3317:
1011:
1024:, which is the result of repeatedly applying the digit sum operation until the remaining value is only a single digit. The decimal digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots can be used for quick
796:
717:
445:
204:
1006:{\displaystyle f(n,q)={\begin{cases}1&{\text{if }}n=1\\f(n,9n-q+1)&{\text{if }}q>\lceil {\frac {9n}{2}}\rceil \\\sum _{i=\max(q-9,1)}^{q}f(n-1,i)&{\text{otherwise}}\end{cases}}}
327:
277:
527:
575:
1082:
or population count; algorithms for performing this operation have been studied, and it has been included as a built-in operation in some computer architectures and some
104:
161:
60:
598:
347:
131:
1419:
1121:
605:
355:
1337:
3351:
1231:
737:
728:
1412:
166:
2219:
1405:
2214:
2229:
1036:
its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the
2209:
285:
2922:
2502:
212:
460:
2224:
3008:
1239:
1188:
3346:
2674:
2324:
1993:
1786:
1051:
algorithms to check the arithmetic operations of early computers. Earlier, in an era of hand calculation,
534:
2850:
2709:
2540:
2354:
2344:
1998:
1978:
780:
2679:
2799:
2422:
2264:
2179:
1988:
1970:
1864:
1854:
1844:
1680:
2704:
2927:
2472:
2093:
1879:
1874:
1869:
1859:
1836:
1151:
2684:
826:
2349:
2259:
1912:
1333:
1298:
Bloch, R. M.; Campbell, R. V. D.; Ellis, M. (1948), "The
Logical Design of the Raytheon Computer",
1060:
3038:
3003:
2789:
2699:
2573:
2548:
2457:
2447:
2169:
2059:
2041:
1961:
1352:
65:
3356:
3298:
2568:
2442:
2073:
1849:
1629:
1556:
3262:
2902:
2553:
2407:
2334:
1489:
1109:
1075:
1068:
1064:
768:
140:
3195:
3089:
3053:
2794:
2517:
2497:
2314:
1983:
1771:
1186:
Bush, L. E. (1940), "An asymptotic formula for the average sum of the digits of integers",
1083:
760:
2274:
1743:
1274:
45:
8:
2917:
2781:
2776:
2744:
2507:
2482:
2477:
2452:
2382:
2378:
2309:
2199:
2031:
1827:
1796:
1056:
745:
580:
3320:
3074:
3069:
2983:
2957:
2855:
2834:
2606:
2487:
2437:
2359:
2329:
2269:
2036:
2016:
1947:
1660:
1315:
1266:
1205:
1166:
332:
116:
2204:
3341:
3316:
3214:
3159:
3013:
2988:
2962:
2417:
2412:
2339:
2319:
2304:
2026:
2008:
1927:
1917:
1902:
1665:
1375:
1126:
1041:
1025:
757:
2739:
3250:
3043:
2629:
2601:
2591:
2583:
2467:
2432:
2427:
2394:
2088:
2051:
1942:
1937:
1932:
1922:
1894:
1781:
1728:
1685:
1624:
1307:
1256:
1248:
1223:
1197:
1020:
The concept of a decimal digit sum is closely related to, but not the same as, the
749:
1733:
3226:
3115:
3048:
2974:
2897:
2871:
2689:
2402:
2194:
2164:
2154:
2149:
1815:
1723:
1670:
1514:
1454:
764:
36:
3231:
3099:
3084:
2948:
2912:
2887:
2763:
2734:
2719:
2596:
2492:
2462:
2189:
2144:
2021:
1619:
1614:
1609:
1581:
1566:
1479:
1464:
1442:
1429:
1146:
1141:
1101:
1095:
1079:
1033:
40:
28:
1378:
3335:
3154:
3138:
3079:
3033:
2729:
2714:
2624:
1907:
1776:
1738:
1695:
1576:
1561:
1551:
1509:
1499:
1474:
1397:
1227:
1108:
are defined by the equality of their digit sums with the digit sums of their
1091:
753:
3190:
3179:
3094:
2932:
2907:
2824:
2724:
2694:
2669:
2653:
2558:
2525:
2248:
2159:
2098:
1675:
1571:
1504:
1484:
1459:
1392:
1261:
1161:
1156:
1136:
1105:
1087:
1067:, these digit sums will have a random distribution closely approximating a
1029:
1021:
712:{\displaystyle \sum _{k=0}^{n}F_{b_{1}}(k)<\sum _{k=0}^{n}F_{b_{2}}(k).}
440:{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}
3149:
3024:
2829:
2293:
2184:
2139:
2134:
1884:
1791:
1690:
1519:
1494:
1469:
20:
791:
The amount of n-digit numbers with digit sum q can be calculated using:
3286:
3267:
2563:
2174:
1319:
1270:
1209:
2892:
2616:
2530:
1648:
1383:
1311:
1252:
1201:
2993:
1131:
1048:
779:
The digit sum can be extended to the negative integers by use of a
2998:
2657:
2651:
723:
454:
1086:. These operations are used in computing applications including
1713:
1104:
are defined in terms of divisibility by their digit sums, and
409:
379:
32:
329:
is one less than the number of digits in the number in base
999:
732:
1055:
suggested using sums of 50 digits taken from mathematical
199:{\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }
1063:; if one assumes that each digit is random, then by the
2377:
799:
608:
583:
537:
463:
358:
335:
288:
215:
169:
143:
119:
68:
48:
2762:
1196:(3), Mathematical Association of America: 154–156,
1373:
1297:
1005:
711:
592:
569:
521:
439:
341:
321:
271:
198:
155:
125:
98:
54:
1761:
1300:Mathematical Tables and Other Aids to Computation
774:
3333:
934:
1647:
1441:
1427:
1306:(24), American Mathematical Society: 286–295,
1222:
741:
322:{\displaystyle k=\lfloor \log _{b}{n}\rfloor }
1413:
272:{\displaystyle F_{b}(n)=\sum _{i=0}^{k}d_{i}}
3249:
1599:
916:
898:
316:
295:
1047:Digit sums are also a common ingredient in
577:and for sufficiently large natural numbers
522:{\displaystyle F_{10}(84001)=8+4+0+0+1=13.}
450:is the value of each digit of the number.
1714:Possessing a specific set of other numbers
1537:
1420:
1406:
3177:
2124:
1332:
1260:
1232:"Strange series and high precision fraud"
1052:
738:On-Line Encyclopedia of Integer Sequences
192:
184:
1345:Journal of the Royal Statistical Society
3334:
3285:
756:digit sums) to derive several rapidly
3284:
3248:
3212:
3176:
3136:
2761:
2650:
2376:
2291:
2246:
2123:
1813:
1760:
1712:
1646:
1598:
1536:
1440:
1401:
1374:
1179:
1044:technique for checking calculations.
722:The sum of the base 10 digits of the
1814:
1338:"The Mathematical Theory of Banking"
1185:
570:{\displaystyle 2\leq b_{1}<b_{2}}
39:. For example, the digit sum of the
3213:
752:(and of the analogous sequence for
133:be a natural number. We define the
13:
3137:
14:
3368:
1367:
3352:Base-dependent integer sequences
3315:
2923:Perfect digit-to-digit invariant
2292:
1394:Simple applications of digit sum
1015:
1326:
1291:
1216:
986:
968:
955:
937:
882:
855:
815:
803:
775:Extension to negative integers
703:
697:
653:
647:
480:
474:
232:
226:
188:
1:
1762:Expressible via specific sums
1240:American Mathematical Monthly
1189:American Mathematical Monthly
1172:
786:
108:
1351:(1): 113–127, archived from
1078:of a number is known as its
742:Borwein & Borwein (1992)
457:, the digit sum of 84001 is
7:
2851:Multiplicative digital root
1115:
783:to represent each integer.
