46:
1102:
423:
409:
416:
3598:
430:
1067:
1062:
1057:
1041:
1036:
1031:
1026:
1010:
1005:
1000:
995:
990:
974:
969:
964:
959:
943:
938:
933:
877:
872:
856:
851:
846:
830:
825:
779:
1093:
1088:
1072:
1046:
1015:
979:
948:
922:
917:
898:
893:
882:
861:
835:
814:
809:
795:
790:
784:
774:
768:
754:
1083:
1078:
1052:
1021:
985:
954:
928:
912:
907:
888:
867:
841:
820:
804:
763:
702:
686:
681:
676:
660:
655:
650:
645:
640:
624:
619:
614:
598:
566:
550:
545:
540:
524:
492:
713:
707:
697:
691:
671:
665:
635:
629:
609:
603:
593:
587:
577:
571:
561:
555:
535:
529:
519:
513:
503:
497:
487:
481:
471:
29:, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered
1546:
125:
is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression
1451:
1231:
1362:
177:
119:
1588:
1700:
367:
The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in
348:
335:
322:
302:
279:
263:
239:
223:
207:
191:
1693:
2500:
1686:
2495:
1669:
2510:
2490:
45:
3203:
2783:
2505:
3289:
2605:
1541:{\displaystyle {\frac {2\pi }{k{\sqrt {1-{\frac {8}{k}}}}}}\tan \left({\frac {\pi }{2}}{\sqrt {1-{\frac {8}{k}}}}\right)}
2955:
2274:
2067:
2990:
2960:
2635:
2625:
3131:
2545:
2279:
2259:
2821:
1164:
2985:
3080:
2703:
2460:
2269:
2251:
2145:
2135:
2125:
1611:
1101:
258:
218:
186:
2965:
3622:
3208:
2753:
2374:
2160:
2155:
2150:
2140:
1418:
1410:
317:
297:
274:
234:
2193:
1367:
which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.
1312:
2450:
3319:
3284:
3070:
2980:
2854:
2829:
2738:
2728:
2340:
2322:
2242:
129:
71:
1559:
3579:
2849:
2723:
2354:
2130:
1910:
1837:
1438:
1406:
202:
2834:
2688:
2615:
1770:
3543:
3183:
3476:
3370:
3334:
3075:
2798:
2778:
2595:
2264:
2052:
2024:
3198:
3062:
3057:
3025:
2788:
2763:
2758:
2733:
2663:
2659:
2590:
2480:
2312:
2108:
2077:
49:
3597:
3601:
3355:
3350:
3264:
3238:
3136:
3115:
2887:
2768:
2718:
2640:
2610:
2550:
2317:
2297:
2228:
1941:
744:
2485:
422:
408:
3495:
3440:
3294:
3269:
3243:
3020:
2698:
2693:
2620:
2600:
2585:
2307:
2289:
2208:
2198:
2183:
1961:
1946:
1665:
1642:
1422:
65:
1645:
3531:
3324:
2910:
2882:
2872:
2864:
2748:
2713:
2708:
2675:
2369:
2332:
2223:
2218:
2213:
2203:
2175:
2062:
2014:
2009:
1966:
1905:
1378:
368:
415:
3507:
3396:
3329:
3255:
3178:
3152:
2970:
2683:
2540:
2475:
2445:
2435:
2430:
2096:
2004:
1951:
1795:
1735:
333:
centered hendecagonal numbers 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... (
26:
346:
centered dodecagonal numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... (
3512:
3380:
3365:
3229:
3193:
3168:
3044:
3015:
3000:
2877:
2773:
2743:
2470:
2425:
2302:
1900:
1895:
1890:
1862:
1847:
1760:
1745:
1723:
1710:
1612:
centered polygonal numbers in OEIS wiki, content "Table of related formulae and values"
311:
3616:
3435:
3419:
3360:
3314:
3010:
2995:
2905:
2630:
2188:
2057:
2019:
1976:
1857:
1842:
1832:
1790:
1780:
1755:
1626:
1414:
291:
3471:
3460:
3375:
3213:
3188:
3105:
3005:
2975:
2950:
2934:
2839:
2806:
2555:
2529:
2440:
2379:
1956:
1852:
1785:
1765:
1740:
1371:
3430:
3305:
3110:
2574:
2465:
2420:
2415:
2165:
2072:
1971:
1800:
1775:
1750:
1622:
357:
3567:
3548:
2844:
2455:
1066:
1061:
1056:
1040:
1035:
1030:
1025:
1009:
1004:
999:
994:
989:
973:
968:
963:
958:
942:
937:
932:
876:
871:
855:
850:
845:
829:
824:
778:
288:
1678:
1092:
1087:
1071:
1045:
1014:
978:
947:
921:
916:
897:
892:
881:
860:
834:
813:
808:
794:
789:
783:
773:
767:
753:
3173:
3100:
3092:
2897:
2811:
1929:
1650:
1434:
1082:
1077:
1051:
1020:
984:
953:
927:
911:
906:
887:
866:
840:
819:
803:
762:
429:
3274:
1389:-gonal number), many centered polygonal numbers are primes. In fact, if
3279:
2938:
1282:
Just as is the case with regular polygonal numbers, the first centered
1150:−1)th triangular number around a central point; therefore, the
738:
732:
701:
685:
680:
675:
659:
654:
649:
644:
639:
623:
618:
613:
597:
565:
549:
544:
539:
523:
491:
1100:
712:
706:
696:
690:
670:
664:
634:
628:
608:
602:
592:
586:
576:
570:
560:
554:
534:
528:
518:
512:
502:
496:
486:
480:
470:
44:
1425:(and not equal to 3), thus both of them cannot be prime numbers.
