A number n such that all the smaller numbers can be written as the sum of distinct proper divisors of n. more
The first 600 practical numbers :
1,
2,
4,
6,
8,
12,
16,
18,
20,
24,
28,
30,
32,
36,
40,
42,
48,
54,
56,
60,
64,
66,
72,
78,
80,
84,
88,
90,
96,
100,
104,
108,
112,
120,
126,
128,
132,
140,
144,
150,
156,
160,
162,
168,
176,
180,
192,
196,
198,
200,
204,
208,
210,
216,
220,
224,
228,
234,
240,
252,
256,
260,
264,
270,
272,
276,
280,
288,
294,
300,
304,
306,
308,
312,
320,
324,
330,
336,
340,
342,
348,
352,
360,
364,
368,
378,
380,
384,
390,
392,
396,
400,
408,
414,
416,
420,
432,
440,
448,
450,
456,
460,
462,
464,
468,
476,
480,
486,
496,
500,
504,
510,
512,
520,
522,
528,
532,
540,
544,
546,
552,
558,
560,
570,
576,
580,
588,
594,
600,
608,
612,
616,
620,
624,
630,
640,
644,
648,
660,
666,
672,
680,
684,
690,
696,
700,
702,
704,
714,
720,
726,
728,
736,
740,
744,
750,
756,
760,
768,
780,
784,
792,
798,
800,
810,
812,
816,
820,
828,
832,
840,
858,
860,
864,
868,
870,
880,
882,
888,
896,
900,
912,
918,
920,
924,
928,
930,
936,
952,
960,
966,
968,
972,
980,
984,
990,
992,
1000,
1008,
1014,
1020,
1024,
1026,
1032,
1036,
1040,
1044,
1050,
1056,
1064,
1080,
1088,
1092,
1100,
1104,
1110,
1116,
1120,
1122,
1128,
1134,
1140,
1144,
1148,
1152,
1160,
1170,
1176,
1184,
1188,
1200,
1204,
1216,
1218,
1224,
1230,
1232,
1240,
1242,
1248,
1254,
1260,
1272,
1280,
1288,
1290,
1296,
1300,
1302,
1312,
1316,
1320,
1326,
1332,
1344,
1350,
1352,
1360,
1368,
1372,
1376,
1380,
1386,
1392,
1400,
1404,
1408,
1410,
1416,
1428,
1440,
1452,
1456,
1458,
1464,
1470,
1472,
1476,
1480,
1482,
1484,
1488,
1496,
1500,
1504,
1512,
1518,
1520,
1530,
1536,
1540,
1548,
1554,
1560,
1566,
1568,
1584,
1590,
1596,
1600,
1620,
1624,
1632,
1638,
1640,
1650,
1656,
1664,
1672,
1674,
1680,
1692,
1696,
1700,
1710,
1716,
1720,
1722,
1728,
1736,
1740,
1760,
1764,
1768,
1770,
1776,
1782,
1792,
1794,
1800,
1806,
1820,
1824,
1830,
1836,
1840,
1848,
1856,
1860,
1872,
1880,
1888,
1890,
1900,
1904,
1908,
1914,
1920,
1932,
1936,
1944,
1950,
1952,
1960,
1968,
1974,
1976,
1980,
1984,
1998,
2000,
2010,
2016,
2024,
2028,
2040,
2046,
2048,
2052,
2058,
2064,
2070,
2072,
2080,
2088,
2100,
2106,
2112,
2120,
2124,
2128,
2130,
2142,
2156,
2160,
2176,
2178,
2184,
2190,
2196,
2200,
2208,
2214,
2220,
2226,
2232,
2240,
2244,
2250,
2256,
2262,
2268,
2280,
2288,
2296,
2300,
2304,
2310,
2320,
2322,
2340,
2352,
2360,
2368,
2376,
2380,
2392,
2394,
2400,
2408,
2412,
2418,
2420,
2430,
2432,
2436,
2440,
2442,
2448,
2460,
2464,
2478,
2480,
2484,
2496,
2500,
2508,
2520,
2538,
2544,
2548,
2550,
2552,
2556,
2560,
2562,
2574,
2576,
2580,
2592,
2600,
2604,
2610,
2624,
2628,
2632,
2640,
2646,
2652,
2660,
2664,
2680,
2688,
2700,
2704,
2706,
2720,
2728,
2730,
2736,
2744,
2752,
2754,
2760,
2772,
2784,
2790,
2800,
2808,
2814,
2816,
2820,
2832,
2838,
2840,
2844,
2850,
2856,
2860,
2862,
2880,
2886,
2898,
2900,
2904,
2912,
2916,
2920,
2928,
2940,
2944,
2952,
2960,
2964,
2968,
2970,
2976,
2982,
2988,
2992,
3000,
3008,
3016,
3024,
3036,
3040,
3042,
3060,
3066,
3072,
3078,
3080,
3096,
3100,
3102,
3108,
3120,
3132,
3136,
3150,
3160,
3168,
3180,
3186,
3192,
3198,
3200,
3204,
3216,
3220,
3224,
3234,
3240,
3248,
3256,
3264,
3276,
3280,
3294,
3300,
3304,
3312,
3318,
3320,
3328,
3330,
3332,
3344,
3348,
3354,
3360,
3366,
3380,
3384,
3388,
3392,
3400,
3402,
3408,
3416,
3420,
3432,
3440,
3444,
3450,
3456,
3468,
3472,
3480,
3486,
3498,
3500,
3504,
3510,
3520,
3528,
3536,
3540,
3552,
3560,
3564,
3570,
3584,
3588,
3600,
3608,
3612,
3618,
3630,
3640.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 829157 values, from 1 to 10000000).
n\r | 0 | 1 |
2 | 829156 | 1 | 2 |
3 | 535172 | 146985 | 147000 | 3 |
4 | 641875 | 1 | 187281 | 0 | 4 |
5 | 309866 | 129847 | 129819 | 129810 | 129815 | 5 |
6 | 535172 | 1 | 147000 | 0 | 146984 | 0 | 6 |
7 | 239022 | 98374 | 98348 | 98318 | 98352 | 98408 | 98335 | 7 |
8 | 432328 | 1 | 93635 | 0 | 209547 | 0 | 93646 | 0 | 8 |
9 | 256842 | 48970 | 48960 | 139135 | 48985 | 49026 | 139195 | 49030 | 49014 | 9 |
10 | 309866 | 1 | 129819 | 0 | 129815 | 0 | 129846 | 0 | 129810 | 0 | 10 |
11 | 155522 | 67360 | 67391 | 67323 | 67365 | 67351 | 67294 | 67448 | 67404 | 67361 | 67338 |
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.