781:signed-digit representation
99:{\displaystyle 9+0+4+5=18.}
10:
3373:
2247:
3311:
3294:
3280:
3258:
3244:
3222:
3208:
3186:
3172:
3145:
3132:
3108:
3062:
3022:
2973:
2947:
2928:Perfect digital invariant
2880:
2864:
2843:
2810:
2775:
2771:
2757:
2665:
2646:
2615:
2582:
2539:
2516:
2503:Superior highly composite
2393:
2389:
2372:
2300:
2287:
2255:
2242:
2130:
2119:
2081:
2072:
2050:
2007:
1969:
1960:
1893:
1835:
1826:
1822:
1809:
1767:
1756:
1719:
1708:
1656:
1642:
1605:
1594:
1547:
1532:
1450:
1436:
1152:Perfect digital invariant
726:0, 1, 2, ... is given by
2541:Euler's totient function
2325:Euler–Jacobi pseudoprime
1600:Other polynomial numbers
1061:random number generation
1040:and is the basis of the
16:Sum of a number's digits
2355:Somer–Lucas pseudoprime
2345:Lucas–Carmichael number
2180:Lazy caterer's sequence
2230:Wedderburn–Etherington
1630:Lucky numbers of Euler
1028:: a natural number is
1007:
964:
713:
679:
629:
594:
571:
523:
441:
343:
323:
273:
258:
200:
157:
156:{\displaystyle b>1}
127:
100:
56:
35:is the sum of all its
2518:Prime omega functions
2335:Frobenius pseudoprime
2125:Combinatorial numbers
1994:Centered dodecahedral
1787:Primary pseudoperfect
1084:programming languages
1076:binary representation
1074:The digit sum of the
1069:Gaussian distribution
1065:central limit theorem
1008:
923:
714:
659:
609:
595:
572:
524:
442:
344:
324:
274:
238:
206:to be the following:
201:
158:
128:
101:
57:
2977:-composition related
2777:Arithmetic functions
2379:Arithmetic functions
2315:Elliptic pseudoprime
1999:Centered icosahedral
1979:Centered tetrahedral
1110:prime factorizations
1057:tables of logarithms
797:
606:
581:
535:
461:
356:
333:
286:
213:
167:
141:
117:
66:
55:{\displaystyle 9045}
46:
3347:Arithmetic dynamics
2903:Kaprekar's constant
2423:Colossally abundant
2310:Catalan pseudoprime
2210:Schröder–Hipparchus
1989:Centered octahedral
1865:Centered heptagonal
1855:Centered pentagonal
1845:Centered triangular
1445:and related numbers
1122:Arithmetic dynamics
746:generating function
3321:Mathematics portal
3263:Aronson's sequence
3009:Smarandache–Wellin
2766:-dependent numbers
2473:Primitive abundant
2360:Strong pseudoprime
2350:Perrin pseudoprime
2330:Fermat pseudoprime
2270:Wolstenholme prime
2094:Squared triangular
1880:Centered decagonal
1875:Centered nonagonal
1870:Centered octagonal
1860:Centered hexagonal
1376:Weisstein, Eric W.
1167:Sum-product number
1026:divisibility tests
1003:
998:
709:
593:{\displaystyle n,}
590:
567:
531:For any two bases
519:
437:
339:
319:
269:
196:
153:
123:
96:
52:
3329:
3328:
3307:
3306:
3276:
3275:
3240:
3239:
3204:
3203:
3168:
3167:
3128:
3127:
3124:
3123:
2943:
2942:
2753:
2752:
2642:
2641:
2638:
2637:
2584:Aliquot sequences
2395:Divisor functions
2368:
2367:
2340:Lucas pseudoprime
2320:Euler pseudoprime
2305:Carmichael number
2283:
2282:
2238:
2237:
2115:
2114:
2111:
2110:
2107:
2106:
2068:
2067:
1956:
1955:
1913:Square triangular
1805:
1804:
1752:
1751:
1704:
1703:
1638:
1637:
1590:
1589:
1528:
1527:
1127:Casting out nines
1042:casting out nines
994:
914:
890:
837:
435:
342:{\displaystyle b}
126:{\displaystyle n}
3364:
3319:
3282:
3281:
3251:Natural language
3246:
3245:
3210:
3209:
3178:Generated via a
3174:
3173:
3134:
3133:
3039:Digit-reassembly
3004:Self-descriptive
2808:
2807:
2773:
2772:
2759:
2758:
2710:Lucas–Carmichael
2700:Harmonic divisor
2648:
2647:
2574:Sparsely totient
2549:Highly cototient
2458:Multiply perfect
2448:Highly composite
2391:
2390:
2374:
2373:
2289:
2288:
2244:
2243:
2225:Telephone number
2121:
2120:
2079:
2078:
2060:Square pyramidal
2042:Stella octangula
1967:
1966:
1833:
1832:
1824:
1823:
1816:Figurate numbers
1811:
1810:
1758:
1757:
1710:
1709:
1644:
1643:
1596:
1595:
1534:
1533:
1438:
1437:
1422:
1415:
1408:
1399:
1398:
1389:
1388:
1361:
1359:
1357:
1342:
1334:Edgeworth, F. Y.
