A number divisible by the sum of its digits. more
The first 600 Harshad numbers :
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
12,
18,
20,
21,
24,
27,
30,
36,
40,
42,
45,
48,
50,
54,
60,
63,
70,
72,
80,
81,
84,
90,
100,
102,
108,
110,
111,
112,
114,
117,
120,
126,
132,
133,
135,
140,
144,
150,
152,
153,
156,
162,
171,
180,
190,
192,
195,
198,
200,
201,
204,
207,
209,
210,
216,
220,
222,
224,
225,
228,
230,
234,
240,
243,
247,
252,
261,
264,
266,
270,
280,
285,
288,
300,
306,
308,
312,
315,
320,
322,
324,
330,
333,
336,
342,
351,
360,
364,
370,
372,
375,
378,
392,
396,
399,
400,
402,
405,
407,
408,
410,
414,
420,
423,
432,
440,
441,
444,
448,
450,
460,
465,
468,
476,
480,
481,
486,
500,
504,
506,
510,
511,
512,
513,
516,
518,
522,
531,
540,
550,
552,
555,
558,
576,
588,
592,
594,
600,
603,
605,
612,
621,
624,
629,
630,
640,
644,
645,
648,
660,
666,
684,
690,
700,
702,
704,
711,
715,
720,
730,
732,
735,
736,
738,
756,
770,
774,
777,
780,
782,
792,
800,
801,
803,
804,
810,
820,
825,
828,
832,
840,
846,
864,
870,
874,
880,
882,
888,
900,
902,
910,
912,
915,
918,
935,
936,
954,
960,
966,
972,
990,
999,
1000,
1002,
1008,
1010,
1011,
1012,
1014,
1015,
1016,
1017,
1020,
1026,
1032,
1035,
1040,
1044,
1050,
1053,
1056,
1062,
1066,
1071,
1080,
1088,
1090,
1092,
1095,
1098,
1100,
1101,
1104,
1107,
1110,
1116,
1120,
1122,
1125,
1128,
1130,
1134,
1140,
1141,
1143,
1148,
1152,
1160,
1161,
1164,
1168,
1170,
1180,
1183,
1185,
1188,
1200,
1204,
1206,
1212,
1215,
1220,
1224,
1230,
1232,
1233,
1236,
1242,
1251,
1260,
1270,
1272,
1274,
1275,
1278,
1296,
1300,
1302,
1304,
1305,
1308,
1310,
1314,
1320,
1323,
1330,
1332,
1341,
1344,
1350,
1360,
1365,
1368,
1380,
1386,
1387,
1394,
1400,
1404,
1410,
1413,
1416,
1417,
1422,
1431,
1440,
1450,
1452,
1455,
1456,
1458,
1476,
1494,
1500,
1503,
1512,
1520,
1521,
1524,
1526,
1530,
1534,
1540,
1545,
1547,
1548,
1558,
1560,
1566,
1584,
1590,
1596,
1602,
1611,
1620,
1630,
1632,
1635,
1638,
1651,
1652,
1656,
1659,
1674,
1679,
1680,
1692,
1701,
1704,
1710,
1720,
1725,
1728,
1729,
1740,
1744,
1746,
1764,
1770,
1782,
1785,
1800,
1810,
1812,
1815,
1818,
1836,
1848,
1853,
1854,
1860,
1872,
1886,
1890,
1896,
1898,
1900,
1904,
1905,
1908,
1920,
1926,
1944,
1950,
1962,
1968,
1974,
1980,
1998,
2000,
2001,
2004,
2007,
2010,
2016,
2020,
2022,
2023,
2024,
2025,
2028,
2030,
2034,
2040,
2043,
2052,
2061,
2064,
2070,
2080,
2085,
2088,
2090,
2100,
2106,
2112,
2115,
2119,
2120,
2124,
2130,
2133,
2136,
2142,
2151,
2156,
2159,
2160,
2170,
2172,
2175,
2176,
2178,
2196,
2200,
2202,
2205,
2208,
2210,
2212,
2214,
2220,
2223,
2232,
2236,
2240,
2241,
2244,
2250,
2260,
2265,
2268,
2280,
2282,
2286,
2289,
2300,
2304,
2310,
2312,
2313,
2316,
2322,
2331,
2340,
2350,
2352,
2353,
2355,
2358,
2376,
2394,
2398,
2400,
2401,
2403,
2408,
2412,
2421,
2424,
2430,
2440,
2445,
2448,
2460,
2464,
2465,
2466,
2470,
2478,
2484,
2490,
2502,
2511,
2520,
2530,
2532,
2534,
2535,
2538,
2556,
2574,
2580,
2584,
2592,
2596,
2600,
2601,
2604,
2608,
2610,
2618,
2620,
2625,
2628,
2640,
2646,
2660,
2664,
2667,
2670,
2682,
2688,
2700,
2704,
2710,
2712,
2715,
2718,
2736,
2752,
2754,
2755,
2760,
2771,
2772,
2790,
2793,
2794,
2800,
2805,
2808,
2820,
2821,
2826,
2844,
2850,
2856,
2862,
2880,
2889,
2912,
2916,
2919,
2924,
2926,
2934,
2940,
2952,
2970,
2976,
2982,
2992,
2997,
3000,
3006,
3012,
3015,
3020,
3024,
3030,
3031,
3032,
3033,
3036,
3038,
3042,
3051,
3055,
3060,
3070,
3072,
3075,
3077,
3078,
3080,
3096,
3097,
3100,
3102.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 61574510 values, from 1 to 1000000000).
n\r | 0 | 1 |
2 | 46531402 | 15043108 | 2 |
3 | 43886027 | 8896987 | 8791496 | 3 |
4 | 32334456 | 7538463 | 14196946 | 7504645 | 4 |
5 | 25003174 | 9244951 | 9556575 | 8738988 | 9030822 | 5 |
6 | 33087798 | 2097998 | 6644615 | 10798229 | 6798989 | 2146881 | 6 |
7 | 14392891 | 7864862 | 7864781 | 7862406 | 7861760 | 7864582 | 7863228 | 7 |
8 | 18782555 | 3781134 | 7108508 | 3760502 | 13551901 | 3757329 | 7088438 | 3744143 | 8 |
9 | 25228402 | 2851971 | 3148443 | 10043760 | 2836091 | 2578608 | 8613865 | 3208925 | 3064445 | 9 |
10 | 17960661 | 2440075 | 7792413 | 2167939 | 7402403 | 7042513 | 6804876 | 1764162 | 6571049 | 1628419 | 10 |
11 | 9429105 | 5184268 | 5245599 | 5201698 | 5228278 | 5221938 | 5206023 | 5241064 | 5186284 | 5251882 | 5178371 |
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.