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digitally balanced numbers
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
2116377349037842 2
3696251568008326912229 3
45804898475365858328364284184 4
541412994130860413613041300014137286 5
6387635630081753968721308615937926572943508 6
72960935292350830574872985118289209029273842929054 7
827070632409693294775223511123097835234396528850841933072 8
9227289122792062324258229892121941462280719239070323274802307252 9
102331696180537823284981804592232664918096032325482180763223254091810637 10
1118779681879860187868918795731879482187958918795671880371187944418817681879265

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.