1330:
1324:
1322:
1295:
1289:
1287:
1286:
1285:
1279:
1273:, archived from
1264:
1236:
1220:
1214:
1212:
1183:
1053:Edgeworth (1888)
1012:
1010:
1009:
1004:
1002:
1001:
995:
992:
963:
958:
915:
910:
902:
891:
888:
838:
835:
750:integer sequence
735:
718:
716:
715:
710:
696:
695:
694:
693:
678:
673:
646:
645:
644:
643:
628:
623:
599:
597:
596:
591:
576:
574:
573:
568:
566:
565:
553:
552:
528:
526:
525:
520:
473:
472:
453:For example, in
446:
444:
443:
438:
436:
434:
433:
424:
423:
422:
417:
416:
400:
399:
398:
397:
373:
368:
367:
348:
346:
345:
340:
328:
326:
325:
320:
315:
307:
306:
278:
276:
275:
270:
268:
267:
257:
252:
225:
224:
205:
203:
202:
197:
195:
187:
179:
178:
162:
160:
159:
154:
132:
130:
129:
124:
105:
103:
102:
97:
61:
59:
58:
53:
3372:
3371:
3367:
3366:
3365:
3363:
3362:
3361:
3332:
3331:
3330:
3325:
3303:
3299:Strobogrammatic
3290:
3272:
3254:
3236:
3218:
3200:
3182:
3164:
3141:
3120:
3104:
3063:Divisor-related
3058:
3018:
2969:
2939:
2876:
2860:
2839:
2806:
2779:
2767:
2749:
2661:
2660:related numbers
2634:
2611:
2578:
2569:Perfect totient
2535:
2512:
2443:Highly abundant
2385:
2364:
2296:
2279:
2251:
2234:
2220:Stirling second
2126:
2103:
2064:
2046:
2003:
1952:
1889:
1850:Centered square
1818:
1801:
1763:
1748:
1715:
1700:
1652:
1651:defined numbers
1634:
1601:
1586:
1557:Double Mersenne
1543:
1524:
1446:
1432:
1430:natural numbers
1426:
1370:
1365:
1364:
1355:
1340:
1331:
1327:
1312:10.2307/2002859
1296:
1292:
1283:
1281:
1277:
1262:1959.13/1043650
1253:10.2307/2324993
1234:
1221:
1217:
1202:10.2307/2304217
1184:
1180:
1175:
1118:
1102:Harshad numbers
1018:
997:
996:
991:
989:
959:
927:
920:
919:
903:
901:
887:
885:
849:
848:
834:
832:
822:
821:
798:
795:
794:
789:
777:
727:
689:
685:
684:
680:
674:
663:
639:
635:
634:
630:
624:
613:
607:
604:
603:
582:
579:
578:
561:
557:
548:
544:
536:
533:
532:
468:
464:
462:
459:
458:
429:
425:
418:
412:
408:
407:
387:
383:
382:
378:
374:
372:
363:
359:
357:
354:
353:
334:
331:
330:
311:
302:
298:
287:
284:
283:
263:
259:
253:
242:
220:
216:
214:
211:
210:
191:
183:
174:
170:
168:
165:
164:
142:
139:
138:
118:
115:
114:
111:
67:
64:
63:
47:
44:
43:
17:
12:
11:
5:
3370:
3360:
3359:
3354:
3349:
3344:
3327:
3326:
3324:
3323:
3312:
3309:
3308:
3305:
3304:
3302:
3301:
3295:
3292:
3291:
3278:
3277:
3274:
3273:
3271:
3270:
3265:
3259:
3256:
3255:
3242:
3241:
3238:
3237:
3235:
3234:
3232:Sorting number
3229:
3227:Pancake number
3223:
3220:
3219:
3206:
3205:
3202:
3201:
3199:
3198:
3193:
3187:
3184:
3183:
3170:
3169:
3166:
3165:
3163:
3162:
3157:
3152:
3146:
3143:
3142:
3139:Binary numbers
3130:
3129:
3126:
3125:
3122:
3121:
3119:
3118:
3112:
3110:
3106:
3105:
3103:
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3066:
3064:
3060:
3059:
3057:
3056:
3051:
3046:
3041:
3036:
3030:
3028:
3020:
3019:
3017:
3016:
3011:
3006:
3001:
2996:
2991:
2986:
2980:
2978:
2971:
2970:
2968:
2967:
2966:
2965:
2954:
2952:
2949:P-adic numbers
2945:
2944:
2941:
2940:
2938:
2937:
2936:
2935:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2890:
2884:
2882:
2878:
2877:
2875:
2874:
2868:
2866:
2865:Coding-related
2862:
2861:
2859:
2858:
2853:
2847:
2845:
2841:
2840:
2838:
2837:
2832:
2827:
2822:
2816:
2814:
2805:
2804:
2803:
2802:
2800:Multiplicative
2797:
2786:
2784:
2769:
2768:
2764:Numeral system
2755:
2754:
2751:
2750:
2748:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2666:
2663:
2662:
2644:
2643:
2640:
2639:
2636:
2635:
2633:
2632:
2627:
2621:
2619:
2613:
2612:
2610:
2609:
2604:
2599:
2594:
2588:
2586:
2580:
2579:
2577:
2576:
2571:
2566:
2561:
2556:
2554:Highly totient
2551:
2545:
2543:
2537:
2536:
2534:
2533:
2528:
2522:
2520:
2514:
2513:
2511:
2510:
2505:
2500:
2495:
2490:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2410:
2408:Almost perfect
2405:
2399:
2397:
2387:
2386:
2370:
2369:
2366:
2365:
2363:
2362:
2357:
2352:
2347:
2342:
2337:
2332:
2327:
2322:
2317:
2312:
2307:
2301:
2298:
2297:
2285:
2284:
2281:
2280:
2278:
2277:
2272:
2267:
2262:
2256:
2253:
2252:
2240:
2239:
2236:
2235:
2233:
2232:
2227:
2222:
2217:
2215:Stirling first
2212:
2207:
2202:
2197:
2192:
2187:
2182:
2177:
2172:
2167:
2162:
2157:
2152:
2147:
2142:
2137:
2131:
2128:
2127:
2117:
2116:
2113:
2112:
2109:
2108:
2105:
2104:
2102:
2101:
2096:
2091:
2085:
2083:
2076:
2070:
2069:
2066:
2065:
2063:
2062:
2056:
2054:
2048:
2047:
2045:
2044:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2005:
2004:
2002:
2001:
1996:
1991:
1986:
1981:
1975:
1973:
1964:
1958:
1957:
1954:
1953:
1951:
1950:
1945:
1940:
1935:
1930:
1925:
1920:
1915:
1910:
1905:
1899:
1897:
1891:
1890:
1888:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1841:
1839:
1830:
1820:
1819:
1807:
1806:
1803:
1802:
1800:
1799:
1794:
1789:
1784:
1779:
1774:
1768:
1765:
1764:
1754:
1753:
1750:
1749:
1747:
1746:
1741:
1736:
1731:
1726:
1720:
1717:
1716:
1706:
1705:
1702:
1701:
1699:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1657:
1654:
1653:
1640:
1639:
1636:
1635:
1633:
1632:
1627:
1622:
1617:
1612:
1606:
1603:
1602:
1592:
1591:
1588:
1587:
1585:
1584:
1579:
1574:
1569:
1564:
1559:
1554:
1548:
1545:
1544:
1530:
1529:
1526:
1525:
1523:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1451:
1448:
1447:
1434:
1433:
1425:
1424:
1417:
1410:
1402:
1396:
1395:
1390:
1369:
1368:External links
1366:
1363:
1362:
1325:
1290:
1247:(7): 622–640,
1228:Borwein, P. B.
1224:Borwein, J. M.