3565:
3529:
3493:
3457:
3417:
3042:
2931:
2657:
2572:
2527:
2404:
2094:
2041:
1993:
1927:
1879:
1817:
1721:
1682:
247:), which are exactly the difference of consecutive cubes, i.e.
16:
Class of series of figurate numbers, each having a central dot
320:
1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... (
352:
339:
326:
306:
300:
1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... (
283:
267:
243:
227:
211:
195:
277:
1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... (
261:
1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... (
1664:(2nd ed.). Oxford University Press. pp. 88â89.
237:
1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (
221:
1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... (
205:
1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... (
189:
1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (
1562:
1454:
1315:
1167:
132:
74:
52:
that centered octagonal numbers are odd squares
3389:
3343:
3303:
3254:
3228:
3161:
3145:
3124:
3091:
3056:
2896:
2863:
2820:
2797:
2674:
2362:
2353:
2331:
2288:
2250:
2241:
2174:
2116:
2107:
68:, plus 1. This can be formalized by the expression
1582:
1540:
1356:
1225:
171:
113:
1226:{\displaystyle C_{k,n}={\frac {kn}{2}}(n-1)+1.}
1405:-gonal numbers which are primes (assuming the
1694:
1401:â 9, then there are infinitely many centered
8:
750:
467:
3562:
3526:
3490:
3454:
3414:
3088:
3053:
3039:
2928:
2671:
2654:
2569:
2524:
2401:
2359:
2247:
2113:
2104:
2091:
2038:
1995:Possessing a specific set of other numbers
1990:
1924:
1876:
1814:
1718:
1701:
1687:
1679:
1134:As can be seen in the above diagrams, the
1569:
1563:
1561:
1521:
1513:
1503:
1477:
1469:
1455:
1453:
1357:{\displaystyle {\frac {k^{2}}{2}}(k-1)+1}
1322:
1316:
1314:
1187:
1172:
1166:
1142:-gonal number can be obtained by placing
444:
375:
133:
131:
75:
73:
1662:The Oxford Mathematics Study Dictionary
1604:
1306:-gonal can be found using the formula:
172:{\displaystyle {\frac {4n(n+1)}{2}}+1}
114:{\displaystyle {\frac {kn(n+1)}{2}}+1}
1632:The Encyclopedia of Integer Sequences
1583:{\displaystyle {\frac {\pi ^{2}}{8}}}
7:
1381:(except the trivial case, i.e. each
1298:-gonal. The next number to be both
37:more dots than the previous layer.
1286:-gonal number is 1. Thus, for any
14:
3596:
3204:Perfect digit-to-digit invariant
1091:
1086:
1081:
1076:
1070:
1065:
1060:
1055:
1050:
1044:
1039:
1034:
1029:
1024:
1019:
1013:
1008:
1003:
998:
993:
988:
983:
977:
972:
967:
962:
957:
952:
946:
941:
936:
931:
926:
920:
915:
910:
905:
896:
891:
886:
880:
875:
870:
865:
859:
854:
849:
844:
839:
833:
828:
823:
818:
812:
807:
802:
793:
788:
782:
777:
772:
766:
761:
752:
711:
705:
700:
695:
689:
684:
679:
674:
669:
663:
658:
653:
648:
643:
638:
633:
627:
622:
617:
612:
607:
601:
596:
591:
585:
575:
569:
564:
559:
553:
548:
543:
538:
533:
527:
522:
517:
511:
501:
495:
490:
485:
479:
469:
428:
421:
414:
407:
60:-gonal number in the series is
1345:
1333:
1267:-gonal number is equal to the
1214:
1202:
154:
142:
96:
84:
1:
2043:Expressible via specific sums
1635:. San Diego: Academic Press.
