A number which is abundant with no abundant proper divisors. more
The first 600 primitive abundant numbers :
12,
18,
20,
30,
42,
56,
66,
70,
78,
88,
102,
104,
114,
138,
174,
186,
196,
222,
246,
258,
272,
282,
304,
308,
318,
354,
364,
366,
368,
402,
426,
438,
464,
474,
476,
498,
532,
534,
550,
572,
582,
606,
618,
642,
644,
650,
654,
678,
748,
762,
786,
812,
822,
834,
836,
868,
894,
906,
942,
945,
978,
992,
1002,
1036,
1038,
1074,
1086,
1146,
1148,
1158,
1182,
1184,
1194,
1204,
1266,
1312,
1316,
1338,
1362,
1374,
1376,
1398,
1430,
1434,
1446,
1484,
1504,
1506,
1542,
1575,
1578,
1614,
1626,
1652,
1662,
1686,
1696,
1698,
1708,
1758,
1842,
1866,
1870,
1876,
1878,
1888,
1902,
1952,
1986,
1988,
2002,
2022,
2044,
2082,
2090,
2094,
2118,
2154,
2202,
2205,
2210,
2212,
2238,
2274,
2298,
2324,
2334,
2382,
2406,
2454,
2470,
2492,
2514,
2526,
2530,
2584,
2586,
2598,
2634,
2658,
2694,
2716,
2742,
2766,
2778,
2802,
2828,
2874,
2884,
2922,
2946,
2990,
2994,
2996,
3018,
3052,
3054,
3126,
3128,
3138,
3164,
3190,
3230,
3246,
3282,
3342,
3378,
3410,
3414,
3426,
3462,
3465,
3496,
3522,
3556,
3558,
3594,
3606,
3642,
3668,
3678,
3702,
3714,
3770,
3786,
3836,
3846,
3858,
3882,
3892,
3918,
3944,
3954,
3966,
4030,
4038,
4062,
4070,
4095,
4098,
4146,
4172,
4206,
4216,
4228,
4254,
4288,
4314,
4362,
4396,
4398,
4408,
4434,
4458,
4506,
4510,
4542,
4544,
4564,
4566,
4614,
4638,
4672,
4676,
4712,
4722,
4730,
4782,
4844,
4854,
4866,
4926,
4938,
4962,
4974,
5012,
5032,
5034,
5056,
5068,
5118,
5142,
5154,
5170,
5178,
5262,
5286,
5298,
5312,
5322,
5336,
5348,
5355,
5404,
5442,
5466,
5514,
5516,
5572,
5574,
5576,
5622,
5624,
5646,
5682,
5696,
5704,
5718,
5775,
5802,
5826,
5830,
5848,
5862,
5898,
5908,
5946,
5982,
5985,
6054,
6078,
6114,
6126,
6186,
6198,
6208,
6232,
6234,
6244,
6294,
6306,
6356,
6366,
6378,
6392,
6412,
6414,
6435,
6464,
6522,
6524,
6536,
6546,
6558,
6582,
6592,
6618,
6654,
6692,
6702,
6738,
6748,
6774,
6808,
6825,
6848,
6906,
6918,
6976,
6978,
7026,
7028,
7086,
7122,
7144,
7158,
7192,
7196,
7206,
7208,
7232,
7245,
7278,
7302,
7338,
7364,
7374,
7386,
7422,
7425,
7494,
7532,
7544,
7554,
7588,
7662,
7674,
7698,
7734,
7746,
7756,
7782,
7806,
7818,
7842,
7868,
7912,
7914,
7924,
7926,
7962,
8024,
8056,
8085,
8166,
8202,
8204,
8238,
8286,
8296,
8394,
8415,
8454,
8538,
8562,
8574,
8596,
8598,
8634,
8682,
8706,
8708,
8718,
8754,
8764,
8826,
8876,
8886,
8898,
8922,
8925,
8934,
8958,
8968,
8994,
9066,
9112,
9135,
9138,
9186,
9258,
9268,
9272,
9294,
9318,
9354,
9402,
9426,
9436,
9474,
9498,
9555,
9582,
9606,
9642,
9654,
9656,
9678,
9714,
9716,
9726,
9762,
9765,
9772,
9822,
9884,
9928,
9942,
9978,
10002,
10014,
10052,
10158,
10182,
10184,
10194,
10254,
10276,
10326,
10338,
10398,
10444,
10446,
10482,
10518,
10554,
10612,
10662,
10698,
10722,
10724,
10734,
10744,
10792,
10806,
10866,
10892,
10938,
10986,
11082,
11096,
11116,
11132,
11166,
11202,
11226,
11228,
11238,
11262,
11274,
11288,
11334,
11406,
11442,
11452,
11478,
11492,
11586,
11598,
11655,
11694,
11706,
11732,
11788,
11838,
11874,
11922,
11958,
11982,
11994,
12018,
12066,
12068,
12102,
12104,
12124,
12162,
12174,
12234,
12292,
12318,
12378,
12404,
12414,
12486,
12498,
12522,
12534,
12572,
12594,
12666,
12678,
12705,
12774,
12786,
12796,
12822,
12846,
12858,
12908,
12915,
12918,
12964,
12966,
13074,
13076,
13192,
13218,
13242,
13278,
13326,
13412,
13422,
13434,
13458,
13506,
13545,
13602,
13614,
13636,
13638,
13686,
13722,
13736,
13748,
13758,
13782,
13854,
13866,
13972,
13998,
14008,
14034,
14046,
14082,
14084,
14106,
14142,
14226,
14252,
14262,
14286,
14298,
14334,
14358,
14394,
14466,
14502,
14538,
14552,
14588,
14622,
14644,
14646,
14682,
14754,
14802,
14805,
14824,
14838,
14862,
15015,
15018,
15028,
15126,
15148,
15186,
15234,
15258,
15294,
15306,
15316,
15342,
15368,
15376,
15474,
15546,
15558,
15596,
15654.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1387244 values, from 12 to 99999956).
n\r | 0 | 1 |
2 | 1379679 | 7565 | 2 |
3 | 1078770 | 154157 | 154317 | 3 |
4 | 299873 | 3783 | 1079806 | 3782 | 4 |
5 | 12976 | 343467 | 343485 | 343691 | 343625 | 5 |
6 | 1071205 | 0 | 154317 | 7565 | 154157 | 0 | 6 |
7 | 261135 | 187630 | 187767 | 187625 | 187760 | 187682 | 187645 | 7 |
8 | 39680 | 1887 | 540023 | 1885 | 260193 | 1896 | 539783 | 1897 | 8 |
9 | 5201 | 51360 | 51415 | 536961 | 51433 | 51418 | 536608 | 51364 | 51484 | 9 |
10 | 6111 | 181 | 343311 | 177 | 343457 | 6865 | 343286 | 174 | 343514 | 168 | 10 |
11 | 7118 | 138049 | 137989 | 138149 | 137909 | 138029 | 137942 | 138001 | 137973 | 138029 | 138056 |
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.