:
1588:
1536:
1522:
1485:
1474:
1427:
1412:
1330:
1316:
26:
2421:Euler's totient function
2205:Euler–Jacobi pseudoprime
1480:Other polynomial numbers
1179:Square pyramidal numbers
1156:Stella octangula numbers
792:Haüy, René-Just (1784),
2235:Somer–Lucas pseudoprime
2225:Lucas–Carmichael number
2060:Lazy caterer's sequence
974:Centered square numbers
2110:Wedderburn–Etherington
1510:Lucky numbers of Euler
582:for three-dimensional
575:
549:
468:
374:
237:HaĂĽy octahedral number
167:
2398:Prime omega functions
2215:Frobenius pseudoprime
2005:Combinatorial numbers
1874:Centered dodecahedral
1667:Primary pseudoperfect
1103:Centered cube numbers
702:(1), Article 03.1.5,
573:
550:
469:
375:
168:
2857:-composition related
2657:Arithmetic functions
2259:Arithmetic functions
2195:Elliptic pseudoprime
1879:Centered icosahedral
1859:Centered tetrahedral
1146:Dodecahedral numbers
798:. See in particular
778:. See in particular
487:
406:
387:= 0, 1, 2, ...) are
302:
281:crystalline minerals
95:
2783:Kaprekar's constant
2303:Colossally abundant
2190:Catalan pseudoprime
2090:Schröder–Hipparchus
1869:Centered octahedral
1745:Centered heptagonal
1735:Centered pentagonal
1725:Centered triangular
1325:and related numbers
1263:8-hypercube numbers
1258:7-hypercube numbers
1253:6-hypercube numbers
1248:5-hypercube numbers
1218:Tesseractic numbers
1174:Tetrahedral numbers
1151:Icosahedral numbers
1067:Dodecagonal numbers
888:2011JSMTE..09..026L
708:2003JIntS...6...15S
590:. For this reason,
479:recurrence relation
397:generating function
247:that lie inside an
222:Centered octahedral
23:
3201:Mathematics portal
3143:Aronson's sequence
2889:Smarandache–Wellin
2646:-dependent numbers
2353:Primitive abundant
2240:Strong pseudoprime
2230:Perrin pseudoprime
2210:Fermat pseudoprime
2150:Wolstenholme prime
1974:Squared triangular
1760:Centered decagonal
1755:Centered nonagonal
1750:Centered octagonal
1740:Centered hexagonal
1141:Octahedral numbers
1047:Heptagonal numbers
1037:Pentagonal numbers
1027:Triangular numbers
846:. OEIS Foundation.
806:Weisstein, Eric W.
599:pentagonal pyramid
588:Euclidean distance
576:
560:octahedral numbers
545:
464:
370:
163:
78:Polyhedral numbers
3209:
3208:
3187:
3186:
3156:
3155:
3120:
3119:
3084:
3083:
3048:
3047:
3008:
3007:
3004:
3003:
2823:
2822:
2633:
2632:
2522:
2521:
2518:
2517:
2464:Aliquot sequences
2275:Divisor functions
2248:
2247:
2220:Lucas pseudoprime
2200:Euler pseudoprime
2185:Carmichael number
2163:
2162:
2118:
2117:
1995:
1994:
1991:
1990:
1987:
1986:
1948:
1947:
1836:
1835:
1793:Square triangular
1685:
1684:
1632:
1631:
1584:
1583:
1518:
1517:
1470:
1469:
1408:
1407:
1275:
1274:
1271:
1270:
1226:
1225:
1208:Pentatope numbers
1187:
1186:
1075:
1074:
1062:Decagonal numbers
1057:Nonagonal numbers
1052:Octagonal numbers
1042:Hexagonal numbers
607:pentagonal number
603:triangular number
459:
368:
229:
228:
161:
3229:
3222:Figurate numbers
3199:
3162:
3131:Natural language
3126:
3090:
3058:Generated via a
3054:
3014:
2919:Digit-reassembly
2884:Self-descriptive
2688:
2653:
2639:
2590:Lucas–Carmichael
