A number n which has a divisors d such that 2⋅n = σ(n) - 2⋅d. more
The first 600 admirable numbers :
12,
20,
24,
30,
40,
42,
54,
56,
66,
70,
78,
84,
88,
102,
104,
114,
120,
138,
140,
174,
186,
222,
224,
234,
246,
258,
270,
282,
308,
318,
354,
364,
366,
368,
402,
426,
438,
464,
474,
476,
498,
532,
534,
582,
606,
618,
642,
644,
650,
654,
672,
678,
762,
786,
812,
822,
834,
836,
868,
894,
906,
942,
945,
978,
992,
1002,
1036,
1038,
1074,
1086,
1146,
1148,
1158,
1182,
1194,
1204,
1266,
1316,
1338,
1362,
1372,
1374,
1398,
1434,
1446,
1484,
1488,
1504,
1506,
1542,
1578,
1614,
1626,
1638,
1652,
1662,
1686,
1698,
1708,
1758,
1842,
1866,
1876,
1878,
1888,
1902,
1952,
1986,
1988,
2002,
2022,
2044,
2082,
2094,
2118,
2154,
2202,
2212,
2238,
2274,
2298,
2324,
2334,
2382,
2406,
2454,
2480,
2492,
2514,
2526,
2586,
2598,
2634,
2658,
2694,
2716,
2742,
2766,
2778,
2802,
2828,
2874,
2884,
2922,
2946,
2994,
2996,
3018,
3052,
3054,
3126,
3138,
3164,
3230,
3246,
3250,
3282,
3342,
3378,
3414,
3426,
3462,
3472,
3522,
3556,
3558,
3594,
3606,
3642,
3668,
3678,
3702,
3714,
3724,
3770,
3786,
3836,
3846,
3858,
3882,
3892,
3918,
3954,
3966,
4030,
4038,
4062,
4095,
4098,
4146,
4172,
4206,
4228,
4254,
4314,
4362,
4396,
4398,
4434,
4458,
4506,
4542,
4564,
4566,
4614,
4638,
4676,
4722,
4730,
4782,
4844,
4854,
4866,
4926,
4938,
4962,
4974,
5012,
5034,
5068,
5118,
5142,
5154,
5178,
5262,
5286,
5298,
5322,
5348,
5404,
5442,
5456,
5466,
5514,
5516,
5572,
5574,
5622,
5624,
5646,
5682,
5718,
5802,
5826,
5830,
5862,
5898,
5908,
5946,
5982,
6054,
6078,
6114,
6126,
6186,
6198,
6200,
6234,
6244,
6294,
6306,
6356,
6366,
6378,
6412,
6414,
6435,
6448,
6522,
6524,
6546,
6558,
6582,
6618,
6654,
6692,
6702,
6738,
6748,
6774,
6906,
6918,
6978,
7026,
7028,
7086,
7122,
7158,
7192,
7196,
7206,
7278,
7302,
7338,
7364,
7374,
7386,
7422,
7425,
7494,
7532,
7554,
7588,
7662,
7674,
7698,
7734,
7746,
7756,
7782,
7806,
7818,
7842,
7868,
7912,
7914,
7924,
7926,
7962,
8166,
8202,
8204,
8238,
8286,
8394,
8415,
8432,
8454,
8538,
8562,
8574,
8596,
8598,
8634,
8682,
8706,
8708,
8718,
8754,
8764,
8826,
8876,
8886,
8898,
8922,
8925,
8934,
8958,
8994,
9066,
9112,
9138,
9186,
9258,
9268,
9294,
9318,
9354,
9402,
9424,
9426,
9436,
9474,
9498,
9555,
9582,
9606,
9642,
9654,
9678,
9714,
9716,
9726,
9762,
9772,
9822,
9884,
9942,
9978,
10002,
10014,
10052,
10158,
10182,
10194,
10254,
10276,
10326,
10338,
10398,
10444,
10446,
10482,
10518,
10554,
10612,
10662,
10698,
10722,
10724,
10734,
10792,
10806,
10866,
10892,
10938,
10986,
11082,
11096,
11116,
11166,
11202,
11226,
11228,
11238,
11262,
11274,
11334,
11406,
11408,
11442,
11452,
11478,
11586,
11598,
11694,
11706,
11732,
11788,
11838,
11874,
11922,
11958,
11982,
11994,
12018,
12066,
12068,
12102,
12124,
12162,
12174,
12234,
12292,
12318,
12378,
12404,
12414,
12486,
12498,
12522,
12534,
12572,
12594,
12666,
12678,
12774,
12786,
12796,
12822,
12846,
12858,
12908,
12918,
12964,
12966,
13074,
13076,
13218,
13242,
13278,
13326,
13412,
13422,
13434,
13458,
13506,
13602,
13614,
13636,
13638,
13686,
13722,
13736,
13748,
13758,
13782,
13854,
13866,
13972,
13998,
14034,
14046,
14082,
14084,
14106,
14142,
14226,
14252,
14262,
14286,
14298,
14334,
14358,
14384,
14394,
14466,
14502,
14538,
14588,
14622,
14644,
14646,
14682,
14754,
14802,
14838,
14862,
15018,
15126,
15148,
15186,
15234,
15258,
15294,
15306,
15316,
15342,
15474,
15546,
15558,
15596,
15654,
15702,
15726,
15764,
15798,
15872,
15882,
15932,
15942,
15954,
15978,
15988,
16026,
16062,
16098,
16122,
16134,
16156,
16158,
16194,
16242,
16256,
16266,
16278,
16314,
16374,
16386,
16436,
16446,
16494,
16518,
16602,
16604,
16662,
16734,
16746,
16772,
16782,
16806,
16818,
16828,
16914,
16996,
16998,
17022,
17058,
17106,
17142,
17164,
17166,
17272,
17274,
17276,
17322,
17332,
17382,
17418,
17454,
17502,
17562,
17634,
17668,
17718,
17742,
17778,
17814,
17816,
17826.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1345934 values, from 12 to 99999956).
n\r | 0 | 1 |
2 | 1345818 | 116 | 2 |
3 | 1071336 | 137208 | 137390 | 3 |
4 | 274563 | 55 | 1071255 | 61 | 4 |
5 | 145 | 336332 | 336371 | 336561 | 336525 | 5 |
6 | 1071220 | 0 | 137390 | 116 | 137208 | 0 | 6 |
7 | 254925 | 181793 | 181928 | 181759 | 181927 | 181822 | 181780 | 7 |
8 | 19689 | 27 | 535773 | 27 | 254874 | 28 | 535482 | 34 | 8 |
9 | 104 | 45715 | 45775 | 535789 | 45755 | 45795 | 535443 | 45738 | 45820 | 9 |
10 | 48 | 7 | 336369 | 4 | 336519 | 97 | 336325 | 2 | 336557 | 6 | 10 |
11 | 76 | 134627 | 134593 | 134724 | 134488 | 134603 | 134480 | 134583 | 134576 | 134586 | 134598 |
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.