Numbers which count the sides of regular polygons constructible with ruler and compass. more
The first 600 constructible polygons :
3,
4,
5,
6,
8,
10,
12,
15,
16,
17,
20,
24,
30,
32,
34,
40,
48,
51,
60,
64,
68,
80,
85,
96,
102,
120,
128,
136,
160,
170,
192,
204,
240,
255,
256,
257,
272,
320,
340,
384,
408,
480,
510,
512,
514,
544,
640,
680,
768,
771,
816,
960,
1020,
1024,
1028,
1088,
1280,
1285,
1360,
1536,
1542,
1632,
1920,
2040,
2048,
2056,
2176,
2560,
2570,
2720,
3072,
3084,
3264,
3840,
3855,
4080,
4096,
4112,
4352,
4369,
5120,
5140,
5440,
6144,
6168,
6528,
7680,
7710,
8160,
8192,
8224,
8704,
8738,
10240,
10280,
10880,
12288,
12336,
13056,
13107,
15360,
15420,
16320,
16384,
16448,
17408,
17476,
20480,
20560,
21760,
21845,
24576,
24672,
26112,
26214,
30720,
30840,
32640,
32768,
32896,
34816,
34952,
40960,
41120,
43520,
43690,
49152,
49344,
52224,
52428,
61440,
61680,
65280,
65535,
65536,
65537,
65792,
69632,
69904,
81920,
82240,
87040,
87380,
98304,
98688,
104448,
104856,
122880,
123360,
130560,
131070,
131072,
131074,
131584,
139264,
139808,
163840,
164480,
174080,
174760,
196608,
196611,
197376,
208896,
209712,
245760,
246720,
261120,
262140,
262144,
262148,
263168,
278528,
279616,
327680,
327685,
328960,
348160,
349520,
393216,
393222,
394752,
417792,
419424,
491520,
493440,
522240,
524280,
524288,
524296,
526336,
557056,
559232,
655360,
655370,
657920,
696320,
699040,
786432,
786444,
789504,
835584,
838848,
983040,
983055,
986880,
1044480,
1048560,
1048576,
1048592,
1052672,
1114112,
1114129,
1118464,
1310720,
1310740,
1315840,
1392640,
1398080,
1572864,
1572888,
1579008,
1671168,
1677696,
1966080,
1966110,
1973760,
2088960,
2097120,
2097152,
2097184,
2105344,
2228224,
2228258,
2236928,
2621440,
2621480,
2631680,
2785280,
2796160,
3145728,
3145776,
3158016,
3342336,
3342387,
3355392,
3932160,
3932220,
3947520,
4177920,
4194240,
4194304,
4194368,
4210688,
4456448,
4456516,
4473856,
5242880,
5242960,
5263360,
5570560,
5570645,
5592320,
6291456,
6291552,
6316032,
6684672,
6684774,
6710784,
7864320,
7864440,
7895040,
8355840,
8388480,
8388608,
8388736,
8421376,
8912896,
8913032,
8947712,
10485760,
10485920,
10526720,
11141120,
11141290,
11184640,
12582912,
12583104,
12632064,
13369344,
13369548,
13421568,
15728640,
15728880,
15790080,
16711680,
16711935,
16776960,
16777216,
16777472,
16842752,
16843009,
17825792,
17826064,
17895424,
20971520,
20971840,
21053440,
22282240,
22282580,
22369280,
25165824,
25166208,
25264128,
26738688,
26739096,
26843136,
31457280,
31457760,
31580160,
33423360,
33423870,
33553920,
33554432,
33554944,
33685504,
33686018,
35651584,
35652128,
35790848,
41943040,
41943680,
42106880,
44564480,
44565160,
44738560,
50331648,
50332416,
50528256,
50529027,
53477376,
53478192,
53686272,
62914560,
62915520,
63160320,
66846720,
66847740,
67107840,
67108864,
67109888,
67371008,
67372036,
71303168,
71304256,
71581696,
83886080,
83887360,
84213760,
84215045,
89128960,
89130320,
89477120,
100663296,
100664832,
101056512,
101058054,
106954752,
106956384,
107372544,
125829120,
125831040,
126320640,
133693440,
133695480,
134215680,
134217728,
134219776,
134742016,
134744072,
142606336,
142608512,
