A number which is not the sum of the proper divisors of any number more
The first 600 untouchable numbers :
2,
5,
52,
88,
96,
120,
124,
146,
162,
188,
206,
210,
216,
238,
246,
248,
262,
268,
276,
288,
290,
292,
304,
306,
322,
324,
326,
336,
342,
372,
406,
408,
426,
430,
448,
472,
474,
498,
516,
518,
520,
530,
540,
552,
556,
562,
576,
584,
612,
624,
626,
628,
658,
668,
670,
708,
714,
718,
726,
732,
738,
748,
750,
756,
766,
768,
782,
784,
792,
802,
804,
818,
836,
848,
852,
872,
892,
894,
896,
898,
902,
926,
934,
936,
964,
966,
976,
982,
996,
1002,
1028,
1044,
1046,
1060,
1068,
1074,
1078,
1080,
1102,
1116,
1128,
1134,
1146,
1148,
1150,
1160,
1162,
1168,
1180,
1186,
1192,
1200,
1212,
1222,
1236,
1246,
1248,
1254,
1256,
1258,
1266,
1272,
1288,
1296,
1312,
1314,
1316,
1318,
1326,
1332,
1342,
1346,
1348,
1360,
1380,
1388,
1398,
1404,
1406,
1418,
1420,
1422,
1438,
1476,
1506,
1508,
1510,
1522,
1528,
1538,
1542,
1566,
1578,
1588,
1596,
1632,
1642,
1650,
1680,
1682,
1692,
1716,
1718,
1728,
1732,
1746,
1758,
1766,
1774,
1776,
1806,
1816,
1820,
1822,
1830,
1838,
1840,
1842,
1844,
1852,
1860,
1866,
1884,
1888,
1894,
1896,
1920,
1922,
1944,
1956,
1958,
1960,
1962,
1972,
1986,
1992,
2008,
2010,
2022,
2024,
2036,
2048,
2050,
2052,
2058,
2062,
2068,
2078,
2096,
2098,
2108,
2118,
2120,
2128,
2136,
2148,
2152,
2158,
2168,
2174,
2178,
2190,
2196,
2198,
2212,
2218,
2226,
2228,
2232,
2248,
2258,
2262,
2302,
2304,
2306,
2316,
2322,
2324,
2330,
2338,
2356,
2364,
2366,
2376,
2388,
2404,
2408,
2410,
2416,
2422,
2430,
2432,
2436,
2446,
2452,
2454,
2464,
2482,
2484,
2490,
2496,
2498,
2500,
2502,
2514,
2518,
2530,
2564,
2568,
2572,
2576,
2586,
2588,
2590,
2600,
2602,
2606,
2608,
2614,
2628,
2640,
2642,
2704,
2718,
2724,
2726,
2736,
2748,
2758,
2760,
2762,
2766,
2774,
2784,
2788,
2808,
2824,
2828,
2850,
2856,
2874,
2876,
2894,
2902,
2914,
2922,
2932,
2944,
2946,
2950,
2952,
2968,
2978,
2982,
2984,
2992,
2994,
2996,
3008,
3018,
3028,
3036,
3060,
3072,
3076,
3078,
3102,
3104,
3114,
3126,
3132,
3136,
3142,
3144,
3152,
3156,
3162,
3174,
3186,
3198,
3202,
3208,
3228,
3234,
3236,
3238,
3246,
3266,
3270,
3276,
3278,
3292,
3296,
3306,
3312,
3318,
3328,
3340,
3356,
3366,
3378,
3384,
3388,
3396,
3400,
3402,
3406,
3412,
3420,
3422,
3428,
3430,
3432,
3448,
3454,
3476,
3484,
3486,
3488,
3504,
3506,
3510,
3524,
3538,
3556,
3564,
3576,
3580,
3588,
3590,
3592,
3600,
3604,
3630,
3636,
3642,
3648,
3650,
3652,
3656,
3666,
3670,
3682,
3684,
3708,
3738,
3744,
3746,
3748,
3752,
3758,
3760,
3774,
3786,
3788,
3792,
3808,
3812,
3816,
3818,
3820,
3828,
3832,
3842,
3860,
3862,
3868,
3872,
3876,
3888,
3900,
3902,
3904,
3936,
3940,
3942,
3954,
3958,
3960,
3972,
3974,
3982,
3986,
4018,
4026,
4032,
4036,
4046,
4048,
4056,
4062,
4068,
4070,
4072,
4078,
4086,
4088,
4098,
4104,
4116,
4120,
4148,
4168,
4170,
4172,
4184,
4188,
4190,
4198,
4206,
4216,
4224,
4238,
4248,
4258,
4268,
4280,
4296,
4302,
4304,
4308,
4312,
4316,
4318,
4320,
4322,
4336,
4344,
4356,
4368,
4370,
4380,
4382,
4386,
4388,
4396,
4402,
4406,
4416,
4430,
4462,
4472,
4476,
4480,
4488,
4490,
4492,
4498,
4500,
4506,
4512,
4530,
4534,
4574,
4580,
4588,
4612,
4614,
4618,
4630,
4648,
4662,
4668,
4672,
4678,
4686,
4688,
4690,
4700,
4710,
4712,
4718,
4738,
4740,
4742,
4748,
4750,
4758,
4764,
4770,
4772,
4782,
4808,
4830,
4838,
4840,
4842,
4850,
4854,
4856,
4860,
4868,
4882,
4884,
4886,
4896,
4898,
4908,
4914,
4916,
4926,
4928,
4942,
4956,
4962,
4964,
4980,
4982,
4984,
4998,
5016,
5030,
5032,
5034,
5044,
5070,
5072,
5092,
5094,
5096,
5106,
5126,
5132,
5136,
5138,
5142,
5144,
5152,
5158,
5160,
5178,
5184,
5202,
5204,
5208,
5214,
5218,
5222,
5224,
5226.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 150232 values, from 2 to 999996).
n\r | 0 | 1 |
2 | 150231 | 1 | 2 |
3 | 65349 | 41873 | 43010 | 3 |
4 | 83603 | 1 | 66628 | 0 | 4 |
5 | 27675 | 35525 | 31933 | 33931 | 21168 | 5 |
6 | 65349 | 0 | 43009 | 0 | 41873 | 1 | 6 |
7 | 22234 | 20439 | 18186 | 26415 | 19465 | 22166 | 21327 | 7 |
8 | 43456 | 0 | 28609 | 0 | 40147 | 1 | 38019 | 0 | 8 |
9 | 21584 | 11939 | 14267 | 26997 | 13237 | 14331 | 16768 | 16697 | 14412 | 9 |
10 | 27674 | 0 | 31933 | 0 | 21168 | 1 | 35525 | 0 | 33931 | 0 | 10 |
11 | 13257 | 17921 | 13451 | 15246 | 13485 | 11571 | 13536 | 14376 | 13084 | 11725 | 12580 |
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.