1215:
1177:
1176:
1174:
1171:
1170:
1169:
1164:
1159:
1154:
1149:
1147:Harshad number
1144:
1142:Hamming weight
1139:
1134:
1129:
1124:
1117:
1114:
1096:computer chess
1080:Hamming weight
1034:if and only if
1017:
1014:
1000:
990:
988:
985:
982:
979:
976:
973:
970:
967:
962:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
926:
922:
921:
918:
913:
909:
906:
900:
897:
894:
886:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
850:
847:
844:
841:
833:
831:
828:
827:
825:
820:
817:
814:
811:
808:
805:
802:
788:
785:
776:
773:
769:transcendental
720:
719:
708:
705:
702:
699:
692:
688:
683:
677:
672:
669:
666:
662:
658:
655:
652:
649:
642:
638:
633:
627:
622:
619:
616:
612:
589:
586:
564:
560:
556:
551:
547:
543:
540:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
471:
467:
448:
447:
432:
428:
421:
415:
411:
406:
403:
396:
393:
390:
386:
381:
377:
371:
366:
362:
338:
318:
314:
310:
305:
301:
297:
294:
291:
280:
279:
266:
262:
256:
251:
248:
245:
241:
237:
234:
231:
228:
223:
219:
194:
190:
186:
182:
177:
173:
152:
149:
146:
122:
110:
107:
95:
92:
89:
86:
83:
80:
77:
74:
71:
51:
41:decimal number
29:natural number
15:
9:
6:
4:
3:
2:
3369:
3358:
3357:Number theory
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3339:
3337:
3322:
3318:
3314:
3313:
3310:
3300:
3297:
3296:
3293:
3288:
3283:
3279:
3269:
3266:
3264:
3261:
3260:
3257:
3252:
3247:
3243:
3233:
3230:
3228:
3225:
3224:
3221:
3216:
3211:
3207:
3197:
3194:
3192:
3189:
3188:
3185:
3181:
3175:
3171:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3144:
3140:
3135:
3131:
3117:
3114:
3113:
3111:
3107:
3101:
3098:
3096:
3093:
3091:
3090:Polydivisible
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3067:
3065:
3061:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3031:
3029:
3026:
3021:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2981:
2979:
2976:
2972:
2964:
2961:
2960:
2959:
2956:
2955:
2953:
2950:
2946:
2934:
2931:
2930:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2885:
2883:
2879:
2873:
2870:
2869:
2867:
2863:
2857:
2854:
2852:
2849:
2848:
2846:
2844:Digit product
2842:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2817:
2815:
2813:
2809:
2801:
2798:
2796:
2793:
2792:
2791:
2788:
2787:
2785:
2783:
2778:
2774:
2770:
2765:
2760:
2756:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2680:Erdős–Nicolas
2678:
2676:
2673:
2671:
2668:
2667:
2664:
2659:
2655:
2649:
2645:
2631:
2628:
2626:
2623:
2622:
2620:
2618:
2614:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2587:
2585:
2581:
2575:
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2546:
2544:
2542:
2538:
2532:
2529:
2527:
2524:
2523:
2521:
2519:
2515:
2509:
2506:
2504:
2501:
2499:
2498:Superabundant
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2400:
2398:
2396:
2392:
2388:
2384:
2380:
2375:
2371:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2302:
2299:
2295:
2290:
2286:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2257:
2254:
2250:
2245:
2241:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2132:
2129:
2122:
2118:
2100:
2097:
2095:
2092:
2090:
2087:
2086:
2084:
2080:
2077:
2075:
2074:4-dimensional
2071:
2061:
2058:
2057:
2055:
2053:
2049:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2014:
2012:
2010:
2006:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1984:Centered cube
1982:
1980:
1977:
1976:
1974:
1972:
1968:
1965:
1963:
1962:3-dimensional
1959:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1900:
1898:
1896:
1892:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1842:
1840:
1838:
1834:
1831:
1829:
1828:2-dimensional
1825:
1821:
1817:
1812:
1808:
1798:
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1772:Nonhypotenuse
1770:
1769:
1766:
1759:
1755:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1721:
1718:
1711:
1707:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1658:
1655:
1650:
1645:
1641:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1607:
1604:
1597:
1593:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1549:
1546:
1541:
1535:
1531:
1521:
1518:
1516:
1513:
1511:
1510:Perfect power
1508:
1506:
1503:
1501:
1500:Seventh power
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1452:
1449:
1444:
1439:
1435:
1431:
1423:
1418:
1416:
1411:
1409:
1404:
1403:
1400:
1393:
1391:
1386:
1385:
1380:
1377:
1372:
1371:
1358:on 2006-09-13
1354:
1350:
1346:
1339:
1335:
1329:
1321:
1317:
1313:
1309:
1305:
1301:
1294:
1280:on 2016-05-09
1276:
1272:
1268:
1263:
1258:
1254:
1250:
1246:
1242:
1241:
1233:
1229:
1225:
1219:
1211:
1207:
1203:
1199:
1195:
1191:
1190:
1182:
1178:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1119:
1113:
1111:
1107:
1106:Smith numbers
1103:
1099:
1097:
1093:
1092:coding theory
1089:
1085:
1081:
1077:
1072:
1070:
1066:
1062:
1059:as a form of
1058:
1054:
1050:
1045:
1043:
1039:
1038:rule of nines
1035:
1031:
1027:
1023:
1013:
983:
980:
977:
974:
971:
965:
960:
952:
949:
946:
943:
940:
931:
928:
924:
911:
907:
904:
895:
892:
879:
876:
873:
870:
867:
864:
861:
858:
852:
845:
842:
839:
829:
823:
818:
812:
809:
806:
800:
792:
784:
782:
772:
770:
766:
762:
759:
755:
751:
747:
743:
739:
734:
730:
725:
706:
700:
690:
686:
681:
675:
670:
667:
664:
660:
656:
650:
640:
636:
631:
625:
620:
617:
614:
610:
602:
601:
600:
587:
584:
562:
558:
554:
549:
545:
541:
538:
529:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
486:
483:
477:
469:
465:
456:
451:
430:
426:
419:
413:
404:
401:
394:
391:
388:
384:
375:
369:
364:
360:
352:
351:
350:
336:
312:
308:
303:
299:
292:
289:
264:
260:
254:
249:
246:
243:
239:
235:
229:
221:
217:
209:
208:
207:
180:
175:
171:
150:
147:
144:
136:
120:
106:
93:
90:
87:
84:
81:
78:
75:
72:
69:
49:
42:
38:
34:
30:
26:
22:
3054:Transposable
2918:Narcissistic
2825:Digital root
2819:
2811:
2745:Super-Poulet
2705:Jordan–Pólya
2654:prime factor
2559:Noncototient
2526:Almost prime
2508:Superperfect
2483:Refactorable
2478:Quasiperfect
2453:Hyperperfect
2294:Pseudoprimes
2265:Wall–Sun–Sun
2200:Ordered Bell
2170:Fuss–Catalan
2082:non-centered
2032:Dodecahedral
2009:non-centered
1895:non-centered
1797:Wolstenholme
1542:× 2 ± 1
1539:
1538:Of the form
1505:Eighth power
1485:Fourth power
1382:
1353:the original
1348:
1344:
1328:
1303:
1299:
1293:
1282:, retrieved
1275:the original
1244:
1238:
1218:
1193:
1187:
1181:
1162:Smith number
1157:Sideways sum
1137:Digital root
1100:
1088:cryptography
1073:
1046:
1037:
1022:digital root
1019:
1016:Applications
793:
790:
778:
721:
530:
452:
449:
281:
134:
112:
24:
18:
3075:Extravagant
3070:Equidigital
3025:permutation
2984:Palindromic
2958:Automorphic
2856:Sum-product
2835:Sum-product
2790:Persistence
2685:Erdős–Woods
2607:Untouchable
2488:Semiperfect
2438:Hemiperfect
2099:Tesseractic
2037:Icosahedral
2017:Tetrahedral
1948:Dodecagonal
1649:Recursively
1520:Prime power
1495:Sixth power
1490:Fifth power
1470:Power of 10
1428:Classes of
1379:"Digit Sum"
33:number base
31:in a given
21:mathematics
3336:Categories
3287:Graphemics
3160:Pernicious
3014:Undulating
2989:Pandigital
2963:Trimorphic
2564:Nontotient
2413:Arithmetic
2027:Octahedral
1928:Heptagonal
1918:Pentagonal
1903:Triangular
1744:Sierpiński