1244:+1)-th consecutive centered
182:These series consist of the
3132:Multiplicative digital root
1646:"Centered polygonal number"
259:centered heptagonal numbers
219:centered pentagonal numbers
187:centered triangular numbers
3639:
1419:centered nonagonal numbers
1411:centered octagonal numbers
1158:-gonal number is equal to
722:Centered hexagonal numbers
318:centered decagonal numbers
310:), which include all even
298:centered nonagonal numbers
275:centered octagonal numbers
235:centered hexagonal numbers
23:centered polygonal numbers
3592:
3575:
3561:
3539:
3525:
3503:
3489:
3467:
3453:
3426:
3413:
3209:Perfect digital invariant
3052:
3038:
2946:
2927:
2784:Superior highly composite
2670:
2653:
2581:
2568:
2536:
2523:
2411:
2400:
2103:
2090:
2048:
2037:
2000:
1989:
1937:
1923:
1886:
1875:
1828:
1813:
1731:
1717:
287:), which are exactly the
25:are a class of series of
2822:Euler's totient function
2606:EulerâJacobi pseudoprime
1881:Other polynomial numbers
1105:As the sum of the first
2636:SomerâLucas pseudoprime
2626:LucasâCarmichael number
2461:Lazy caterer's sequence
439:Centered square numbers
203:centered square numbers
33:-gonal number contains
2511:WedderburnâEtherington
1911:Lucky numbers of Euler
1584:
1542:
1407:Bunyakovsky conjecture
1358:
1236:The difference of the
1227:
1126:
356:), which are also the
173:
115:
53:
2799:Prime omega functions
2616:Frobenius pseudoprime
2406:Combinatorial numbers
2275:Centered dodecahedral
2068:Primary pseudoperfect
1585:
1543:
1359:
1228:
1104:
174:
116:
48:
3258:-composition related
3058:Arithmetic functions
2660:Arithmetic functions
2596:Elliptic pseudoprime
2280:Centered icosahedral
2260:Centered tetrahedral
1560:
1452:
1313:
1302:-gonal and centered
1294:-gonal and centered
1275:-gonal number plus (
1165:
130:
72:
3184:Kaprekar's constant
2704:Colossally abundant
2591:Catalan pseudoprime
2491:SchröderâHipparchus
2270:Centered octahedral
2146:Centered heptagonal
2136:Centered pentagonal
2126:Centered triangular
1726:and related numbers
1122:− (n−1)
64:times the previous
3602:Mathematics portal
3544:Aronson's sequence
3290:SmarandacheâWellin
3047:-dependent numbers
2754:Primitive abundant
2641:Strong pseudoprime
2631:Perrin pseudoprime
2611:Fermat pseudoprime
2551:Wolstenholme prime
2375:Squared triangular
2161:Centered decagonal
2156:Centered nonagonal
2151:Centered octagonal
2141:Centered hexagonal
1660:F. Tapson (1999).
1643:Weisstein, Eric W.
1580:
1538:
1445:-gonal numbers is
1429:Sum of reciprocals
1423:triangular numbers
1354:
1248:-gonal numbers is
1223:
1127:
1117:-th hex number is
169:
111:
54:
3610:
3609:
3588:
3587:
3557:
3556:
3521:
3520:
3485:
3484:
3449:
3448:
3409:
3408:
3405:
3404:
3224:
3223:
3034:
3033:
2923:
2922:
2919:
2918:
2865:Aliquot sequences
2676:Divisor functions
2649:
2648:
2621:Lucas pseudoprime
2601:Euler pseudoprime
2586:Carmichael number
2564:
2563:
2519:
2518:
2396:
2395:
2392:
2391:
2388:
2387:
2349:
2348:
2237:
2236:
2194:Square triangular
2086:
2085:
2033:
2032:
1985:
1984:
1919:
1918:
1871:
1870:
1809:
1808:
1578:
1531:
1529:
1511:
1490:
1487:
1485:
1441:for the centered
1331:
1200:
1099:
1098:
719:
718:
436:
435:
161:
103:
66:triangular number
3630:
3623:Figurate numbers
3600:
3563:
3532:Natural language
3527:
3491:
3459:Generated via a
3455:
3415:
3320:Digit-reassembly
3285:Self-descriptive
3089:
3054:
3040:
2991:LucasâCarmichael
2981:Harmonic divisor
2929:
2855:Sparsely totient
2830:Highly cototient
2739:Multiply perfect
2729:Highly composite
2672:
2655:
2570:
2525:
2506:Telephone number
2402:
2360:
2341:Square pyramidal
2323:Stella octangula
2248:
2114:
2105:
2097:Figurate numbers
2092:
2039:
1991:
1925:
1877:
1815:
1719:
1703:
1696:
1689:
1680:
1675:
1656:
1655:
1636:
1614:
1609:
1589:
1587:
1586:
1581:
1579:
1574:
1573:
1564:
1547:
1545:
1544:
1539:
1537:
1533:
1532:
1530:
1522:
1514:
1512:
1504:
1491:
1489:
1488:
1486:
1478:
1470:
1464:
1456:
1379:polygonal number
1363:
1361:
1360:
1355:
1332:
1327:
1326:
1317:
1232:
1230:
1229:
1224:
1201:
1196:
1188:
1183:
1182:
1125:
1095:
1090:
1085:
1080:
1074:
1069:
1064:
1059:
1054:
1048:
1043:
1038:
1033:
1028:
1023:
1017:
1012:
1007:
1002:
997:
992:
987:
981:
976:
971:
966:
961:
956:
950:
945:
940:
935:
930:
924:
919:
914:
909:
900:
895:
890:
884:
879:
874:
869:
863:
858:
853:
848:
843:
837:
832:
827:
822:
816:
811:
806:
797:
792:
786:
781:
776:
770:
765:
756:
726:
725:
715:
709:
704:
699:
693:
688:
683:
678:
673:
667:
662:
657:
652:
647:
642:
637:
631:
626:
621:
616:
611:
605:
600:
595:
589:
579:
573:
568:
563:
557:
552:
547:
542:
537:
531:
526:
521:
515:
505:
499:
494:
489:
483:
473:
443:
442:
432:
425:
418:
411:
374:
373:
369:Polygonal number
355:
342:
329:
309:
286:
270:
246:
230:
214:
198:
178:
176:
175:
170:
162:
157:
134:
120:
118:
117:
112:
104:
99:
76:
27:figurate numbers
3638:
3637:
3633:
3632:
3631:
3629:
3628:
3627:
3613:
3612:
3611:
3606:
3584:
3580:Strobogrammatic
3571:
3553:
3535:
3517:
3499:
3481:
3463:
3445:
3422:
3401:
3385:
3344:Divisor-related
3339:
3299:
3250:
3220:
3157:
3141:
3120:
3087:
3060:
3048:
3030:
2942:
2941:related numbers
2915:
2892:
2859:
2850:Perfect totient
2816:
2793:
2724:Highly abundant
2666:
2645:
2577:
2560:
2532:
2515:
2501:Stirling second
2407:
2384:
2345:
2327:
2284:
2233:
2170:
2131:Centered square
2099:
2082:
2044:
2029:
1996:
1981:
1933:
1932:defined numbers
1915:
1882:
1867:
1838:Double Mersenne
1824:
1805:
1727:
1713:
1711:natural numbers
1707:
1672:
1659:
1641:
1640:
1621:
1618:
1617:
1610:
1606:
1601:
1565:
1558:
1557:
1502:
1498:
1465:
1457:
1450:
1449:
1431:
1318:
1311:
1310:
1189:
1168:
1163:
1162:
1146:copies of the (
1132:
1118:
1109:hex numbers is
1075:
1049:
1018:
982:
951:
925:
885:
864:
838:
817:
787:
771:
724:
710:
694:
668:
632:
606:
590:
574:
558:
532:
516:
500:
484:
441:
401:
399:
394:
392:
387:
385:
380:
378:
347:
334:
321:
312:perfect numbers
301:
278:
262:
238:
222:
206:
190:
135:
128:
127:
77:
70:
69:
43:
17:
12:
11:
5:
3636:
3634:
3626:
3625:
3615:
3614:
3608:
3607:
3605:
3604:
3593:
3590:
3589:
3586:
3585:
3583:
3582:
3576:
3573:
3572:
3566:
3559:
3558:
3555:
3554:
3552:
3551:
3546:
3540:
3537:
3536:
3530:
3523:
3522:
3519:
3518:
3516:
3515:
3513:Sorting number
3510:
3508:Pancake number
3504:
3501:
3500:
3494:
3487:
3486:
3483:
3482:
3480:
3479:
3474:
3468:
3465:
3464:
3458:
3451:
3450:
3447:
3446:
3444:
3443:
3438:
3433:
3427:
3424:
3423:
3420:Binary numbers
3418:
3411:
3410:
3407:
3406:
3403:
3402:
3400:
3399:
3393:
3391:
3387:
3386:
3384:
3383:
3378:
3373:
3368:
3363:
3358:
3353:
3347:
3345:
3341:
3340:
3338:
3337:
3332:
3327:
3322:
3317:
3311:
3309:
3301:
3300:
3298:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3261:
3259:
3252:
3251:
3249:
3248:
3247:
3246:
3235:
3233:
3230:P-adic numbers
3226:
3225:
3222:
3221:
3219:
3218:
3217:
3216:
3206:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3165:
3163:
3159:
3158:
3156:
3155:
3149:
3147:
3146:Coding-related
3143:
3142:
3140:
3139:
3134:
3128:
3126:
3122:
3121:
3119:
3118:
3113:
3108:
3103:
3097:
3095:
3086:
3085:
3084:
3083:
3081:Multiplicative
3078:
3067:
3065:
3050:
3049:
3045:Numeral system
3043:
3036:
3035:
3032:
3031:
3029:
3028:
3023:
3018:
3013:
3008:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2947:
2944:
2943:
2932:
2925:
2924:
2921:
2920:
2917:
2916:
2914:
2913:
2908:
2902:
2900:
2894:
2893:
2891:
2890:
2885:
2880:
2875:
2869:
2867:
2861:
2860:
2858:
2857:
2852:
2847:
2842:
2837:
2835:Highly totient
2832:
2826:
2824:
2818:
2817:
2815:
2814:
2809:
2803:
2801:
2795:
2794:
2792:
2791:
2786:
2781:
2776:
2771:
2766:
2761:
2756:
2751:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2691:
2689:Almost perfect
2686:
2680:
2678:
2668:
2667:
2658:
2651:
2650:
2647:
2646:
2644:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2582:
2579:
2578:
2573:
2566:
2565:
2562:
2561:
2559:
2558:
2553:
2548:
2543:
2537:
2534:
2533:
2528:
2521:
2520:
2517:
2516:
2514:
2513:
2508:
2503:
2498:
2496:Stirling first
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2412:
2409:
2408:
2405:
2398:
2397:
2394:
2393:
2390:
2389:
2386:
2385:
2383:
2382:
2377:
2372:
2366:
2364:
2357:
2351:
2350:
2347:
2346:
2344:
2343:
2337:
2335:
2329:
2328:
2326:
2325:
2320:
2315:
2310:
2305:
2300:
2294:
2292:
2286:
2285:
2283:
2282:
2277:
2272:
2267:
2262:
2256:
2254:
2245:
2239:
2238:
2235:
2234:
2232:
2231:
2226:
2221:
2216:
2211:
2206:
2201:
2196:
2191:
2186:
2180:
2178:
2172:
2171:
2169:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2122:
2120:
2111:
2101:
2100:
2095:
2088:
2087:
2084:
2083:
2081:
2080:
2075:
2070:
2065:
2060:
2055:
2049:
2046:
2045:
2042:
2035:
2034:
2031:
2030:
2028:
2027:
2022:
2017:
2012:
2007:
2001:
1998:
1997:
1994:
1987:
1986:
1983:
1982:
1980:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1938:
1935:
1934:
1928:
1921:
1920:
1917:
1916:
1914:
1913:
1908:
1903:
1898:
1893:
1887:
1884:
1883:
1880:
1873:
1872:
1869:
1868:
1866:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1829:
1826:
1825:
1818:
1811:
1810:
1807:
1806:
1804:
1803:
1798:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1738:
1732:
1729:
1728:
1722:
1715:
1714:
1708:
1706:
1705:
1698:
1691:
1683:
1677:
1676:
1671:0-19-914-567-9
1670:
1657:
1638:
1616:
1615:
1603:
1602:
1600:
1597:
1596:
1595:
1577:
1572:
1568:
1554:
1553:
1536:
1528:
1525:
1520:
1517:
1510:
1507:
1501:
1497:
1494:
1484:
1481:
1476:
1473:
1468:
1463:
1460:
1430:
1427:
1415:square numbers
1385:is the second
1365:
1364:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1330:
1325:
1321:
1234:
1233:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1199:
1195:
1192:
1186:
1181:
1178:
1175:
1171:
1131:
1128:
1097:
1096:
903:
901:
800:
798:
759:
757:
749:
748:
745:
742:
739:
736:
733:
730:
723:
720:
717:
716:
583:
582:
580:
509:
508:
506:
477:
476:
474:
466:
465:
462:
461:
459:
456:
455:
453:
450:
449:
447:
440:
437:
434:
433:
426:
419:
412:
404:
403:
396:
389:
382:
362:
361:
344:
331:
315:
295:
272:
256:
232:
216:
200:
168:
165:
160:
156:
153:
150:
147:
144:
141:
138:
110:
107:
102:
98:
95:
92:
89:
86:
83:
80:
56:Each centered
42:
39:
15:
13:
10:
9:
6:
4:
3:
2:
3635:
3624:
3621:
3620:
3618:
3603:
3599:
3595:
3594:
3591:
3581:
3578:
3577:
3574:
3569:
3564:
3560:
3550:
3547:
3545:
3542:
3541:
3538:
3533:
3528:
3524:
3514:
3511:
3509:
3506:
3505:
3502:
3497:
3492:
3488:
3478:
3475:
3473:
3470:
3469:
3466:
3462:
3456:
3452:
3442:
3439:
3437:
3434:
3432:
3429:
3428:
3425:
3421:
3416:
3412:
3398:
3395:
3394:
3392:
3388:
3382:
3379:
3377:
3374:
3372:
3371:Polydivisible
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3348:
3346:
3342:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3312:
3310:
3307:
3302:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3262:
3260:
3257:
3253:
3245:
3242:
3241:
3240:
3237:
3236:
3234:
3231:
3227:
3215:
3212:
3211:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3166:
3164:
3160:
3154:
3151:
3150:
3148:
3144:
3138:
3135:
3133:
3130:
3129:
3127:
3125:Digit product
3123:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3098:
3096:
3094:
3090:
3082:
3079:
3077:
3074:
3073:
3072:
3069:
3068:
3066:
3064:
3059:
3055:
3051:
3046:
3041:
3037:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2961:ErdĆsâNicolas
2959:
2957:
2954:
2952:
2949:
2948:
2945:
2940:
2936:
2930:
2926:
2912:
2909:
2907:
2904:
2903:
2901:
2899:
2895:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2868:
2866:
2862:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2828:
2827:
2825:
2823:
2819:
2813:
2810:
2808:
2805:
2804:
2802:
2800:
2796:
2790:
2787:
2785:
2782:
2780:
2779:Superabundant
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2681:
2679:
2677:
2673:
2669:
2665:
2661:
2656:
2652:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2583:
2580:
2576:
2571:
2567:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2538:
2535:
2531:
2526:
2522:
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:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2413:
2410:
2403:
2399:
2381:
2378:
2376:
2373:
2371:
2368:
2367:
2365:
2361:
2358:
2356:
2355:4-dimensional
2352:
2342:
2339:
2338:
2336:
2334:
2330:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2293:
2291:
2287:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2265:Centered cube
2263:
2261:
2258:
2257:
2255:
2253:
2249:
2246:
2244:
2243:3-dimensional
2240:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2181:
2179:
2177:
2173:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2121:
2119:
2115:
2112:
2110:
2109:2-dimensional
2106:
2102:
2098:
2093:
2089:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2053:Nonhypotenuse
2051:
2050:
2047:
2040:
2036:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2002:
1999:
1992:
1988:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1939:
1936:
1931:
1926:
1922:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1888:
1885:
1878:
1874:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1830:
1827:
1822:
1816:
1812:
1802:
1799:
1797:
1794:
1792:
1791:Perfect power
1789:
1787:
1784:
1782:
1781:Seventh power
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1733:
1730:
1725:
1720:
1716:
1712:
1704:
1699:
1697:
1692:
1690:
1685:
1684:
1681:
1673:
1667:
1663:
1658:
1653:
1652:
1647:
1644:
