A number whose divisors can be partitioned into two set with the same sum. more
The first 600 Zumkeller numbers :
6,
12,
20,
24,
28,
30,
40,
42,
48,
54,
56,
60,
66,
70,
78,
80,
84,
88,
90,
96,
102,
104,
108,
112,
114,
120,
126,
132,
138,
140,
150,
156,
160,
168,
174,
176,
180,
186,
192,
198,
204,
208,
210,
216,
220,
222,
224,
228,
234,
240,
246,
252,
258,
260,
264,
270,
272,
276,
280,
282,
294,
300,
304,
306,
308,
312,
318,
320,
330,
336,
340,
342,
348,
350,
352,
354,
360,
364,
366,
368,
372,
378,
380,
384,
390,
396,
402,
408,
414,
416,
420,
426,
432,
438,
440,
444,
448,
456,
460,
462,
464,
468,
474,
476,
480,
486,
490,
492,
496,
498,
500,
504,
510,
516,
520,
522,
528,
532,
534,
540,
544,
546,
550,
552,
558,
560,
564,
570,
572,
580,
582,
588,
594,
600,
606,
608,
612,
616,
618,
620,
624,
630,
636,
640,
642,
644,
650,
654,
660,
666,
672,
678,
680,
684,
690,
696,
700,
702,
704,
708,
714,
720,
726,
728,
732,
736,
740,
744,
750,
756,
760,
762,
768,
770,
780,
786,
792,
798,
804,
810,
812,
816,
820,
822,
828,
832,
834,
836,
840,
852,
858,
860,
864,
868,
870,
876,
880,
888,
894,
896,
906,
910,
912,
918,
920,
924,
928,
930,
936,
940,
942,
945,
948,
952,
960,
966,
972,
978,
980,
984,
990,
992,
996,
1000,
1002,
1008,
1014,
1020,
1026,
1032,
1036,
1038,
1040,
1044,
1050,
1056,
1060,
1064,
1068,
1074,
1080,
1086,
1088,
1092,
1100,
1104,
1110,
1116,
1120,
1122,
1128,
1134,
1140,
1144,
1146,
1148,
1158,
1160,
1164,
1170,
1176,
1180,
1182,
1184,
1188,
1190,
1194,
1200,
1204,
1212,
1216,
1218,
1220,
1224,
1230,
1232,
1236,
1240,
1242,
1248,
1254,
1260,
1266,
1272,
1280,
1284,
1288,
1290,
1300,
1302,
1308,
1312,
1316,
1320,
1326,
1330,
1332,
1338,
1340,
1344,
1350,
1356,
1360,
1362,
1368,
1372,
1374,
1376,
1380,
1386,
1392,
1398,
1400,
1404,
1408,
1410,
1416,
1420,
1428,
1430,
1434,
1440,
1446,
1452,
1456,
1460,
1464,
1470,
1472,
1476,
1480,
1482,
1484,
1488,
1496,
1500,
1504,
1506,
1512,
1518,
1520,
1524,
1530,
1536,
1540,
1542,
1548,
1554,
1560,
1566,
1572,
1575,
1578,
1580,
1584,
1590,
1596,
1608,
1610,
1614,
1620,
1624,
1626,
1632,
1638,
1640,
1644,
1650,
1652,
1656,
1660,
1662,
1664,
1668,
1672,
1674,
1680,
1686,
1692,
1696,
1698,
1700,
1704,
1708,
1710,
1716,
1720,
1722,
1728,
1734,
1736,
1740,
1750,
1752,
1758,
1760,
1768,
1770,
1776,
1780,
1782,
1788,
1792,
1794,
1806,
1812,
1820,
1824,
1830,
1836,
1840,
1842,
1848,
1856,
1860,
1866,
1870,
1872,
1876,
1878,
1880,
1884,
1888,
1890,
1896,
1900,
1902,
1904,
1908,
1914,
1920,
1932,
1938,
1940,
1944,
1950,
1952,
1956,
1960,
1968,
1974,
1976,
1980,
1984,
1986,
1988,
1992,
1998,
2000,
2002,
2004,
2010,
2016,
2020,
2022,
2024,
2028,
2030,
2040,
2044,
2046,
2052,
2058,
2060,
2064,
2070,
2072,
2076,
2080,
2082,
2088,
2090,
2094,
2100,
2106,
2112,
2118,
2120,
2124,
2128,
2130,
2136,
2140,
2142,
2148,
2154,
2156,
2160,
2166,
2170,
2172,
2176,
2180,
2184,
2190,
2196,
2200,
2202,
2205,
2208,
2210,
2212,
2214,
2220,
2226,
2232,
2238,
2240,
2244,
2250,
2256,
2260,
2262,
2268,
2274,
2280,
2288,
2292,
2296,
2298,
2300,
2310,
2316,
2320,
2322,
2324,
2328,
2334,
2340,
2346,
2352,
2360,
2364,
2368,
2370,
2376,
2380,
2382,
2388,
2392,
2394,
2400,
2406,
2408,
2412,
2418,
2420,
2424,
2430,
2432,
2436,
2440,
2442,
2448,
2454,
2460,
2464,
2470,
2472,
2478,
2480,
2484,
2490,
2492,
2496,
2508,
2514,
2520,
2526,
2530,
2532,
2538,
2540,
2544,
2548,
2550,
2552,
2556,
2560,
2562,
2568,
2574,
2576,
2580,
2584,
2586,
2590,
2598,
2600,
2604,
2610,
2616,
2620,
2622,
2624,
2628,
2632,
2634,
2640,
2646.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 23051 values, from 6 to 100000).
n\r | 0 | 1 |
2 | 22843 | 208 | 2 |
3 | 15383 | 3836 | 3832 | 3 |
4 | 14597 | 104 | 8246 | 104 | 4 |
5 | 7371 | 3920 | 3924 | 3923 | 3913 | 5 |
6 | 15175 | 0 | 3832 | 208 | 3836 | 0 | 6 |
7 | 5377 | 2927 | 2918 | 2976 | 2912 | 2975 | 2966 | 7 |
8 | 8184 | 51 | 4108 | 53 | 6413 | 53 | 4138 | 51 | 8 |
9 | 4216 | 1284 | 1277 | 5583 | 1271 | 1276 | 5584 | 1281 | 1279 | 9 |
10 | 7164 | 1 | 3924 | 0 | 3913 | 207 | 3919 | 0 | 3923 | 0 | 10 |
11 | 2873 | 2013 | 2010 | 2022 | 2023 | 2015 | 2022 | 2010 | 2016 | 2020 | 2027 |
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.