1666:Jacobsthal
1465:Power of 3
1460:Power of 2
1284:2009-03-02
1173:References
1032:by 3 or 9
787:Properties
758:converging
109:Definition
3044:Parasitic
2893:Factorion
2820:Digit sum
2812:Digit sum
2630:Fortunate
2617:Primorial
2531:Semiprime
2468:Practical
2433:Descartes
2428:Deficient
2418:Betrothed
2260:Wieferich
2089:Pentatope
2052:pyramidal
1943:Decagonal
1938:Nonagonal
1933:Octagonal
1923:Hexagonal
1782:Practical
1729:Congruent
1661:Fibonacci
1625:Loeschian
1384:MathWorld
1030:divisible
993:otherwise
975:−
944:−
925:∑
917:⌉
899:⌈
871:−
661:∑
611:∑
542:≤
402:−
317:⌋
309:
296:⌊
240:∑
189:→
137:for base
135:digit sum
62:would be
25:digit sum
3342:Addition
3116:Friedman
3049:Primeval
2994:Repdigit
2951:-related
2898:Kaprekar
2872:Meertens
2795:Additive
2782:dynamics
2690:Friendly
2602:Sociable
2592:Amicable
2403:Abundant
2383:dynamics
2205:Schröder
2195:Narayana
2165:Eulerian
2155:Delannoy
2150:Dedekind
1971:centered
1837:centered
1724:Amenable
1681:Narayana
1671:Leonardo
1567:Mersenne
1515:Powerful
1455:Achilles
1336:(1888),
1230:(1992),
1132:Checksum
1116:See also
1049:checksum
889:if
836:if
765:rational
748:of this
744:use the
724:integers
3289:related
3253:related
3217:related
3215:Sorting
3100:Vampire
3085:Harshad
3027:related
2999:Repunit
2913:Lychrel
2888:Dudeney
2740:Størmer
2735:Sphenic
2720:Regular
2658:divisor
2597:Perfect
2493:Sublime
2463:Perfect
2190:Motzkin
2145:Catalan
1686:Padovan
1620:Leyland
1615:Idoneal
1610:Hilbert
1582:Woodall
1320:2002859
1271:2324993
1210:2304217
736:in the
733:A007953
731::
455:base 10
3155:Odious
3080:Frugal
3034:Cyclic
3023:Digit-
2730:Smooth
2715:Pronic
2675:Cyclic
2652:Other
2625:Euclid
2275:Wilson
2249:Primes
1908:Square
1777:Polite
1739:Riesel
1734:Knödel
1696:Perrin
1577:Thabit
1562:Fermat
1552:Cullen
1475:Square
1443:Powers
1318:
1269:
1208:
1094:, and
771:sums.
761:series
754:binary
349:, and
282:where
37:digits
23:, the
3196:Prime
3191:Lucky
3180:sieve
3109:Other
3095:Smith
2975:Digit
2933:Happy
2908:Keith
2881:Other
2725:Rough
2695:Giuga
2160:Euler
2022:Cubic
1676:Lucas
1572:Proth
1356:(PDF)
1341:(PDF)
1316:JSTOR
1278:(PDF)
1267:JSTOR
1235:(PDF)
1206:JSTOR
763:with
478:84001
27:of a
3150:Evil
2830:Self
2780:and
2670:Blum
2381:and
2185:Lobb
2140:Cake
2135:Bell
1885:Star
1792:Ulam
1691:Pell
1480:Cube
896:>
767:and
729:OEIS
657:<
555:<
148:>
113:Let
50:9045
3268:Ban
2656:or
2175:Lah
1308:doi
1257:hdl
1249:doi
1198:doi
935:max
517:13.
410:mod
380:mod
300:log
94:18.
19:In
3338::
1381:.
1349:51
1347:,
1343:,
1314:,
1302:,
1265:,
1255:,
1245:99
1243:,
1237:,
1226:;
1204:,
1194:47
1192:,
1112:.
1098:.
1090:,
1071:.
740:.
470:10
163:,
1540:a
1421:e
1414:t
1407:v
1387:.
1360:.
1323:.
1310::
1304:3
1288:.
1259::
1251::
1213:.
1200::
987:)
984:i
981:,
978:1
972:n
969:(
966:f
961:q
956:)
953:1
950:,
947:9
941:q
938:(
932:=
929:i
912:2
908:n
905:9
893:q
883:)
880:1
877:+
874:q
868:n
865:9
862:,
859:n
856:(
853:f
846:1
843:=
840:n
830:1
824:{
819:=
816:)
813:q
810:,
807:n
804:(
801:f
707:.
704:)
701:k
698:(
691:2
687:b
682:F
676:n
671:0
668:=
665:k
654:)
651:k
648:(
641:1
637:b
632:F
626:n
621:0
618:=
615:k
588:,
585:n
563:2
559:b
550:1
546:b
539:2
514:=
511:1
508:+
505:0
502:+
499:0
496:+
493:4
490:+
487:8
484:=
481:)
475:(
466:F
431:i
427:b
420:i
414:b
405:n
395:1
392:+
389:i
385:b
376:n
370:=
365:i
361:d
337:b
313:n
304:b
293:=
290:k
265:i
261:d
255:k
250:0
247:=
244:i
236:=
233:)
230:n
227:(
222:b
218:F
193:N
185:N
181::
176:b
172:F
151:1
145:b
121:n
91:=
88:5
85:+
82:4
79:+
76:0
73:+
70:9
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