1639:
1634:
1631:
1628:
1627:Simon Plouffe
1624:
1620:
1619:
1613:
1608:
1605:
1598:
1593:
1575:
1570:
1566:
1556:
1555:
1551:
1534:
1526:
1523:
1518:
1515:
1508:
1505:
1499:
1495:
1492:
1482:
1479:
1474:
1471:
1466:
1461:
1458:
1448:
1447:
1446:
1444:
1440:
1436:
1428:
1426:
1424:
1420:
1416:
1412:
1409:). Since all
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1373:
1368:
1351:
1348:
1342:
1339:
1336:
1328:
1323:
1319:
1309:
1308:
1307:
1305:
1301:
1297:
1293:
1289:
1285:
1280:
1278:
1274:
1270:
1266:
1263:-th centered
1262:
1257:
1255:
1251:
1247:
1243:
1240:-th and the (
1239:
1220:
1217:
1211:
1208:
1205:
1197:
1193:
1190:
1184:
1179:
1176:
1173:
1169:
1161:
1160:
1159:
1157:
1153:
1149:
1145:
1141:
1137:
1129:
1124:
1121:
1116:
1112:
1108:
1103:
1094:
1089:
1084:
1079:
1073:
1068:
1063:
1058:
1053:
1047:
1042:
1037:
1032:
1027:
1022:
1016:
1011:
1006:
1001:
996:
991:
986:
980:
975:
970:
965:
960:
955:
949:
944:
939:
934:
929:
923:
918:
913:
908:
904:
902:
899:
894:
889:
883:
878:
873:
868:
862:
857:
852:
847:
842:
836:
831:
826:
821:
815:
810:
805:
801:
799:
796:
791:
785:
780:
775:
769:
764:
760:
758:
755:
751:
746:
743:
740:
737:
734:
731:
728:
727:
721:
714:
708:
703:
698:
692:
687:
682:
677:
672:
666:
661:
656:
651:
646:
641:
636:
630:
625:
620:
615:
610:
604:
599:
594:
588:
584:
581:
578:
572:
567:
562:
556:
551:
546:
541:
536:
530:
525:
520:
514:
510:
507:
504:
498:
493:
488:
482:
478:
475:
472:
468:
463:
460:
457:
454:
451:
448:
445:
438:
431:
427:
424:
420:
417:
413:
410:
406:
405:
397:
390:
383:
376:
372:
370:
365:
359:
354:
350:
345:
341:
337:
332:
328:
324:
319:
316:
313:
308:
304:
299:
296:
293:
290:
285:
281:
276:
273:
269:
265:
260:
257:
254:
250:
245:
241:
236:
233:
229:
225:
220:
217:
213:
209:
204:
201:
197:
193:
188:
185:
184:
183:
180:
166:
163:
158:
151:
148:
145:
139:
136:
124:
108:
105:
100:
93:
90:
87:
81:
78:
67:
63:
59:
51:
47:
40:
38:
36:
32:
28:
24:
19:
3335:Transposable
3199:Narcissistic
3106:Digital root
3026:Super-Poulet
2986:JordanâPĂłlya
2935:prime factor
2840:Noncototient
2807:Almost prime
2789:Superperfect
2764:Refactorable
2759:Quasiperfect
2734:Hyperperfect
2575:Pseudoprimes
2546:WallâSunâSun
2481:Ordered Bell
2451:FussâCatalan
2363:non-centered
2313:Dodecahedral
2290:non-centered
2176:non-centered
2117:
2078:Wolstenholme
1823:× 2 ± 1
1820:
1819:Of the form
1786:Eighth power
1766:Fourth power
1661:
1649:
1637:: Fig. M3826
1633:
1630:
1607:
1591:
1549:
1442:
1432:
1402:
1398:
1394:
1390:
1386:
1382:
1377:cannot be a
1374:
1372:prime number
1369:
1366:
1303:
1299:
1295:
1291:
1290:, 1 is both
1287:
1283:
1281:
1276:
1272:
1271:-th regular
1268:
1264:
1260:
1258:
1253:
1249:
1245:
1241:
1237:
1235:
1155:
1154:th centered
1151:
1147:
1143:
1139:
1138:th centered
1135:
1133:
1123:
1119:
1114:
1110:
1106:
366:
363:
358:star numbers
252:
248:
181:
122:
61:
57:
55:
34:
30:
22:
20:
18:
3356:Extravagant
3351:Equidigital
3306:permutation
3265:Palindromic
3239:Automorphic
3137:Sum-product
3116:Sum-product
3071:Persistence
2966:ErdĆsâWoods
2888:Untouchable
2769:Semiperfect
2719:Hemiperfect
2380:Tesseractic
2318:Icosahedral
2298:Tetrahedral
2229:Dodecagonal
1930:Recursively
1801:Prime power
1776:Sixth power
1771:Fifth power
1751:Power of 10
1709:Classes of
1623:Neil Sloane
1439:reciprocals
364:and so on.