2580:Harmonic divisor
2528:
2454:Sparsely totient
2429:Highly cototient
2338:Multiply perfect
2328:Highly composite
2271:
2254:
2169:
2124:
2105:Telephone number
2001:
1959:
1940:Square pyramidal
1922:Stella octangula
1847:
1713:
1704:
1696:Figurate numbers
1691:
1638:
1590:
1524:
1476:
1414:
1318:
1302:
1295:
1288:
1279:
1238:
1198:
1086:
957:
944:Figurate numbers
937:
930:
923:
914:
907:
906:
881:
857:
848:
847:
829:
820:
819:
818:
797:
789:
783:
777:
752:
746:
745:
744:
733:
727:
725:
724:
722:
693:
684:
675:
673:
660:Figurate Numbers
647:
614:Delannoy numbers
584:taxicab geometry
554:
552:
551:
546:
538:
537:
473:
471:
470:
465:
460:
458:
457:
456:
434:
433:
432:
410:
379:
377:
376:
371:
369:
364:
363:
359:
343:
342:
306:
253:Delannoy numbers
172:
170:
169:
164:
162:
157:
156:
152:
136:
135:
99:
83:Delannoy numbers
60:
49:Publication year
31:
24:
3237:
3236:
3232:
3231:
3230:
3228:
3227:
3226:
3212:
3211:
3210:
3205:
3183:
3179:Strobogrammatic
3170:
3152:
3134:
3116:
3098:
3080:
3062:
3044:
3021:
3000:
2984:
2943:Divisor-related
2938:
2898:
2849:
2819:
2756:
2740:
2719:
2686:
2659:
2647:
2629:
2541:
2540:related numbers
2514:
2491:
2458:
2449:Perfect totient
2415:
2392:
2323:Highly abundant
2265:
2244:
2176:
2159:
2131:
2114:
2100:Stirling second
2006:
1983:
1944:
1926:
1883:
1832:
1769:
1730:Centered square
1698:
1681:
1643:
1628:
1595:
1580:
1532:
1531:defined numbers
1514:
1481:
1466:
1437:Double Mersenne
1423:
1404:
1326:
1312:
1310:natural numbers
1306:
1276:
1267:
1222:
1183:
1160:
1122:
1071:
1013:
946:
941:
911:
910:
859:
858:
851:
831:
830:
823:
804:
803:
791:
790:
786:
775:
754:
753:
749:
742:
735:
734:
730:
720:
718:
691:
686:
685:
678:
671:
649:
648:
639:
634:
568:
529:
485:
484:
448:
435:
424:
411:
404:
403:
334:
330:
326:
307:
300:
299:
289:
265:
245:integer lattice
241:figurate number
225:
127:
123:
119:
100:
93:
92:
81:
34:
17:
16:Figurate number
12:
11:
5:
3235:
3233:
3225:
3224:
3214:
3213:
3207:
3206:
3204:
3203:
3192:
3189:
3188:
3185:
3184:
3182:
3181:
3175:
3172:
3171:
3165:
3158:
3157:
3154:
3153:
3151:
3150:
3145:
3139:
3136:
3135:
3129:
3122:
3121:
3118:
3117:
3115:
3114:
3112:Sorting number
3109:
3107:Pancake number
3103:
3100:
3099:
3093:
3086:
3085:
3082:
3081:
3079:
3078:
3073:
3067:
3064:
3063:
3057:
3050:
3049:
3046:
3045:
3043:
3042:
3037:
3032:
3026:
3023:
3022:
3019:Binary numbers
3017:
3010:
3009:
3006:
3005:
3002:
3001:
2999:
2998:
2992:
2990:
2986:
2985:
2983:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2946:
2944:
2940:
2939:
2937:
2936:
2931:
2926:
2921:
2916:
2910:
2908:
2900:
2899:
2897:
2896:
2891:
2886:
2881:
2876:
2871:
2866:
2860:
2858:
2851:
2850:
2848:
2847:
2846:
2845:
2834:
2832:
2829:P-adic numbers
2825:
2824:
2821:
2820:
2818:
2817:
2816:
2815:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2764:
2762:
2758:
2757:
2755:
2754:
2748:
2746:
2745:Coding-related
2742:
2741:
2739:
2738:
2733:
2727:
2725:
2721:
2720:
2718:
2717:
2712:
2707:
2702:
2696:
2694:
2685:
2684:
2683:
2682:
2680:Multiplicative
2677:
2666:
2664:
2649:
2648:
2644:Numeral system
2642:
2635:
2634:
2631:
2630:
2628:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2546:
2543:
2542:
2531:
2524:
2523:
2520:
2519:
2516:
2515:
2513:
2512:
2507:
2501:
2499:
2493:
2492:
2490:
2489:
2484:
2479:
2474:
2468:
2466:
2460:
2459:
2457:
2456:
2451:
2446:
2441:
2436:
2434:Highly totient
2431:
2425:
2423:
2417:
2416:
2414:
2413:
2408:
2402:
2400:
2394:
2393:
2391:
2390:
2385:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2290:
2288:Almost perfect
2285:
2279:
2277:
2267:
2266:
2257:
2250:
2249:
2246:
2245:
2243:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2207:
2202:
2197:
2192:
2187:
2181:
2178:
2177:
2172:
2165:
2164:
2161:
2160:
2158:
2157:
2152:
2147:
2142:
2136:
2133:
2132:
2127:
2120:
2119:
2116:
2115:
2113:
2112:
2107:
2102:
2097:
2095:Stirling first
2092:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2027:
2022:
2017:
2011:
2008:
2007:
2004:
1997:
1996:
1993:
1992:
1989:
1988:
1985:
1984:
1982:
1981:
1976:
1971:
1965:
1963:
1956:
1950:
1949:
1946:
1945:
1943:
1942:
1936:
1934:
1928:
1927:
1925:
1924:
1919:
1914:
1909:
1904:
1899:
1893:
1891:
1885:
1884:
1882:
1881:
1876:
1871:
1866:
1861:
1855:
1853:
1844:
1838:
1837:
1834:
1833:
1831:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1790:
1785:
1779:
1777:
1771:
1770:
1768:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1721:
1719:
1710:
1700:
1699:
1694:
1687:
1686:
1683:
1682:
1680:
1679:
1674:
1669:
1664:
1659:
1654:
1648:
1645:
1644:
1641:
1634:
1633:
1630:
1629:
1627:
1626:
1621:
1616:
1611:
1606:
1600:
1597:
1596:
1593:
1586:
1585:
1582:
1581:
1579:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1537:
1534:
1533:
1527:
1520:
1519:
1516:
1515:
1513:
1512:
1507:
1502:
1497:
1492:
1486:
1483:
1482:
1479:
1472:
1471:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1428:
1425:
1424:
1417:
1410:
1409:
1406:
1405:
1403:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1362:
1357:
1352:
1347:
1342:
1337:
1331:
1328:
1327:
1321:
1314:
1313:
1307:
1305:
1304:
1297:
1290:
1282:
1273:
1272:
1269:
1268:
1266:
1265:
1260:
1255:
1250:
1244:
1242:
1235:
1228:
1227:
1224:
1223:
1221:
1220:
1215:
1210:
1204:
1202:
1195:
1189:
1188:
1185:
1184:
1182:
1181:
1176:
1170:
1168:
1162:
1161:
1159:
1158:
1153:
1148:
1143:
1138:
1132:
1130:
1124:
1123:
1121:
1120:
1115:
1110:
1105:
1100:
1094:
1092:
1083:
1077:
1076:
1073:
1072:
1070:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1032:Square numbers
1029:
1023:
1021:
1015:
1014:
1012:
1011:
1006:
1001:
996:
991:
986:
981:
976:
971:
965:
963:
954:
948:
947:
942:
940:
939:
932:
925:
917:
909:
908:
849:
821:
802:. As cited by
800:pp. 13–14
784:
773:
747:
728:
676:
669:
636:
635:
633:
630:
567:
564:
556:
555:
544:
541:
536:
532:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
475:
474:
463:
455:
451:
447:
444:
441:
438:
431:
427:
423:
420:
417:
414:
393:
392:
381:
380:
367:
362:
358:
355:
352:
349:
346:
341:
337:
333:
329:
325:
322:
319:
316:
313:
310:
288:
285:
269:René Just Haüy
264:
261:
257:René Just Haüy
227:
226:
224:
223:
220:
214:
212:
205:
204:
178:
174:
173:
160:
155:
151:
148:
145:
142:
139:
134:
130:
126:
122:
118:
115:
112:
109:
106:
103:
90:
86:
85:
75:
68:
67:
62:
54:
53:
50:
46:
45:
43:René Just Haüy
40:
36:
35:
32:
15:
13:
10:
9:
6:
4:
3:
2:
3234:
3223:
3220:
3219:
3217:
3202:
3198:
3194:
3193:
3190:
3180:
3177:
3176:
3173:
3168:
3163:
3159:
3149:
3146:
3144:
3141:
3140:
3137:
3132:
3127:
3123:
3113:
3110:
3108:
3105:
3104:
3101:
3096:
3091:
3087:
3077:
3074:
3072:
3069:
3068:
3065:
3061:
3055:
3051:
3041:
3038:
3036:
3033:
3031:
3028:
3027:
3024:
3020:
3015:
3011:
2997:
2994:
2993:
2991:
2987:
2981:
2978:
2976:
2973:
2971:
2970:Polydivisible
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2947:
2945:
2941:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2909:
2906:
2901:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2861:
2859:
2856:
2852:
2844:
2841:
2840:
2839:
2836:
2835:
2833:
2830:
2826:
2814:
2811:
2810:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2759:
2753:
2750:
2749:
2747:
2743:
2737:
2734:
2732:
2729:
2728:
2726:
2724:Digit product
2722:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2697:
2695:
2693:
2689:
2681:
2678:
2676:
2673:
2672:
2671:
2668:
2667:
2665:
2663:
2658:
2654:
2650:
2645:
2640:
2636:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2560:Erdős–Nicolas
2558:
2556:
2553:
2551:
2548:
2547:
2544:
2539:
2535:
2529:
2525:
2511:
2508:
2506:
2503:
2502:
2500:
2498:
2494:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2469:
2467:
2465:
2461:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2426:
2424:
2422:
2418:
2412:
2409:
2407:
2404:
2403:
2401:
2399:
2395:
2389:
2386:
2384:
2381:
2379:
2378:Superabundant
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2280:
2278:
2276:
2272:
2268:
2264:
2260:
2255:
2251:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2182:
2179:
2175:
2170:
2166:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2137:
2134:
2130:
2125:
2121:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2012:
2009:
2002:
1998:
1980:
1977:
1975:
1972:
1970:
1967:
1966:
1964:
1960:
1957:
1955:
1954:4-dimensional
1951:
1941:
1938:
1937:
1935:
1933:
1929:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1894:
1892:
1890:
1886:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1864:Centered cube
1862:
1860:
1857:
1856:
1854:
1852:
1848:
1845:
1843:
1842:3-dimensional
1839:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1780:
1778:
1776:
1772:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1722:
1720:
1718:
1714:
1711:
1709:
1708:2-dimensional
1705:
1701:
1697:
1692:
1688:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1652:Nonhypotenuse
1650:
1649:
1646:
1639:
1635:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1601:
1598:
1591:
1587:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1538:
1535:
1530:
1525:
1521:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1487:
1484:
1477:
1473:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1429:
1426:
1421:
1415:
1411:
1401:
1398:
1396:
1393:
1391:
1390:Perfect power
1388:
1386:
1383:
1381:
1380:Seventh power
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1332:
1329:
1324:
1319:
1315:
1311:
1303:
1298:
1296:
1291:
1289:
1284:
1283:
1280:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1245:
1243:
1239:
1236:
1234:
1229:
1219:
1216:
1214:
1211:
1209:
1206:
1205:
1203:
1199:
1196:
1194:
1193:4-dimensional
1190:
1180:
1177:
1175:
1172:
1171:
1169:
1167:
1163:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1133:
1131:
1129:
1125:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1095:
1093:
1091:
1087:
1084:
1082:
1081:3-dimensional
1078:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1024:
1022:
1020:
1016:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
966:
964:
962:
958:
955:
953:
952:2-dimensional
949:
945:
938:
933:
931:
926:
924:
919:
918:
915:
905:
901:
897:
893:
889:
885:
880:
875:
871:
867:
863:
856:
854:
850:
845:
844:
838:
834:
828:
826:
822:
816:
815:
810:
807:
801:
795:
788:
785:
781:
776:
774:9789400753808
770:
766:
762:
758:
751:
748:
741:
740:
732:
729:
717:
713:
709:
705:
701:
697:
690:
683:
681:
677:
672:
670:9789814355483
666:
662:
661:
656:
652:
646:
644:
642:
638:
631:
629:
627:
623:
619:
615:
610:
608:
604:
600:
595:
593:
589:
585:
581:
572:
565:
563:
561:
542:
539:
534:
530:
526:
523:
517:
514:
511:
505:
502:
496:
490:
483:
482:
481:
480:
461:
453:
445:
442:
439:
429:
421:
418:
415:
402:
401:
400:
398:
390:
389:
388:
386:
365:
360:
356:
353:
350:
347:
344:
339:
335:
331:
327:
320:
317:
314:
311:
298:
297:
296:
294:
286:
284:
282:
278:
274:
270:
262:
260:
258:
254:
250:
246:
242:
238:
234:
221:
219:
216:
215:
213:
210:
206:
202:
198:
194:
190:
186:
182:
179:
175:
158:
153:
149:
146:
143:
140:
137:
132:
128:
124:
120:
113:
110:
107:
104:
91:
87:
84:
79:
76:
73:
69:
66:
63:
55:
51:
47:
44:
41:
37:
30:
25:
19:
2934:Transposable
2798:Narcissistic
2705:Digital root
2625:Super-Poulet
2585:Jordan–Pólya
2534:prime factor
2439:Noncototient
2406:Almost prime
2388:Superperfect
2363:Refactorable
2358:Quasiperfect
2333:Hyperperfect
2174:Pseudoprimes
2145:Wall–Sun–Sun
2080:Ordered Bell
2050:Fuss–Catalan
1962:non-centered
1912:Dodecahedral
1889:non-centered
1868:
1775:non-centered
1677:Wolstenholme
1422:× 2 ± 1
1419:
1418:Of the form
1385:Eighth power
1365:Fourth power
1241:non-centered
1201:non-centered
1136:Cube numbers
1128:non-centered
1107:
1019:non-centered
1009:Star numbers
872:(9): 09026,
869:
865:
840:
812:
793:
787:
756:
750:
738:
731:
721:September 8,
719:, retrieved
699:
695:
659:
655:Deza, Michel
625:
621:
617:
616:of the form
611:
596:
577:
557:
476:
394:
384:
382:
292:
290:
273:mineralogist
266:
236:
232:
230:
18:
2955:Extravagant
2950:Equidigital
2905:permutation
2864:Palindromic
2838:Automorphic
2736:Sum-product
2715:Sum-product
2670:Persistence
2565:Erdős–Woods
2487:Untouchable
2368:Semiperfect
2318:Hemiperfect
1979:Tesseractic
1917:Icosahedral
1897:Tetrahedral
1828:Dodecagonal
1529:Recursively
1400:Prime power
1375:Sixth power
1370:Fifth power
1350:Power of 10
1308:Classes of
1233:dimensional
796:(in French)
651:Deza, Elena
580:metric ball
271:, a French
177:First terms
72:Subsequence
39:Named after
3167:Graphemics
