A number which in a given base b contains all the digits from 0 to b-1 an equal number of times. more
The first 600 digitally balanced numbers :
2,
9,
10,
11,
12,
15,
19,
21,
35,
37,
38,
41,
42,
44,
49,
50,
52,
56,
75,
78,
99,
108,
114,
120,
135,
139,
141,
142,
147,
149,
150,
153,
154,
156,
163,
165,
166,
169,
170,
172,
177,
178,
180,
184,
195,
197,
198,
201,
202,
204,
209,
210,
212,
216,
225,
226,
228,
232,
240,
260,
266,
268,
278,
290,
294,
302,
304,
308,
312,
316,
318,
332,
344,
348,
380,
384,
396,
410,
412,
416,
420,
424,
426,
434,
438,
450,
460,
462,
468,
500,
502,
508,
518,
520,
524,
527,
528,
532,
534,
535,
539,
541,
542,
544,
550,
551,
552,
555,
557,
558,
563,
565,
566,
569,
570,
572,
574,
578,
582,
583,
586,
587,
588,
589,
590,
595,
596,
597,
598,
600,
601,
602,
604,
611,
612,
613,
614,
617,
618,
620,
622,
624,
625,
626,
628,
630,
632,
647,
651,
652,
653,
654,
658,
659,
660,
661,
662,
665,
666,
668,
675,
676,
677,
678,
681,
682,
684,
689,
690,
692,
694,
696,
698,
707,
709,
710,
713,
714,
716,
721,
722,
724,
728,
737,
738,
740,
742,
744,
752,
775,
779,
781,
782,
787,
789,
790,
793,
794,
796,
803,
805,
806,
809,
810,
812,
817,
818,
820,
824,
835,
837,
838,
841,
842,
844,
849,
850,
852,
856,
865,
866,
868,
872,
880,
894,
898,
899,
901,
902,
905,
906,
908,
913,
914,
916,
920,
929,
930,
932,
936,
944,
954,
961,
962,
964,
968,
970,
976,
978,
990,
992,
1014,
1022,
1054,
1070,
1102,
1110,
1138,
1142,
1178,
1190,
1202,
1210,
1294,
1298,
1334,
1346,
1358,
1366,
1394,
1398,
1454,
1470,
1478,
1490,
1634,
1646,
1654,
1670,
1726,
1730,
1758,
1766,
1778,
1790,
1826,
1830,
1914,
1922,
1934,
1946,
1982,
1986,
2014,
2022,
2054,
2070,
2079,
2095,
2102,
2103,
2107,
2109,
2110,
2127,
2134,
2135,
2139,
2141,
2142,
2146,
2151,
2154,
2155,
2157,
2158,
2163,
2165,
2166,
2169,
2170,
2172,
2191,
2199,
2203,
2205,
2206,
2215,
2219,
2221,
2222,
2226,
2227,
2229,
2230,
2233,
2234,
2236,
2247,
2251,
2253,
2254,
2259,
2261,
2262,
2265,
2266,
2268,
2275,
2277,
2278,
2281,
2282,
2284,
2289,
2290,
2292,
2296,
2319,
2327,
2331,
2333,
2334,
2343,
2347,
2349,
2350,
2355,
2357,
2358,
2361,
2362,
2364,
2375,
2379,
2381,
2382,
2386,
2387,
2389,
2390,
2393,
2394,
2396,
2402,
2403,
2405,
2406,
2409,
2410,
2412,
2417,
2418,
2420,
2424,
2426,
2430,
2439,
2443,
2445,
2446,
2451,
2453,
2454,
2457,
2458,
2460,
2467,
2469,
2470,
2473,
2474,
2476,
2481,
2482,
2484,
2488,
2499,
2501,
2502,
2505,
2506,
2508,
2513,
2514,
2516,
2520,
2529,
2530,
2532,
2536,
2538,
2542,
2544,
2558,
2566,
2575,
2582,
2583,
2586,
2587,
2589,
2590,
2599,
2603,
2605,
2606,
2611,
2613,
2614,
2617,
2618,
2620,
2631,
2635,
2637,
2638,
2642,
2643,
2645,
2646,
2649,
2650,
2652,
2659,
2661,
2662,
2665,
2666,
2668,
2673,
2674,
2676,
2678,
2680,
2690,
2695,
2699,
2701,
2702,
2707,
2709,
2710,
2713,
2714,
2716,
2723,
2725,
2726,
2729,
2730,
2732,
2737,
2738,
2740,
2744,
2755,
2757,
2758,
2761,
2762,
2764,
2766,
2769,
2770,
2772,
2776,
2778,
2785,
2786,
2788,
2790,
2792,
2800,
2823,
2826,
2827,
2829,
2830,
2835,
2837,
2838,
2841,
2842,
2844,
2851,
2853,
2854,
2857,
2858,
2860,
2865,
2866,
2868,
2872,
2882,
2883,
2885,
2886,
2889,
2890,
2892,
2897,
2898,
2900,
2902,
2904,
2910,
2913,
2914,
2916,
2920,
2926,
2928,
2930,
2947,
2949,
2950,
2953,
2954,
2956,
2961,
2962,
2964,
2968,
2977,
2978,
2980,
2984,
2992,
3009,
3010,
3012,
3016,
3024,
3040,
3087,
3095,
3099,
3101,
3102,
3111,
3115,
3117,
3118,
3123,
3125,
3126,
3129,
3130,
3132,
3143,
3147,
3149,
3150,
3155,
3157,
3158,
3161,
3162.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 20675576 values, from 2 to 199999936).
n\r | 0 | 1 |
2 | 11637734 | 9037842 | 2 |
3 | 6962515 | 6800832 | 6912229 | 3 |
4 | 5804898 | 4753658 | 5832836 | 4284184 | 4 |
5 | 4141299 | 4130860 | 4136130 | 4130001 | 4137286 | 5 |
6 | 3876356 | 3008175 | 3968721 | 3086159 | 3792657 | 2943508 | 6 |
7 | 2960935 | 2923508 | 3057487 | 2985118 | 2892090 | 2927384 | 2929054 | 7 |
8 | 2707063 | 2409693 | 2947752 | 2351112 | 3097835 | 2343965 | 2885084 | 1933072 | 8 |
9 | 2272891 | 2279206 | 2324258 | 2298921 | 2194146 | 2280719 | 2390703 | 2327480 | 2307252 | 9 |
10 | 2331696 | 1805378 | 2328498 | 1804592 | 2326649 | 1809603 | 2325482 | 1807632 | 2325409 | 1810637 | 10 |
11 | 1877968 | 1879860 | 1878689 | 1879573 | 1879482 | 1879589 | 1879567 | 1880371 | 1879444 | 1881768 | 1879265 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.