Numbers that count the maximal number of pieces in which a circle can be divided into by n linear cuts. more
The first 600 pancake numbers :
1,
2,
4,
7,
11,
16,
22,
29,
37,
46,
56,
67,
79,
92,
106,
121,
137,
154,
172,
191,
211,
232,
254,
277,
301,
326,
352,
379,
407,
436,
466,
497,
529,
562,
596,
631,
667,
704,
742,
781,
821,
862,
904,
947,
991,
1036,
1082,
1129,
1177,
1226,
1276,
1327,
1379,
1432,
1486,
1541,
1597,
1654,
1712,
1771,
1831,
1892,
1954,
2017,
2081,
2146,
2212,
2279,
2347,
2416,
2486,
2557,
2629,
2702,
2776,
2851,
2927,
3004,
3082,
3161,
3241,
3322,
3404,
3487,
3571,
3656,
3742,
3829,
3917,
4006,
4096,
4187,
4279,
4372,
4466,
4561,
4657,
4754,
4852,
4951,
5051,
5152,
5254,
5357,
5461,
5566,
5672,
5779,
5887,
5996,
6106,
6217,
6329,
6442,
6556,
6671,
6787,
6904,
7022,
7141,
7261,
7382,
7504,
7627,
7751,
7876,
8002,
8129,
8257,
8386,
8516,
8647,
8779,
8912,
9046,
9181,
9317,
9454,
9592,
9731,
9871,
10012,
10154,
10297,
10441,
10586,
10732,
10879,
11027,
11176,
11326,
11477,
11629,
11782,
11936,
12091,
12247,
12404,
12562,
12721,
12881,
13042,
13204,
13367,
13531,
13696,
13862,
14029,
14197,
14366,
14536,
14707,
14879,
15052,
15226,
15401,
15577,
15754,
15932,
16111,
16291,
16472,
16654,
16837,
17021,
17206,
17392,
17579,
17767,
17956,
18146,
18337,
18529,
18722,
18916,
19111,
19307,
19504,
19702,
19901,
20101,
20302,
20504,
20707,
20911,
21116,
21322,
21529,
21737,
21946,
22156,
22367,
22579,
22792,
23006,
23221,
23437,
23654,
23872,
24091,
24311,
24532,
24754,
24977,
25201,
25426,
25652,
25879,
26107,
26336,
26566,
26797,
27029,
27262,
27496,
27731,
27967,
28204,
28442,
28681,
28921,
29162,
29404,
29647,
29891,
30136,
30382,
30629,
30877,
31126,
31376,
31627,
31879,
32132,
32386,
32641,
32897,
33154,
33412,
33671,
33931,
34192,
34454,
34717,
34981,
35246,
35512,
35779,
36047,
36316,
36586,
36857,
37129,
37402,
37676,
37951,
38227,
38504,
38782,
39061,
39341,
39622,
39904,
40187,
40471,
40756,
41042,
41329,
41617,
41906,
42196,
42487,
42779,
43072,
43366,
43661,
43957,
44254,
44552,
44851,
45151,
45452,
45754,
46057,
46361,
46666,
46972,
47279,
47587,
47896,
48206,
48517,
48829,
49142,
49456,
49771,
50087,
50404,
50722,
51041,
51361,
51682,
52004,
52327,
52651,
52976,
53302,
53629,
53957,
54286,
54616,
54947,
55279,
55612,
55946,
56281,
56617,
56954,
57292,
57631,
57971,
58312,
58654,
58997,
59341,
59686,
60032,
60379,
60727,
61076,
61426,
61777,
62129,
62482,
62836,
63191,
63547,
63904,
64262,
64621,
64981,
65342,
65704,
66067,
66431,
66796,
67162,
67529,
67897,
68266,
68636,
69007,
69379,
69752,
70126,
70501,
70877,
71254,
71632,
72011,
72391,
72772,
73154,
73537,
73921,
74306,
74692,
75079,
75467,
75856,
76246,
76637,
77029,
77422,
77816,
78211,
78607,
79004,
79402,
79801,
80201,
80602,
81004,
81407,
81811,
82216,
82622,
83029,
83437,
83846,
84256,
84667,
85079,
85492,
85906,
86321,
86737,
87154,
87572,
87991,
88411,
88832,
89254,
89677,
90101,
90526,
90952,
91379,
91807,
92236,
92666,
93097,
93529,
93962,
94396,
94831,
95267,
95704,
96142,
96581,
97021,
97462,
97904,
98347,
98791,
99236,
99682,
100129,
100577,
101026,
101476,
101927,
102379,
102832,
103286,
103741,
104197,
104654,
105112,
105571,
106031,
106492,
106954,
107417,
107881,
108346,
108812,
109279,
109747,
110216,
110686,
111157,
111629,
112102,
112576,
113051,
113527,
114004,
114482,
114961,
115441,
115922,
116404,
116887,
117371,
117856,
118342,
118829,
119317,
119806,
120296,
120787,
121279,
121772,
122266,
122761,
123257,
123754,
124252,
124751,
125251,
125752,
126254,
126757,
127261,
127766,
128272,
128779,
129287,
129796,
130306,
130817,
131329,
131842,
132356,
132871,
133387,
133904,
134422,
134941,
135461,
135982,
136504,
137027,
137551,
138076,
138602,
139129,
139657,
140186,
140716,
141247,
141779,
142312,
142846,
143381,
143917,
144454,
144992,
145531,
146071,
146612,
147154,
147697,
148241,
148786,
149332,
149879,
150427,
150976,
151526,
152077,
152629,
153182,
153736,
154291,
154847,
155404,
155962,
156521,
157081,
157642,
158204,
158767,
159331,
159896,
160462,
161029,
161597,
162166,
162736,
163307,
163879,
164452,
165026,
165601,
166177,
166754,
167332,
167911,
168491,
169072,
169654,
170237,
170821,
171406,
171992,
172579,
173167,
173756,
174346,
174937,
175529,
176122,
176716,
177311,
177907,
178504,
179102,
179701.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 44721360 values, from 1 to 999999997764121).
n\r | 0 | 1 |
2 | 22360680 | 22360680 | 2 |
3 | 0 | 29814240 | 14907120 | 3 |
4 | 11180340 | 11180340 | 11180340 | 11180340 | 4 |
5 | 0 | 17888544 | 17888544 | 0 | 8944272 | 5 |
6 | 0 | 14907120 | 7453560 | 0 | 14907120 | 7453560 | 6 |
7 | 6388766 | 12777531 | 12777531 | 0 | 12777532 | 0 | 0 | 7 |
8 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 8 |
9 | 0 | 9938080 | 14907120 | 0 | 9938080 | 0 | 0 | 9938080 | 0 | 9 |
10 | 0 | 8944272 | 8944272 | 0 | 4472136 | 0 | 8944272 | 8944272 | 0 | 4472136 | 10 |
11 | 8131156 | 8131157 | 8131157 | 0 | 8131156 | 4065578 | 0 | 8131156 | 0 | 0 | 0 |
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.