3040:Pernicious
2894:Undulating
2869:Pandigital
2843:Trimorphic
2444:Nontotient
2293:Arithmetic
1907:Octahedral
1808:Heptagonal
1798:Pentagonal
1783:Triangular
1624:Sierpiński
1546:Jacobsthal
1345:Power of 3
1340:Power of 2
780:p. 10
632:References
249:octahedron
2924:Parasitic
2773:Factorion
2700:Digit sum
2692:Digit sum
2510:Fortunate
2497:Primorial
2411:Semiprime
2348:Practical
2313:Descartes
2308:Deficient
2298:Betrothed
2140:Wieferich
1969:Pentatope
1932:pyramidal
1823:Decagonal
1818:Nonagonal
1813:Octagonal
1803:Hexagonal
1662:Practical
1609:Congruent
1541:Fibonacci
1505:Loeschian
1166:pyramidal
904:119308823
879:1106.1078
814:MathWorld
515:−
443:−
3216:Category
2996:Friedman
2929:Primeval
2874:Repdigit
2831:-related
2778:Kaprekar
2752:Meertens
2675:Additive
2662:dynamics
2570:Friendly
2482:Sociable
2472:Amicable
2283:Abundant
2263:dynamics
2085:Schröder
2075:Narayana
2045:Eulerian
2035:Delannoy
2030:Dedekind
1851:centered
1717:centered
1604:Amenable
1561:Narayana
1551:Leonardo
1447:Mersenne
1395:Powerful
1335:Achilles
1090:centered
961:centered
657:(2012),
277:polycube
65:Infinity
61:of terms
3169:related
3133:related
3097:related
3095:Sorting
2980:Vampire
2965:Harshad
2907:related
2879:Repunit
2793:Lychrel
2768:Dudeney
2620:Størmer
2615:Sphenic
2600:Regular
2538:divisor
2477:Perfect
2373:Sublime
2343:Perfect
2070:Motzkin
2025:Catalan
1566:Padovan
1500:Leyland
1495:Idoneal
1490:Hilbert
1462:Woodall
1231:Higher
884:Bibcode
835:(ed.).
716:1971435
704:Bibcode
287:Formula
263:History
218:A001845
89:Formula
3035:Odious
2960:Frugal
2914:Cyclic
2903:Digit-
2610:Smooth
2595:Pronic
2555:Cyclic
2532:Other
2505:Euclid
2155:Wilson
2129:Primes
1788:Square
1657:Polite
1619:Riesel
1614:Knödel
1576:Perrin
1457:Thabit
1442:Fermat
1432:Cullen
1355:Square
1323:Powers
902:
771:
714:
667:
57:Total
3076:Prime
3071:Lucky
3060:sieve
2989:Other
2975:Smith
2855:Digit
2813:Happy
2788:Keith
2761:Other
2605:Rough
2575:Giuga
2040:Euler
1902:Cubic
1556:Lucas
1452:Proth
900:S2CID
874:arXiv
743:(PDF)
692:(PDF)
239:is a
211:index
203:, 377
3030:Evil
2710:Self
2660:and
2550:Blum
2261:and
2065:Lobb
2020:Cake
2015:Bell
1765:Star
1672:Ulam
1571:Pell
1360:Cube
870:2011
841:The
769:ISBN
723:2014
665:ISBN
395:The
209:OEIS
52:1801
3148:Ban
2536:or
2055:Lah
892:doi
761:doi
620:(3,
235:or
201:231
197:129
59:no.
3218::
898:,
890:,
882:,
868:,
864:,
852:^
839:.
824:^
811:.
767:,
712:MR
710:,
698:,
694:,
679:^
653:;
640:^
562:.
543:2.
283:.
259:.
231:A
199:,
195:,
193:63
191:,
189:25
187:,
183:,
74:of
1420:a
1301:e
1294:t
1287:v
936:e
929:t
922:v
894::
886::
876::
817:.
782:.
763::
726:.
706::
700:6
674:.
626:n
622:n
618:D
540:+
535:2
531:n
527:4
524:+
521:)
518:1
512:n
509:(
506:C
503:=
500:)
497:n
494:(
491:C
462:.
454:4
450:)
446:x
440:1
437:(
430:3
426:)
422:x
419:+
416:1
413:(
385:n
366:3
361:)
357:3
354:+
351:n
348:2
345:+
340:2
336:n
332:2
328:(
324:)
321:1
318:+
315:n
312:2
309:(
293:n
185:7
181:1
159:3
154:)
150:3
147:+
144:n
141:2
138:+
133:2
129:n
125:2
121:(
117:)
114:1
111:+
108:n
105:2
102:(
80:,
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