Numbers which count the maximal number of pieces in which a cake can be divided into by n planar cuts. more
The first 600 cake numbers :
1,
2,
4,
8,
15,
26,
42,
64,
93,
130,
176,
232,
299,
378,
470,
576,
697,
834,
988,
1160,
1351,
1562,
1794,
2048,
2325,
2626,
2952,
3304,
3683,
4090,
4526,
4992,
5489,
6018,
6580,
7176,
7807,
8474,
9178,
9920,
10701,
11522,
12384,
13288,
14235,
15226,
16262,
17344,
18473,
19650,
20876,
22152,
23479,
24858,
26290,
27776,
29317,
30914,
32568,
34280,
36051,
37882,
39774,
41728,
43745,
45826,
47972,
50184,
52463,
54810,
57226,
59712,
62269,
64898,
67600,
70376,
73227,
76154,
79158,
82240,
85401,
88642,
91964,
95368,
98855,
102426,
106082,
109824,
113653,
117570,
121576,
125672,
129859,
134138,
138510,
142976,
147537,
152194,
156948,
161800,
166751,
171802,
176954,
182208,
187565,
193026,
198592,
204264,
210043,
215930,
221926,
228032,
234249,
240578,
247020,
253576,
260247,
267034,
273938,
280960,
288101,
295362,
302744,
310248,
317875,
325626,
333502,
341504,
349633,
357890,
366276,
374792,
383439,
392218,
401130,
410176,
419357,
428674,
438128,
447720,
457451,
467322,
477334,
487488,
497785,
508226,
518812,
529544,
540423,
551450,
562626,
573952,
585429,
597058,
608840,
620776,
632867,
645114,
657518,
670080,
682801,
695682,
708724,
721928,
735295,
748826,
762522,
776384,
790413,
804610,
818976,
833512,
848219,
863098,
878150,
893376,
908777,
924354,
940108,
956040,
972151,
988442,
1004914,
1021568,
1038405,
1055426,
1072632,
1090024,
1107603,
1125370,
1143326,
1161472,
1179809,
1198338,
1217060,
1235976,
1255087,
1274394,
1293898,
1313600,
1333501,
1353602,
1373904,
1394408,
1415115,
1436026,
1457142,
1478464,
1499993,
1521730,
1543676,
1565832,
1588199,
1610778,
1633570,
1656576,
1679797,
1703234,
1726888,
1750760,
1774851,
1799162,
1823694,
1848448,
1873425,
1898626,
1924052,
1949704,
1975583,
2001690,
2028026,
2054592,
2081389,
2108418,
2135680,
2163176,
2190907,
2218874,
2247078,
2275520,
2304201,
2333122,
2362284,
2391688,
2421335,
2451226,
2481362,
2511744,
2542373,
2573250,
2604376,
2635752,
2667379,
2699258,
2731390,
2763776,
2796417,
2829314,
2862468,
2895880,
2929551,
2963482,
2997674,
3032128,
3066845,
3101826,
3137072,
3172584,
3208363,
3244410,
3280726,
3317312,
3354169,
3391298,
3428700,
3466376,
3504327,
3542554,
3581058,
3619840,
3658901,
3698242,
3737864,
3777768,
3817955,
3858426,
3899182,
3940224,
3981553,
4023170,
4065076,
4107272,
4149759,
4192538,
4235610,
4278976,
4322637,
4366594,
4410848,
4455400,
4500251,
4545402,
4590854,
4636608,
4682665,
4729026,
4775692,
4822664,
4869943,
4917530,
4965426,
5013632,
5062149,
5110978,
5160120,
5209576,
5259347,
5309434,
5359838,
5410560,
5461601,
5512962,
5564644,
5616648,
5668975,
5721626,
5774602,
5827904,
5881533,
5935490,
5989776,
6044392,
6099339,
6154618,
6210230,
6266176,
6322457,
6379074,
6436028,
6493320,
6550951,
6608922,
6667234,
6725888,
6784885,
6844226,
6903912,
6963944,
7024323,
7085050,
7146126,
7207552,
7269329,
7331458,
7393940,
7456776,
7519967,
7583514,
7647418,
7711680,
7776301,
7841282,
7906624,
