Numbers which count the ways in which a set of identical objects can be partitioned. more
The partition number up to 10
15 :
1,
2,
3,
5,
7,
11,
15,
22,
30,
42,
56,
77,
101,
135,
176,
231,
297,
385,
490,
627,
792,
1002,
1255,
1575,
1958,
2436,
3010,
3718,
4565,
5604,
6842,
8349,
10143,
12310,
14883,
17977,
21637,
26015,
31185,
37338,
44583,
53174,
63261,
75175,
89134,
105558,
124754,
147273,
173525,
204226,
239943,
281589,
329931,
386155,
451276,
526823,
614154,
715220,
831820,
966467,
1121505,
1300156,
1505499,
1741630,
2012558,
2323520,
2679689,
3087735,
3554345,
4087968,
4697205,
5392783,
6185689,
7089500,
8118264,
9289091,
10619863,
12132164,
13848650,
15796476,
18004327,
20506255,
23338469,
26543660,
30167357,
34262962,
38887673,
44108109,
49995925,
56634173,
64112359,
72533807,
82010177,
92669720,
104651419,
118114304,
133230930,
150198136,
169229875,
190569292,
214481126,
241265379,
271248950,
304801365,
342325709,
384276336,
431149389,
483502844,
541946240,
607163746,
679903203,
761002156,
851376628,
952050665,
1064144451,
1188908248,
1327710076,
1482074143,
1653668665,
1844349560,
2056148051,
2291320912,
2552338241,
2841940500,
3163127352,
3519222692,
3913864295,
4351078600,
4835271870,
5371315400,
5964539504,
6620830889,
7346629512,
8149040695,
9035836076,
10015581680,
11097645016,
12292341831,
13610949895,
15065878135,
16670689208,
18440293320,
20390982757,
22540654445,
24908858009,
27517052599,
30388671978,
33549419497,
37027355200,
40853235313,
45060624582,
49686288421,
54770336324,
60356673280,
66493182097,
73232243759,
80630964769,
88751778802,
97662728555,
107438159466,
118159068427,
129913904637,
142798995930,
156919475295,
172389800255,
189334822579,
207890420102,
228204732751,
250438925115,
274768617130,
301384802048,
330495499613,
362326859895,
397125074750,
435157697830,
476715857290,
522115831195,
571701605655,
625846753120,
684957390936,
749474411781,
819876908323,
896684817527,
980462880430,
1071823774337,
1171432692373,
1280011042268,
1398341745571,
1527273599625,
1667727404093,
1820701100652,
1987276856363,
2168627105469,
2366022741845,
2580840212973,
2814570987591,
3068829878530,
3345365983698,
3646072432125,
3972999029388,
4328363658647,
4714566886083,
5134205287973,
5590088317495,
6085253859260,
6622987708040,
7206841706490,
7840656226137,
8528581302375,
9275102575355,
10085065885767,
10963707205259,
11916681236278,
12950095925895,
14070545699287,
15285151248481,
16601598107914,
18028182516671,
19573856161145,
21248279009367,
23061871173849,
25025873760111,
27152408925615,
29454549941750,
31946390696157,
34643126322519,
37561133582570,
40718063627362,
44132934884255,
47826239745920,
51820051838712,
56138148670947,
60806135438329,
65851585970275,
71304185514919,
77195892663512,
83561103925871,
90436839668817,
97862933703585,
105882246722733,
114540884553038,
123888443077259,
133978259344888,
144867692496445,
156618412527946,
169296722391554,
182973889854026,
197726516681672,
213636919820625,
230793554364681,
249291451168559,
269232701252579,
290726957916112,
313891991306665,
338854264248680,
365749566870782,
394723676655357,
425933084409356,
459545750448675,
495741934760846,
534715062908609,
576672674947168,
621837416509615,
670448123060170,
722760953690372,
779050629562167,
839611730366814,
904760108316360,
974834369944625.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 25000 values, from 1 to 7.95⋅10170).
n\r | 0 | 1 |
2 | 12604 | 12396 | 2 |
3 | 8326 | 8334 | 8340 | 3 |
4 | 6247 | 6152 | 6357 | 6244 | 4 |
5 | 9121 | 3829 | 4149 | 3934 | 3967 | 5 |
6 | 4208 | 4110 | 4172 | 4118 | 4224 | 4168 | 6 |
7 | 6816 | 3017 | 3020 | 3072 | 3073 | 2947 | 3055 | 7 |
8 | 3133 | 3132 | 3168 | 3179 | 3114 | 3020 | 3189 | 3065 | 8 |
9 | 2775 | 2740 | 2827 | 2774 | 2753 | 2753 | 2777 | 2841 | 2760 | 9 |
10 | 4679 | 1932 | 2086 | 1949 | 1957 | 4442 | 1897 | 2063 | 1985 | 2010 | 10 |
11 | 4425 | 2085 | 2029 | 1966 | 2111 | 2087 | 2096 | 1996 | 2133 | 2076 | 1996 |
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.