The Catalan numbers up to 1015 :
1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 100000 values, from 1 to 1.78⋅1060198).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 49985 | 15 | 2 | ||||||||
| 3 | 48466 | 766 | 768 | 3 | |||||||
| 4 | 49865 | 15 | 120 | 0 | 4 | ||||||
| 5 | 47086 | 567 | 655 | 802 | 890 | 5 | |||||
| 6 | 48455 | 2 | 766 | 11 | 764 | 2 | 6 | ||||
| 7 | 46162 | 619 | 658 | 653 | 642 | 661 | 605 | 7 | |||
| 8 | 49309 | 1 | 15 | 0 | 556 | 14 | 105 | 0 | 8 | ||
| 9 | 44626 | 250 | 261 | 1920 | 264 | 247 | 1920 | 252 | 260 | 9 | |
| 10 | 47076 | 1 | 654 | 0 | 887 | 10 | 566 | 1 | 802 | 3 | 10 |
| 11 | 44206 | 576 | 590 | 574 | 590 | 565 | 562 | 598 | 581 | 591 | 567 |
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.
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