A number of the form n! / (k! ⋅ (n-k)!). more
The first 600 nontrivial binomial coefficients :
6,
10,
15,
20,
21,
28,
35,
36,
45,
55,
56,
66,
70,
78,
84,
91,
105,
120,
126,
136,
153,
165,
171,
190,
210,
220,
231,
252,
253,
276,
286,
300,
325,
330,
351,
364,
378,
406,
435,
455,
462,
465,
495,
496,
528,
560,
561,
595,
630,
666,
680,
703,
715,
741,
780,
792,
816,
820,
861,
903,
924,
946,
969,
990,
1001,
1035,
1081,
1128,
1140,
1176,
1225,
1275,
1287,
1326,
1330,
1365,
1378,
1431,
1485,
1540,
1596,
1653,
1711,
1716,
1770,
1771,
1820,
1830,
1891,
1953,
2002,
2016,
2024,
2080,
2145,
2211,
2278,
2300,
2346,
2380,
2415,
2485,
2556,
2600,
2628,
2701,
2775,
2850,
2925,
2926,
3003,
3060,
3081,
3160,
3240,
3276,
3321,
3403,
3432,
3486,
3570,
3654,
3655,
3741,
3828,
3876,
3916,
4005,
4060,
4095,
4186,
4278,
4368,
4371,
4465,
4495,
4560,
4656,
4753,
4845,
4851,
4950,
4960,
5005,
5050,
5151,
5253,
5356,
5456,
5460,
5565,
5671,
5778,
5886,
5984,
5985,
5995,
6105,
6188,
6216,
6328,
6435,
6441,
6545,
6555,
6670,
6786,
6903,
7021,
7140,
7260,
7315,
7381,
7503,
7626,
7750,
7770,
7875,
8001,
8008,
8128,
8256,
8385,
8436,
8515,
8568,
8646,
8778,
8855,
8911,
9045,
9139,
9180,
9316,
9453,
9591,
9730,
9870,
9880,
10011,
10153,
10296,
10440,
10585,
10626,
10660,
10731,
10878,
11026,
11175,
11325,
11440,
11476,
11480,
11628,
11781,
11935,
12090,
12246,
12341,
12376,
12403,
12561,
12650,
12720,
12870,
12880,
13041,
13203,
13244,
13366,
13530,
13695,
13861,
14028,
14190,
14196,
14365,
14535,
14706,
14878,
14950,
15051,
15180,
15225,
15400,
15504,
15576,
15753,
15931,
16110,
16215,
16290,
16471,
16653,
16836,
17020,
17205,
17296,
17391,
17550,
17578,
17766,
17955,
18145,
18336,
18424,
18528,
18564,
18721,
18915,
19110,
19306,
19448,
19503,
19600,
19701,
19900,
20100,
20301,
20349,
20475,
20503,
20706,
20825,
20910,
21115,
21321,
21528,
21736,
21945,
22100,
22155,
22366,
22578,
22791,
23005,
23220,
23426,
23436,
23653,
23751,
23871,
24090,
24310,
24531,
24753,
24804,
24976,
25200,
25425,
25651,
25878,
26106,
26235,
26334,
26335,
26565,
26796,
27028,
27132,
27261,
27405,
27495,
27720,
27730,
27966,
28203,
28441,
28680,
28920,
29161,
29260,
29403,
29646,
29890,
30135,
30381,
30628,
30856,
30876,
31125,
31375,
31465,
31626,
31824,
31878,
32131,
32385,
32509,
32640,
32896,
33153,
33411,
33649,
33670,
33930,
34191,
34220,
34453,
34716,
34980,
35245,
35511,
35778,
35960,
35990,
36046,
36315,
36585,
36856,
37128,
37401,
37675,
37820,
37950,
38226,
38503,
38760,
38781,
39060,
39340,
39621,
39711,
39903,
40186,
40470,
40755,
40920,
41041,
41328,
41616,
41664,
41905,
42195,
42486,
42504,
42778,
43071,
43365,
43660,
43680,
43758,
43956,
44253,
44551,
44850,
45150,
45451,
45753,
45760,
46056,
46360,
46376,
46665,
46971,
47278,
47586,
47895,
47905,
48205,
48516,
48620,
48828,
49141,
49455,
49770,
50086,
50116,
50388,
50403,
50721,
51040,
51360,
51681,
52003,
52326,
52360,
52394,
52650,
52975,
53130,
53301,
53628,
53956,
54264,
54285,
54615,
54740,
54946,
55278,
55611,
55945,
56280,
56616,
56953,
57155,
57291,
57630,
57970,
58311,
58653,
58905,
58996,
59340,
59640,
59685,
60031,
60378,
60726,
61075,
61425,
61776,
62128,
62196,
62481,
62835,
63190,
63546,
63903,
64261,
64620,
64824,
64980,
65341,
65703,
65780,
66045,
66066,
66430,
66795,
67161,
67525,
67528,
67896,
68265,
68635,
69006,
69378,
69751,
70125,
70300,
70500,
70876,
71253,
71631,
72010,
72390,
72771,
73150,
73153,
73536,
73815,
73920,
74305,
74613,
74691,
75078,
75466,
75582,
75855,
76076,
76245,
76636,
77028,
77421,
77520,
77815,
78210,
78606,
79003,
79079,
79401,
79800,
80200,
80601,
80730,
81003,
81406,
81810,
82160,
82215,
82251,
82621,
83028,
83436,
83845,
84255,
84666,
85078,
85320,
85491,
85905,
86320,
86736,
87153,
87571,
87990,
88410,
88560,
88831,
89253,
89676,
90100,
90525,
90951,
91378,
91390,
91806,
91881,
92235,
92378,
92665,
93096,
93528,
93961,
94395,
94830,
95266,
95284,
95703,
96141,
96580,
97020,
97461,
97903,
98280,
98346,
98770,
98790,
99235,
99681,
100128,
100576,
100947,
101025,
101270,
101475,
101926,
102340.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 4517490 values, from 6 to 10000002437316).
n\r | 0 | 1 |
2 | 2269130 | 2248360 | 2 |
3 | 2998338 | 1505013 | 14139 | 3 |
4 | 1144753 | 1124176 | 1124377 | 1124184 | 4 |
5 | 1816364 | 1797848 | 332 | 894777 | 8169 | 5 |
6 | 1502859 | 749140 | 10398 | 1495479 | 755873 | 3741 | 6 |
7 | 1298175 | 1284761 | 82 | 1283443 | 5676 | 645 | 644708 | 7 |
8 | 572605 | 562083 | 562188 | 562091 | 572148 | 562093 | 562189 | 562093 | 8 |
9 | 1000275 | 1495549 | 4711 | 998814 | 4710 | 4700 | 999249 | 4754 | 4728 | 9 |
10 | 914283 | 896864 | 249 | 447296 | 6133 | 902081 | 900984 | 83 | 447481 | 2036 | 10 |
11 | 826399 | 821288 | 3937 | 813168 | 410859 | 830 | 813273 | 3655 | 38 | 3586 | 820457 |
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.