3568:Graphemics
3441:Pernicious
3295:Undulating
3270:Pandigital
3244:Trimorphic
2845:Nontotient
2694:Arithmetic
2308:Octahedral
2209:Heptagonal
2199:Pentagonal
2184:Triangular
2025:SierpiĆski
1947:Jacobsthal
1746:Power of 3
1741:Power of 2
1599:References
1417:, and all
1370:Whereas a
393:pentagonal
379:triangular
3325:Parasitic
3174:Factorion
3101:Digit sum
3093:Digit sum
2911:Fortunate
2898:Primorial
2812:Semiprime
2749:Practical
2714:Descartes
2709:Deficient
2699:Betrothed
2541:Wieferich
2370:Pentatope
2333:pyramidal
2224:Decagonal
2219:Nonagonal
2214:Octagonal
2204:Hexagonal
2063:Practical
2010:Congruent
1942:Fibonacci
1906:Loeschian
1651:MathWorld
1567:π
1519:−
1506:π
1496:
1475:−
1462:π
1421:are also
1413:are also
1340:−
1209:−
400:hexagonal
314:except 6,
3617:Category
3397:Friedman
3330:Primeval
3275:Repdigit
3232:-related
3179:Kaprekar
3153:Meertens
3076:Additive
3063:dynamics
2971:Friendly
2883:Sociable
2873:Amicable
2684:Abundant
2664:dynamics
2486:Schröder
2476:Narayana
2446:Eulerian
2436:Delannoy
2431:Dedekind
2252:centered
2118:centered
2005:Amenable
1962:Narayana
1952:Leonardo
1848:Mersenne
1796:Powerful
1736:Achilles
1629:(1995).
1130:Formulas
398:centered
391:centered
384:centered
377:centered
121:, where
41:Examples
3570:related
3534:related
3498:related
3496:Sorting
3381:Vampire
3366:Harshad
3308:related
3280:Repunit
3194:Lychrel
3169:Dudeney
3021:StĂžrmer
3016:Sphenic
3001:Regular
2939:divisor
2878:Perfect
2774:Sublime
2744:Perfect
2471:Motzkin
2426:Catalan
1967:Padovan
1901:Leyland
1896:Idoneal
1891:Hilbert
1863:Woodall
402:number
395:number
388:number
381:number
353:A003154
351::
340:A069125
338::
327:A062786
325::
307:A060544
305::
292:squares
284:A016754
282::
268:A069099
266::
244:A003215
242::
228:A005891
226::
212:A001844
210::
196:A005448
194::
3436:Odious
3361:Frugal
3315:Cyclic
3304:Digit-
3011:Smooth
2996:Pronic
2956:Cyclic
2933:Other
2906:Euclid
2556:Wilson
2530:Primes
2189:Square
2058:Polite
2020:Riesel
2015:Knödel
1977:Perrin
1858:Thabit
1843:Fermat
1833:Cullen
1756:Square
1724:Powers
1668:
1625:&
1113:, the
386:square
3477:Prime
3472:Lucky
3461:sieve
3390:Other
3376:Smith
3256:Digit
3214:Happy
3189:Keith
3162:Other
3006:Rough
2976:Giuga
2441:Euler
2303:Cubic
1957:Lucas
1853:Proth
1590:, if
1548:, if
1397:â 8,
1393:â„ 3,
1279:-1).
1256:+1).
255:â 1),
50:Proof
3431:Evil
3111:Self
3061:and
2951:Blum
2662:and
2466:Lobb
2421:Cake
2416:Bell
2166:Star
2073:Ulam
1972:Pell
1761:Cube
1666:ISBN
1433:The
1259:The
349:OEIS
336:OEIS
323:OEIS
303:OEIS
280:OEIS
264:OEIS
240:OEIS
224:OEIS
208:OEIS
192:OEIS
21:The
3549:Ban
2937:or
2456:Lah
1594:= 8
1552:â 8
1493:tan
1437:of
1435:sum
747:37
464:25
458:13
289:odd
251:â (
3619::
1648:.
1252:(2
1221:1.
741:19
452:5
446:1
371:.
343:),
330:),
271:),
231:),
215:),
199:),
179:.
1821:a
1702:e
1695:t
1688:v
1674:.
1654:.
1592:k
1576:8
1571:2
1550:k
1535:)
1527:k
1524:8
1516:1
1509:2
1500:(
1483:k
1480:8
1472:1
1467:k
1459:2
1443:k
1403:k
1399:k
1395:k
1391:k
1387:p
1383:p
1375:p
1352:1
1349:+
1346:)
1343:1
1337:k
1334:(
1329:2
1324:2
1320:k
1304:k
1300:k
1296:k
1292:k
1288:k
1284:k
1277:n
1273:k
1269:n
1265:k
1261:n
1254:n
1250:k
1246:k
1242:n
1238:n
1218:+
1215:)
1212:1
1206:n
1203:(
1198:2
1194:n
1191:k
1185:=
1180:n
1177:,
1174:k
1170:C
1156:k
1152:n
1148:n
1144:k
1140:k
1136:n
1120:n
1115:n
1111:n
1107:n
735:7
729:1
360:,
294:,
253:n
249:n
167:1
164:+
159:2
155:)
152:1
149:+
146:n
143:(
140:n
137:4
123:n
109:1
106:+
101:2
97:)
94:1
91:+
88:n
85:(
82:n
79:k
62:k
58:k
35:k
31:k
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