143163392,
167772160,
167774720,
168427520,
168430090,
178257920,
178260640,
178954240,
201326592,
201329664,
202113024,
202116108,
213909504,
213912768,
214745088,
251658240,
251662080,
252641280,
252645135,
267386880,
267390960,
268431360,
268435456,
268439552,
269484032,
269488144,
285212672,
285217024,
286326784,
286331153,
335544320,
335549440,
336855040,
336860180,
356515840,
356521280,
357908480,
402653184,
402659328,
404226048,
404232216,
427819008,
427825536,
429490176,
503316480,
503324160,
505282560,
505290270,
534773760,
534781920,
536862720,
536870912,
536879104,
538968064,
538976288,
570425344,
570434048,
572653568,
572662306,
671088640,
671098880,
673710080,
673720360,
713031680,
713042560,
715816960,
805306368,
805318656,
808452096,
808464432,
855638016,
855651072,
858980352,
858993459,
1006632960,
1006648320,
1010565120,
1010580540,
1069547520,
1069563840,
1073725440,
1073741824,
1073758208,
1077936128,
1077952576,
1140850688,
1140868096,
1145307136,
1145324612,
1342177280,
1342197760,
1347420160,
1347440720,
1426063360,
1426085120,
1431633920,
1431655765,
1610612736,
1610637312,
1616904192,
1616928864,
1711276032,
1711302144,
1717960704,
1717986918,
2013265920,
2013296640,
2021130240,
2021161080,
2139095040,
2139127680,
2147450880,
2147483648,
2147516416,
2155872256,
2155905152,
2281701376,
2281736192,
2290614272,
2290649224,
2684354560,
2684395520,
2694840320,
2694881440,
2852126720,
2852170240,
2863267840,
2863311530,
3221225472,
3221274624,
3233808384,
3233857728,
3422552064,
3422604288,
3435921408,
3435973836,
4026531840,
4026593280,
4042260480,
4042322160,
4278190080,
4278255360,
4294901760,
4294967295,
4294967296,
4295032832,
4311744512,
4311810304,
4563402752,
4563472384,
4581228544,
4581298448,
5368709120,
5368791040,
5389680640,
5389762880,
5704253440,
5704340480,
5726535680,
5726623060,
6442450944,
6442549248,
6467616768,
6467715456,
6845104128,
6845208576,
6871842816,
6871947672,
8053063680,
8053186560,
8084520960,
8084644320,
8556380160,
8556510720,
8589803520,
8589934590,
8589934592,
8590065664,
8623489024,
8623620608,
9126805504,
9126944768,
9162457088,
9162596896,
10737418240,
10737582080,
10779361280,
10779525760,
11408506880,
11408680960,
11453071360,
11453246120,
12884901888,
12885098496,
12935233536,
12935430912,
13690208256,
13690417152,
13743685632,
13743895344,
16106127360,
16106373120,
16169041920,
16169288640,
17112760320,
17113021440,
17179607040,
17179869180,
17179869184,
17180131328,
17246978048,
17247241216,
18253611008,
18253889536,
18324914176,
18325193792,
21474836480,
21475164160.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1094 values, from 3 to 900719925264384).
n\r | 0 | 1 |
2 | 1063 | 31 | 2 |
3 | 536 | 279 | 279 | 3 |
4 | 1032 | 15 | 31 | 16 | 4 |
5 | 528 | 140 | 142 | 141 | 143 | 5 |
6 | 520 | 7 | 271 | 16 | 272 | 8 | 6 |
7 | 0 | 185 | 183 | 187 | 178 | 181 | 180 | 7 |
8 | 1000 | 7 | 15 | 8 | 32 | 8 | 16 | 8 | 8 |
9 | 0 | 93 | 91 | 268 | 94 | 94 | 268 | 92 | 94 | 9 |
10 | 512 | 3 | 136 | 2 | 139 | 16 | 137 | 6 | 139 | 4 | 10 |
11 | 0 | 110 | 108 | 109 | 108 | 111 | 111 | 108 | 109 | 109 | 111 |
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.