7972328,
8038395,
8104826,
8171622,
8238784,
8306313,
8374210,
8442476,
8511112,
8580119,
8649498,
8719250,
8789376,
8859877,
8930754,
9002008,
9073640,
9145651,
9218042,
9290814,
9363968,
9437505,
9511426,
9585732,
9660424,
9735503,
9810970,
9886826,
9963072,
10039709,
10116738,
10194160,
10271976,
10350187,
10428794,
10507798,
10587200,
10667001,
10747202,
10827804,
10908808,
10990215,
11072026,
11154242,
11236864,
11319893,
11403330,
11487176,
11571432,
11656099,
11741178,
11826670,
11912576,
11998897,
12085634,
12172788,
12260360,
12348351,
12436762,
12525594,
12614848,
12704525,
12794626,
12885152,
12976104,
13067483,
13159290,
13251526,
13344192,
13437289,
13530818,
13624780,
13719176,
13814007,
13909274,
14004978,
14101120,
14197701,
14294722,
14392184,
14490088,
14588435,
14687226,
14786462,
14886144,
14986273,
15086850,
15187876,
15289352,
15391279,
15493658,
15596490,
15699776,
15803517,
15907714,
16012368,
16117480,
16223051,
16329082,
16435574,
16542528,
16649945,
16757826,
16866172,
16974984,
17084263,
17194010,
17304226,
17414912,
17526069,
17637698,
17749800,
17862376,
17975427,
18088954,
18202958,
18317440,
18432401,
18547842,
18663764,
18780168,
18897055,
19014426,
19132282,
19250624,
19369453,
19488770,
19608576,
19728872,
19849659,
19970938,
20092710,
20214976,
20337737,
20460994,
20584748,
20709000,
20833751,
20959002,
21084754,
21211008,
21337765,
21465026,
21592792,
21721064,
21849843,
21979130,
22108926,
22239232,
22370049,
22501378,
22633220,
22765576,
22898447,
23031834,
23165738,
23300160,
23435101,
23570562,
23706544,
23843048,
23980075,
24117626,
24255702,
24394304,
24533433,
24673090,
24813276,
24953992,
25095239,
25237018,
25379330,
25522176,
25665557,
25809474,
25953928,
26098920,
26244451,
26390522,
26537134,
26684288,
26831985,
26980226,
27129012,
27278344,
27428223,
27578650,
27729626,
27881152,
28033229,
28185858,
28339040,
28492776,
28647067,
28801914,
28957318,
29113280,
29269801,
29426882,
29584524,
29742728,
29901495,
30060826,
30220722,
30381184,
30542213,
30703810,
30865976,
31028712,
31192019,
31355898,
31520350,
31685376,
31850977,
32017154,
32183908,
32351240,
32519151,
32687642,
32856714,
33026368,
33196605,
33367426,
33538832,
33710824,
33883403,
34056570,
34230326,
34404672,
34579609,
34755138,
34931260,
35107976,
35285287,
35463194,
35641698,
35820800.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 181713 values, from 1 to 999999021407449).
n\r | 0 | 1 |
2 | 136284 | 45429 | 2 |
3 | 60570 | 60572 | 60571 | 3 |
4 | 68142 | 22715 | 68142 | 22714 | 4 |
5 | 36342 | 36343 | 36343 | 36342 | 36343 | 5 |
6 | 45427 | 15144 | 45429 | 15143 | 45428 | 15142 | 6 |
7 | 25959 | 77877 | 25959 | 0 | 25959 | 25959 | 0 | 7 |
8 | 56785 | 11358 | 56785 | 11357 | 11357 | 11357 | 11357 | 11357 | 8 |
9 | 20190 | 20191 | 20191 | 20190 | 20191 | 20190 | 20190 | 20190 | 20190 | 9 |
10 | 27256 | 9086 | 27258 | 9086 | 27257 | 9086 | 27257 | 9085 | 27256 | 9086 | 10 |
11 | 16519 | 16520 | 16520 | 0 | 49558 | 16519 | 0 | 0 | 16520 | 49557 | 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.