A number divisible by the number of its divisors. more
The first 600 tau or refactorable numbers :
1,
2,
8,
9,
12,
18,
24,
36,
40,
56,
60,
72,
80,
84,
88,
96,
104,
108,
128,
132,
136,
152,
156,
180,
184,
204,
225,
228,
232,
240,
248,
252,
276,
288,
296,
328,
344,
348,
360,
372,
376,
384,
396,
424,
441,
444,
448,
450,
468,
472,
480,
488,
492,
504,
516,
536,
560,
564,
568,
584,
600,
612,
625,
632,
636,
640,
664,
672,
684,
708,
712,
720,
732,
776,
792,
804,
808,
824,
828,
852,
856,
864,
872,
876,
880,
882,
896,
904,
936,
948,
972,
996,
1016,
1040,
1044,
1048,
1056,
1068,
1089,
1096,
1112,
1116,
1152,
1164,
1176,
1192,
1200,
1208,
1212,
1224,
1236,
1248,
1250,
1256,
1260,
1284,
1304,
1308,
1332,
1336,
1344,
1356,
1360,
1368,
1384,
1408,
1432,
1440,
1448,
1476,
1520,
1521,
1524,
1528,
1544,
1548,
1572,
1576,
1592,
1620,
1632,
1644,
1656,
1664,
1668,
1680,
1688,
1692,
1784,
1788,
1800,
1812,
1816,
1824,
1832,
1840,
1864,
1884,
1908,
1912,
1920,
1928,
1944,
1956,
1980,
2000,
2004,
2008,
2016,
2025,
2056,
2076,
2088,
2104,
2124,
2148,
2152,
2160,
2168,
2172,
2176,
2178,
2196,
2208,
2216,
2232,
2240,
2248,
2264,
2292,
2316,
2320,
2340,
2344,
2364,
2388,
2412,
2432,
2456,
2480,
2488,
2504,
2532,
2536,
2556,
2560,
2601,
2628,
2640,
2648,
2664,
2676,
2688,
2696,
2700,
2724,
2748,
2772,
2776,
2784,
2792,
2796,
2824,
2844,
2868,
2872,
2892,
2904,
2936,
2944,
2952,
2960,
2976,
2984,
2988,
3012,
3032,
3042,
3060,
3064,
3084,
3096,
3112,
3120,
3156,
3168,
3176,
3204,
3208,
3228,
3240,
3249,
3252,
3272,
3276,
3280,
3324,
3352,
3360,
3368,
3372,
3384,
3396,
3420,
3440,
3448,
3456,
3464,
3492,
3512,
3516,
3528,
3544,
3552,
3592,
3600,
3636,
3656,
3684,
3688,
3704,
3708,
3712,
3732,
3736,
3744,
3756,
3760,
3804,
3816,
3832,
3852,
3896,
3924,
3928,
3936,
3968,
3972,
3992,
4024,
4032,
4044,
4050,
4056,
4068,
4072,
4080,
4128,
4140,
4164,
4168,
4184,
4188,
4224,
4236,
4240,
4248,
4284,
4308,
4320,
4328,
4376,
4392,
4404,
4456,
4476,
4480,
4500,
4504,
4512,
4548,
4552,
4560,
4568,
4572,
4596,
4616,
4668,
4696,
4716,
4720,
4736,
4744,
4761,
4764,
4788,
4792,
4808,
4812,
4824,
4856,
4860,
4880,
4896,
4904,
4908,
4928,
4932,
4936,
4952,
4992,
5000,
5004,
5028,
5040,
5048,
5052,
5088,
5112,
5128,
5144,
5148,
5172,
5176,
5196,
5202,
5220,
5224,
5248,
5256,
5268,
5272,
5280,
5288,
5292,
5316,
5360,
5364,
5384,
5388,
5416,
5436,
5464,
5472,
5484,
5504,
5520,
5528,
5532,
5556,
5580,
5604,
5608,
5625,
5652,
5664,
5672,
5680,
5688,
5748,
5752,
5760,
5796,
5816,
5824,
5840,
5844,
5856,
5864,
5868,
5892,
5912,
5944,
5976,
5988,
6000,
6008,
6012,
6016,
6036,
6048,
6056,
6088,
6108,
6144,
6152,
6160,
6184,
6228,
6240,
6252,
6276,
6296,
6320,
6376,
6408,
6432,
6444,
6472,
6488,
6492,
6498,
6516,
6528,
6561,
6564,
6568,
6584,
6616,
6624,
6632,
6640,
6660,
6684,
6712,
6720,
6732,
6756,
6784,
6804,
6816,
6824,
6828,
6852,
6856,
6872,
6876,
6904,
6912,
6924,
6936,
6948,
6960,
6984,
7008,
7016,
7040,
7044,
7048,
7064,
7092,
7096,
7116,
7120,
7164,
7188,
7212,
7256,
7272,
7280,
7284,
7288,
7296,
7308,
7352,
7356,
7380,
7392,
7404,
7416,
7428,
7432,
7440,
7496,
7500,
7524,
7528,
7552,
7569,
7572,
7576,
7584,
7596,
7616,
7624,
7680,
7692,
7704,
7716,
7736,
7740,
7760,
7764,
7768,
7776,
7808,
7812,
7816,
7836,
7848,
7864,
7908,
7920,
7928,
7932,
7956,
7968,
7976,
8028,
8064,
8072,
8076,
8080,
8100,
8104,
8124,
8136,
8152,
8160,
8168,
8172,
8196,
8240,
8244,
8248,
8264,
8292,
8312,
8320,
8352,
8388,
8392,
8400,
8408,
8412,
8460,
8488,
8504,
8508,
8512,
8544,
8552,
8560,
8576,
8604,
8628,
8649,
8664,
8676,
8696,
8712,
8720,
8724,
8728,
8736,
8744,
8776,
8796,
8824,
8832.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 32608999 values, from 1 to 1000000000).
n\r | 0 | 1 |
2 | 32607186 | 1813 | 2 |
3 | 19162388 | 6723708 | 6722903 | 3 |
4 | 32605862 | 1813 | 1324 | 0 | 4 |
5 | 7035083 | 6393929 | 6393219 | 6392938 | 6393830 | 5 |
6 | 19160588 | 13 | 6722903 | 1800 | 6723695 | 0 | 6 |
7 | 2776071 | 4971946 | 4972230 | 4972303 | 4972197 | 4971974 | 4972278 | 7 |
8 | 22246944 | 1813 | 1324 | 0 | 10358918 | 0 | 0 | 0 | 8 |
9 | 9095537 | 2240962 | 2240998 | 5032699 | 2241364 | 2240890 | 5034152 | 2241382 | 2241015 | 9 |
10 | 7034706 | 718 | 6393219 | 0 | 6393112 | 377 | 6393211 | 0 | 6392938 | 718 | 10 |
11 | 1278244 | 3133538 | 3132991 | 3133008 | 3132776 | 3133161 | 3132800 | 3133464 | 3133172 | 3132827 | 3133018 |
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.